Modeling, Simulation and Optimization of Complex Processes

Bock · Kostina · Phu · Rannacher (Eds.) Modeling, Simulation and Optimization of Complex Processes

Hans Georg Bock · Ekaterina Kostina Hoang Xuan Phu · Rolf Rannacher Editors

Modeling, Simulation and Optimization of Complex Processes Proceedings of the International Conference on High Performance Scientific Computing, March 10–14, 2003, Hanoi, Vietnam

With 231 Figures, and 34 Tables

123

Editors Hans Georg Bock Universität Heidelberg Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR) Im Neuenheimer Feld 368 69120 Heidelberg, Germany e-mail: [email protected]

Hoang Xuan Phu Institute of Mathematics Vietnamese Academy of Science and Technology (VAST) 18 Hoang Quoc Viet Road 10307 Hanoi,Vietnam e-mail: [email protected]

Ekaterina Kostina Universität Heidelberg Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR) Im Neuenheimer Feld 368 69120 Heidelberg, Germany e-mail: [email protected]

Rolf Rannacher Universität Heidelberg Institut für Angewandte Mathematik Im Neuenheimer Feld 294 68120 Heidelberg, Germany e-mail: [email protected]

Library of Congress Control Number: 2004115281

Mathematics Subject Classification: 49-06, 60-06, 68-06, 70-06, 76-06, 85-06, 90-06, 93-06, 94-06

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= [3, 8], DZ = 2, 7] DT = [−1000, 1000]

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x = 2t x = 2t t = x/2

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%-

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.

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Jψ1 (u) = ψ1 · Jψ2 ×y (u) = ψ2 ·

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y × [T (v, p) · n] dσ.

(

∂S ∂S

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*

−(ρ|g||ω|−1 ω, ϕ)D − (∇ · v, q)D ,

ϕ1 ϕ2 & A1 (u; ϕ) H1ψ (D) := H1 (D) ∩ {(v, V, ω) : ∇ · v = 0 Ω, V = ψ1 , ω = ψ2 } .

#,%-

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! " # $%%&' $(&) * ' ! N −1/2 N −1 ' N ' * ' * + * * ! * * * , - . / * 0 * $1& $%2& * # $%%&' $(&) 3456" $%1& 7 * *

#

$ %& *+++!""

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µ

e−iωt

∇ × E − iωµH = 0, ∇ × H + iωεE = 0.

E H ! ε(x), x = (x1 , x2 , x3 ) " # $ # x2 " ! ε(x) = ε(x1 , x3 ) x1 L > 0 ε(x1 + nL, x3 ) = ε(x1 , x3 ) ∀ x1 , x3 ∈

Ê,

n .

! ε(x) " ε(x) ≥ 0 ε(x) > 0 ε(x) = 0 % ε {(x1 , x3 ) : b2 < x3 < b1 }

ε1 ε2 ε(x1 , x3 ) = ε1 Ω1 = {(x1 , x3 ) : x3 ≥ b1 }, ε(x1 , x3 ) = ε2 Ω2 = {(x1 , x3 ) : x3 ≤ b2 }.

% ε1 > 0 ε2

# Ω2 $ &

# & ' ( & & % (

E x2 ' E = (0, u, 0)T ∈ 3 u = u(x1 , x3 ) &

Ê

∆u + k 2 (x)u = 0

Ê2.

)

k 2 (x) = ω 2 ε(x)µ * H x2 ' H = (0, u, 0)T ∈ 3 u = u(x1 , x3 ) ' 1 2 ∇u + u = 0

div . + k 2 (x)

Ê

Ê

, ! $ # ( ) + & % !

Ω x1 #

ε(x)

! ! "

# $

% #

&'( )*

+ &'( ,

$

)-* .

&'( $ &'( '

#

/

$

$ &'(

(

R2 u ∈ H01 (Ω)

Ω

0

A(u, v) = F (v) ∀v ∈ H01 (Ω),

1

F ∈ H −1 (Ω) H01 (Ω) A(u, v) = [p(x)∇u · ∇v + r(x)uv] dx ∀u, v ∈ H01 (Ω).

Ω

.

p ∈ C 1 (Ω), r ∈ C 0 (Ω), p(x) > 0

Ω

'

r(x) ≥ 0

Ω

Mj , 0 ≤ j ≤ J Xj ⊂ H01 (Ω) nj

/ (

Mj

Ω

Mj

#

j

#

O(n2j ) ! " ! ! ! " # $ ! ! " ! ! " ! " % ! &! ! ' ' ! ( ) " ! ! * ! " ' ' " + & ! , - ! ! . " / 0 ! " (" ) 1 ! ! ! 2 ! 3 ! ' $ ( ) ! ! ! ! ! " ! 1 ! N 3 ! " O(N ) 4 Nj ! ! Mj 5 z ∈ Nj " ϕzj ! ! Xj " 6 1 z 0 !j ! " / 0 7 N 8 !j = z ∈ Nj : z " z ∈ Nj−1 ϕz = ϕz N j j−1 . xk !j = xk , k = 1, · · · , n 5 " N ˜ j ϕkj = ϕj j j ! xkj 5 0 ≤ j ≤ J " Aj : Xj → Xj (Aj w, v) = A(w, v),

∀w, v ∈ Xj ,

" (·, ·) L2 (Ω) 9 & Qj , Pj : XJ → Xj (Qj w, v) = (w, v),

A(Pj w, v) = A(w, v),

∀v ∈ Xj , ∀w ∈ XJ .

@ @ 1 @ @ @

C @ C @ 4 C4 @ @ 1 C 1 @ @ @ @C

C @ C @ 2 C 3 @ @ C @ C

C @ 2 C 3@ C @ C 3 2C 2 @ 3 C 2 C 3@ @C C @ CC @

AJ uJ (m+1)

uJ

(m)

= uJ

= fJ

(m)

+ BJ (fJ − AJ uJ ).

Bj : Xj → Xj , 0 ≤ j ≤ J

B0 = A−1 0 . j > 0 g ∈ Xj 3

v1 = Rj g ! v2 = v1 + Bj−1 Qj−1 (g − Aj v1 ), " v3 = v2 + Rjt (g − Aj v2 ).

Bj g = v

Pjk : XJ → Xjk := span

k ϕj

A(Pjk w, ϕkj ) = A(w, ϕkj ) ∀w ∈ XJ .

Rj # $ % n ˜j " $ # Rj = I − (I − Pjk ) A−1 j . k=1

& '

( % ) * + , +-, + , ( % ) ./ 0 % . % % / . /

( ) % 1 0

2

+3,

Mj , 0 ≤ j ≤ J

K ∈

Mj

hK ≤ ChK Mj J

K ∈ Mj−1 δ < 1

I − BJ AJ A < δ.

xkj ∈ N!j ϕkj ! Xj Ejk Mj ϕkj hkj Mj

xkj ! " " J

" $3/2 hli /hkj ≤C

xkj ∈ N!j ,

i=j+1 xl ∈N ei ,xl ∈E k i

i−1

i

j

"

hli /hkj

$1/2

≤C

xli ∈ N!i.

j=1 xk ∈N ej ,xl ∈E k j

i

j

# $ Πj : XJ → Xj J

|(Πj v − Πj−1 v)(z)|2 ≤ CA(v, v)

∀v ∈ XJ .

j=1 z∈N ej

!

% % &

' ()* + uI = eiαx −iβx α = k1 sin θ, β = k1 cos θ −π/2 < θ < π/2 , - u u ()* uα = ue−iαx x1 L > 0 . Γj = {(x1 , x3 ) : 0 < x1 < L, x3 = bj }, j = 1, 2 , Ω = {(x1 , x3 ) : 0 < x1 < L b2 < x3 < b1 }. 1

1

3

u Ω1 Ω2 uI Ω1 n αn = 2πn/L n ∈ Z j = 1, 2 βjn

=

βjn (α)

1/2 k 2 − (αn + α)2 = 1/2 j i (αn + α)2 − kj2

kj2 ≥ (αn + α)2 kj2 < (αn + α)2 .

kj2 = (αn + α)2 n ∈ Z, j = 1, 2 uI Ω1 Ω1 u = uI +

!

n

An1 ei(αn +α)x1 +iβ1 x3 , x ∈ Ω1 .

n∈Z

" Ω2 u=

#!

An2 ei(αn +α)x1 −iβ2 x3 , x ∈ Ω2 . n

n∈Z

%

$% f f = n∈Z f (n) ei(α +α)x & ' ( Tj )*+ n

1

(Tj f )(x1 ) =

iβjn f (n) ei(αn +α)x1 , 0 < x1 < L , j = 1, 2.

,!

n∈Z

% u Ωj , j = 1, 2 ! #! ∂(u − uI ) − T1 (u − uI ) = 0 ∂ν

Γ1 ,

∂u − T2 u = 0 ∂ν

Γ2 ,

-.!

ν ∂Ω )*+ / -& % *! -.! $ $ $ ωj |ωj | → +∞ $ $ ω 0 ε(x) %1

Ω 2 *! -.! $ ' 345 Ω 345 6 δ1 δ2 Ω1 Ω2 345

Ω s(x3 ) = s1 (x3 ) + is2 (x3 )

Ê

s1 , s2 ∈ C( ), s1 ≥ 1, s2 ≥ 0, s(x3 ) = 1 b2 ≤ x3 ≤ b1 .

s1 ≡ 1

s1

! " #

Ω1PML = {(x1 , x3 ) : 0 < x1 < L b1 < x3 < b1 + δ1 }, Ω2PML = {(x1 , x3 ) : 0 < x1 < L b2 − δ2 < x3 < b2 },

$ ∂ L := ∂x1

1 ∂ ∂ ∂ s(x3 ) + + k 2 (x)s(x3 ). ∂x1 ∂x3 s(x3 ) ∂x3

! %

L(ˆ u − uI ) = 0 Lˆ u=0

Ω1PML , Ω2PML .

& '

! % uˆ Ω u+k 2 (x)ˆ u = 0 D = {(x1 , x3 ) : 0 < x1 < L, b2 −δ2 <

( % ∆ˆ x3 < b1 + δ1 } )

Lˆ u = −g D

*

%+ uˆ(0, x3 ) = e−iαL uˆ(L, x3 ) b2 − δ2 < x3 < b1 +δ1 ) uˆ = uI Γ1PML = {(x1 , x3 ) : 0 < x1 < L, x3 = b1 + δ1 } uˆ = 0 Γ2PML = {(x1 , x3 ) : 0 < x1 < L, x3 = b2 − δ2 }

g=

−LuI Ω1PML , 0

.

! " %

uˆ u % Ω ,

- " & u ˆ = uI +

$ R R " n x3 n x3 An1 eiβ1 b1 s(τ )dτ + B1n e−iβ1 b1 s(τ )dτ ei(αn +α)x1 Ω1PML .

n∈Z

%

(n)

uˆ(x1 , b1 ) = uI (x1 , b1 ) + n∈Z u ˆα (b1 )ei(αn +α)x1 Γ1 n n A1 , B1 u ˆ = uI Γ1PML

An1 + B1n = uˆnα (b1 ) An1 e

R b1 +δ1

iβ1n

b1

s(τ )dτ

+ B1n e

−iβ1n

R b1 +δ1 b1

s(τ )dτ

= 0.

uˆ = uI +

ζ n (x3 ) 1 u ˆ(n) (b1 )ei(αn +α)x1 ζ1n (b1 ) α

Ω1PML ,

n∈Z

ζ1n (x3 )

=e

−iβ1n

Rb

1 +δ1 x3

s(τ )dτ

−e

iβ1n

Rb

1 +δ1 x3

s(τ )dτ

u ˆ=

ζ n (x3 ) 2 u ˆ(n) (b2 )ei(αn +α)x1 ζ2n (b2 ) α

Ω2PML ,

n∈Z

ζ2n (x3 ) = e

−iβ2n

R x3

b2 −δ2

s(τ )dτ

−e

iβ2n

R x3

s(τ )dτ

b2 −δ2

% (n) i(αn +α)x1 e n∈Z f PML & # $ % ! Tj !

f

"!

f =

PML f (x1 ) = iβjn coth(−iβjn σj )f (n) ei(αn +α)x1 , Tj

'

n∈Z

coth(τ ) =

eτ +e−τ eτ −e−τ

b1 +δ1

σ1 =

s(τ )dτ,

σ2 =

b1

b2

s(τ )dτ.

(

b2 −δ2

)

∂(ˆ u − uI ) − T1PML(ˆ u − uI ) = 0 ∂ν 2 * ∆n j = |kj n

+ βj

Γ1 ,

∂u ˆ − T2PML u ˆ=0 ∂ν

Γ2 .

− (αn + α)2 |1/2 Uj = {n : kj2 > (αn + α)2 } j = 1, 2 = ∆nj n ∈ Uj βjn = i∆nj n ∈ / Uj *

n ∆− j = min{∆j : n ∈ Uj },

n ∆+ / Uj }. j = min{∆j : n ∈

! ,(- ! )

ϕ, ψ ϕα = ϕe−iαx

x1

1

, ψα = ψe−iαx1

PML ¯ ϕ)ψdx1 ≤ Mj xϕL2 (Γj )xψL2 (Γj ),

(Tj ϕ − Tj

Γj

Mj = max e

σj

2∆− j

− 2σI ∆ j j −1

2∆+ j

,

+ 2σR ∆ j j −1

e

σjR , σjI

, σjR , σjI

σj = σjR + iσjI

σjR , σjI

Mj

s(x3 )

!

⎧ " $m ⎨ 1 + σ m x3 −b1 1 " δ1 $m s(x3 ) = ⎩ 1 + σ m b2 −x3 2 δ2 " #

σjR

!

x3 ≥ b1

!

x3 ≤ b2

σjm = 1+ δj , m+1

σjI =

,

m ≥ 1.

σjm δj . m+1

$%

& # ' δj ! ' σjm σjm

" !' '# !

( # !

M1, M2 uˆ |u − uˆ|Ω :=

|b(u − u ˆ, ψ)| 1 0=ψ∈H 1 (Ω) xψH (Ω) sup

ˆ 1 u ˆ 2 u ≤ CM ˆ − uI L2 (Γ1 ) + CM ˆ L2 (Γ2 )

Cˆ =

1 + (b2 − b1 )−1

)

X(D) = {w ∈ H 1 (D) : wα = we−iαx1

x1

L}

aD : X(D) × X(D) → C ∂ϕ ∂ ψ¯ 1 ∂ϕ ∂ ψ¯ 2 ¯ aD (ϕ, ψ) = + − k (x)s(x3 )ϕψ dx. s(x3 ) ∂x1 ∂x1 s(x3 ) ∂x3 ∂x3 G

* !

X0 (D) = {w ∈ X(D), w = 0 Γ1PML ∪ Γ2PML}

uˆ ∈ X(D) uˆ = uI Γ1PML, uˆ = 0 Γ2PML

¯ g ψdx

aD (ˆ u, ψ) =

∀ ψ ∈ X0 (D).

D

Mh D ! T ∈ Mh " T # Ω1PML $ Ω2PML Ω % % #

&'# x1 $ & (0, z) $ (L, z) $ ( ( Vh (D) ⊂ X(D) % ¯ → Vh (D) # Vh0 (D) = Vh (D) & X0 (D) Ih : C(D) % # # % ##) # ' uˆh ∈ Vh (D) uˆh = Ih uI Γ1PML, uˆh = 0 Γ2PML

aD (ˆ uh , ψh ) =

A(x) =

A11 0 0 A22

=

g ψ¯h dx

∀ ψh ∈ Vh0 (D).

D

0 s(x3 ) 0 1/s(x3 )

,

B(x) = k 2 (x)s(x3 ).

% L a D

L = div (A(x)∇) + B(x), aD (ϕ, ψ) = D A(x)∇ϕ∇ψ¯ − B(x)ϕψ¯ dx.

T ∈ Mh $ hT Bh ΓjPML$ j = 1, 2 e ∈ Bh$ he T ∈ Mh $

RT := Lˆ uh |T + g|T =

L(ˆ uh |T − uI |T ) T ⊂ Ω1PML , Lˆ uh |T .

e ∈ Bh T1 T2 ∈ Mh $

% *# e Je = (A∇ˆ uh |T1 − A∇ˆ uh |T2 ) · νe ,

( ( νe e # T2 T1 Γleft = {(x1 , x3 ) : x1 = 0, b2 − δ2 < x3 < b1 + δ1 } Γright = {(x1 , x3 ) : x1 = L, b2 − δ2 < x3 < b1 + δ1 } + e = Γleft ∩ ∂T

T ∈ Mh e Γright T ( ' ∂ ∂ −iαL ) , Je = A11 ∂x (ˆ u | ) − e · (ˆ u | h T h T ∂x 1 ' 1 ( ∂ ∂ ) . Je = A11 eiαL · ∂x (ˆ u | ) − (ˆ u | h T h T ∂x1 1

T ∈ Mh ηT $1/2 ( ' "1 , ηT = max ρ(x3 ) · hT RT L2 (T ) + he Je 2L2 (e) 2 x∈T˜ e⊂T

T˜ T |s(x3 )|e−Rj (x3 ) x ∈ ΩjPML , ρ(x3 ) = 1 x ∈ Ω.

Rj (x3 ) (j = 1, 2) x3 x3 − + R1 (x3 ) = min ∆1 s2 (τ )dτ, ∆1 s1 (τ )dτ , b1 b1 b2

R2 (x3 ) = min ∆− 2

s2 (τ )dτ, ∆+ 2

x3

x3 ≥ b1 ,

b2

s1 (τ )dτ

x3 ≤ b2 .

,

x3

C > 0 Mh ˆ 1 ˆ ˆ 2 ˆ |u − uˆh |Ω ≤ CM uh − uI L2 (Γ1 ) + CM uh L2 (Γ2 ) 1/2 2 ˆ +CM3 Ih uI − uI L2 (Γ1PML ) + C ηT , T ∈Mh

Cˆ

M3 = max

Mj (j = 1, 2) −

I

−

I

−∆1 σ1 2∆− 1e

1 − e−2∆1 σ1

,

+

R

+

R

−∆1 σ1 2∆+ 1e

1 − e−2∆1 σ1

.

! "#$ σjR σjI Mj % % e−Rj (x3 ) "#$ ΩjPML & & "#$ ' & "#$ "#$

!

" ! ! " " # δj " σjm $ $%&'' ( ! ) EPML * + EFEM ! EPML = M1 u ˆh − uI L2 (Γ1 ) + M2 u ˆh L2 (Γ2 ) , $1/2 " EFEM = M3 u ˆh − uI L2 (Γ1PML ) + ηT2 .

$%' $%,'

T ∈Mh

EPML EFEM " - . ! *

δj σjm Mj L1/2 ≤ 10−8 ! # ! * + . / " * ! * " "

$%,' 0 " T ∈ Mh ! * !) η˜T = ηT + M3 Ih uI − uI L2 (Γ1PML ∩∂T ) .

1 ! ! !

2 TOL > 0 m = 2, δ1 = δ2 = δ • • •

3

δ σjm Mj L1/2 ≤ 10−8 j = 1, 24 D = Ω2PML ∪ Γ2 ∪ Ω ∪ Γ1 ∪ Ω1PML Mh

D4 EFEM > TOL

5 * Mh ! " η˜

T

>

1 2

maxT ∈Mh η˜T

* T ∈ Mh

5 $%%' Mh 5 Mh ! 1 ! ! " δ # " σjm " Mj L1/2 ≤ 10−8 j = 1, 2 +

3

2.25

x3

ε1

0 ε2 −1

0

0.3

1.2

1.6

2

x1

1 total efficiency

Efficiency

0.8

th

efficiency of 0 order reflected mode

0.6

0.4 st

efficiency of −1 order reflected mode 0.2

0

0

2000

4000 6000 8000 Number of nodal points

10000

µ = 1

! " " # $ ε1 = 1, ε2 = (0.22 + 6.71i)2 % θ = π/6% ω = π " L = 2 ! & " ' !( ) & " !( ! ) " " # * # + ! " " " , "( ! ) " *- " "

(

! ,+,

*+.

! " " " %

"

/0 % % "% 1 2 ( 3 45 6 4 4 7 % % -8*+ 9-:,; /$0 < % 1 2 $ " * " 5 7 4 " % % :8- 9--;

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)$, ! - '( )*

&

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# $

$

# % % n & '( ) $ Oxyz $ z ≥ 0 * ri (xi , yi , 0) Fi (Xi , Yi , Ni )

Φi = (p − pi )S i = 1, . . . , n p pi S R(Rx , Ry , Rz ) M(Mx , My , Mz ) (Xi2 + Yi2 )1/2 ≤ f Ni ,

i = 1, . . . , n,

f ! Ni ≥ 0 " Rz +

Rx +

%

Xi = 0,

% (Ni − Φi ) = 0, Mx + yi (Ni − Φi ) = 0, % My − xi (Ni − Φi ) = 0, Ry +

Yi = 0,

Mz +

(xi Yi − yi Xi ) = 0.

# $

% i = 1, . . . , n Xi , Yi Ni # $ i = 1, . . . , n & # ' ( & Ni ≥ 0 z = 0 $ ' ) Xi Yi $ i = 1, . . . , n z = 0 * & K Oxy % xi Φi + My x∗ = % , Φi − Rz

% yi Φi − Mx y∗ = % . Φi − Rz

+

coD

Φi > Rz ,

K ∈ coD,

D = {r1 , . . . , rn },

"

Q=

x,y

Mz −xR P y +yRx , ρi Ni

*1/2 ) ρi = (x − xi )2 + (y − yi )2 , sup

&

x, y !

Oxy !

#

sup Q ≤ f,

%

$

i = 1, . . . , n.

" '(

Ni > 0 ! ( ' (

$ ( ) (

n > 3!

Ni

" ( (

! " ! *! " & ( ! + + ( ( (! " , - , . ( ! /' ( ( (

Ni (! 0! "

' ( (

( ! 1 ' $ (

Ni

! " (

sup Q ≤ f,

2

x,y,Ni

sup

x, y Ni ! (

&

3 ( '(! ! 3

(

(Rx , Ry , Rz ) O!

( (

(Mx , My , Mz )

! 4 Φi = (p − pi )S, i = 1, . . . , n! #! 4

K!

! 4& ! ! (! !

& (

Ni *! " $ ! 3 0 ((( (

2 2 ( &! $ 2 * 0 (

pi !

"

" # ! "

a

b

!

h r m (n = 4) p0 ! $ % $ $ % & & " '

m ≤ 2π(p − p0 )ar2 (gh)−1 , (

m ≤ 4πf (p − p0 )r2 g −1 .

a = 0.344m, b = 0.151m, h = 0.095m,

r = 0.034m, p0 = 0.1p, f = 0.5

" 48kg

" ! 30kg

67kg

) & *+ ,- ! " " ./0 $ ( 1 ! " )2 *3- ! " & & & 4

$

4 ! "

( !

"

" " 5 " "

( " " 6 " " 5 " ! " *3- 7 8 "

9

!

" # $ ! " ω % # ! α # "& ' " (

"

v1 = Lω L ! " # !

$ α → π/2 "

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! ! "#"$# % & '( ! ) ( * ! + ,-

(0.0139%) ! " "

# " $$

% $ &' (

) y = 1.25* + Us = 0mm · s−1 )(*, 3mm · s−1 )!*, 6mm · s−1 )-* 9mm · s−1 )* $

$ 304 )(./* 0.0139% ' ( + I = 100A U = 10.6V

!" 3mm · s #!" 6mm · s $! 9mm · s ! % % & ' ( )( Us = 0mm · s−1

−1

−1

−1

2300K

!

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304 #

*

+ , %

U T % % (m s ) (K) (mm) (mm) --( .- -)) ) -.( ./ /)0 -.( 0 -00 . 0 )( -) .-- .-//) )

(mm s−1 )

max −1

max

0.0139%

! z

"

1 s # 50 K $ % 25 K $ %

2300K & ∂γ ' ( ∂T ) & (

! ) t = 1.0 s& t = 1.06 s& t = 1, 12 s t = 1.18 s

#

! * ( &

( 4 mm& #

1 ms−1 + ν = 6.81 × 10−7 m2 s−1 ,& - Re ≈ 5800

&

& #

( ≈ 1 ms−1 #

#

* T > 2000 K

+ +

" ) . ! & /& 0& 1 &

& x = 0

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! " #$ % ! " #$ &$ ' ! " ( "

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# $

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0

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6 1 ##3 #

! ! 78

5 9

':++

! "

5 !

#

); ;/

6 !

t ∈ [0, te ]

<

8

D

L 6 x ∈ [0, L]

t0 ρ v = q = ρv p T x t # x h(x) # %&& &;(

T (x, t)! " #$%&! ∂t ρ + ∂x q = 0 , $ λ(q) ρv|v| , ∂t q + ∂x p + ∂x (ρv 2 ) + gρ∂x h = − ' 2D p = γ(T )z(p, T )ρ . ( ) γ T ! z(p, T ) z = 1 * +, ! - . - -.-! z(p, T ) = 1 + 0.257(p/pc) − 0.533

(p/pc ) , (T /Tc)

%

pc Tc / "! * ! 0 λ(q) ' 12! 1 = −2 log10 λ(q)

k 2.51 + Re(q) λ(q) 3.71D

.

3

) k

! η ! Re(q) 4 ! Re(q) =

D q ≈ 106 . η

5

6 $1( / ! A = π4 D2 ! 1/A 7 ! ! ρ ← Aρ 28! ! q ← Aq 28! γ ← γ/A! $ ( ' 2 ∂t q ∂x (ρv2 ) 9 #$%& v = q/ρ ! A∂x p + gρ∂x h = −

λ(q) q|q| . 2D ρ

+

q pin pout

Operating Range of a Turbo-Compressor 50

eta_ad = 0.60 0.65 0.70 0.75 0.80 0.85

45 40

n = 6500 5850 5200 4550 3900 3250 measurement

Adiabatic Head

35 30 25 20 15 10 5 0 2

4

6

8

10

12

Volumetric Flowrate

!" Nth = c1 qHad

c 1 , c2

κ = c1 q c2 z(pin , Tin )Tin κ−1

Had

1

pout pin

κ−1 κ

2 −1 κ

.

n

!

ηad

"

q

Had

#

$

B = c3 N b(n, N )

b

N

#

&'

0.5(

B

N = Nth /ηad

%

# ! )

*

!

#

B = c˜3 Nth

c˜3 = c3 b/ηad

+ %

c ∈ (0, 1]

pout = cpin ,

qout = qin .

pout = pin ,

qout = qin

(open) ,

qin = qout = 0

(closed) .

∆p ∈ [∆pmin , ∆pmax ]

pout = pin − ∆p ,

qout = qin

(open) ,

qin = qout = 0

(closed) .

G = (N , A)

N+ N− N0 N = N+ ∪ N− ∪ N0 .

! Api

Acn Acs Avl Arg A = Api ∪ Acn ∪ Acs ∪ Avl ∪ Arg . 3 45 6 3 45 6 passive

"

active (controlled)

# a ∈ A i, j ∈ N ij ∈ A$ % & i j $

' te costcs = ca Ba (t) dt . ( 0

a∈Acs

) * ' + & t > te $ ,+ ' t = te $ - ' + La mmin ≤ ρa (x, te ) dx . . a∈Api

0

! "

Γa = {0, La} a ∈ Api

a ∈ Api # I = [0, te] $ ΓI = {0, 1, 2, . . . , te }

∆t = 1 # x = (x0 , . . . , xt ) with xt = (pt , qt , st , ut ) , t = 0, . . . , te . %& ' t

pt = (pit )i∈N ( qt = (qit , qjt )ij∈A

st = (sat )a∈A

(t−1, t)

ut = (uat )a∈A

) e

sijt = ρjt , ij ∈ Api , uat = ∆pat , a ∈ Acs , sat = Nat , a ∈ Acs , uat = ∆pat , a ∈ Arg .

%*

+ sat uat # ,-. $ . a = ij ∈ Api t ∈ {1, . . . , te} t− = t − 1) ρjt − ρjt− qjt − qit + =0, ∆t La 2 pjt − pit hj − hi λ(qjt ) qjt Aa + gρjt + =0, La La 2Da ρjt pjt − γ(Tjt )z(pjt , Tjt )ρjt = 0 .

%/

01 0% # 2

a = ij ∈ Acs Bat = c˜3a Nat

κ = c1a c2a c˜3a qjt z(pit , Tit )Tit κ−1

1

pjt pit

κ−1 κ

2

−1

.

00

# 3

4 Γa ΓI costcs =

a∈Acs

ca

te Bat− + Bat ∆t , 2 t=1

mmin ≤

ij∈Api

Lij

ρite + ρjte . 2

05

#

$

( $ j ∈ N0 t ∈ {1, . . . , te}

P2 P1

CsA

Vl1

CsB

CsC

C3

Rg1 C1

C2

qijt =

i: ij∈A

qjkt .

k: jk∈A

x0 = x ˆ0

i ∈ N+ ∪ N− ij ∈ A : i ∈ N+ j ∈ N−

!

"# $% & ' () * !

+ , -"$ !,

./ .1/

, ! 0 ,

* &

$ 23455 324

52 & 62 & $ 147

12 8

*

! 7

a ∈ Api La = 10 km * -"$ 5597 56

55:2

*

;

,

& 5:46 < 452

3

10−6

8 0 + &

= 0 ,

0 27>2245>22 +

Cs A Cs B Cs C Rg 1

20

Cs A Cs B Cs C Rg 1

20

15

15

10

10

5

5

0

0 0

6

12

18

24

30

36

42

48

6

12

18

24

30

Cs A Cs B Cs C Rg 1

20

0

15

10

10

5

5

0

42

48

42

48

42

48

Cs A Cs B Cs C Rg 1

20

15

36

0 0

6

12

18

24

30

36

42

48

6

12

18

24

30

Cs A Cs B Cs C Rg 1

20

0

Cs A Cs B Cs C Rg 1

20

15

15

10

10

5

5

0

36

0 0

6

12

18

24

30

36

42

48

0

6

12

18

24

30

36

! " !! "

# $ % &

!" # $ %

& ' % " ! $ ! " ( % % % % % % ) $

* + , % ( - (. % # %

/ % % % (% % % " % 0 #

$ % % % % % % 1 % 2 % % % ( ( 3 & $ # % % ( & %

' %* % *# % %%#

& / % 245 % %

' & ( & 6 ,7 8 523 ,7 9 ) 6 % % : ' 5 8 7 8 1 7* % & $ % -

% +

1 # $ %

+ 1 #

!

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$ $ ) , . 1/ 1 %011

! , . %$ ! " $+

! " # $ % & ! " ' ' ( ) *! $ " # # + $ ! ,$

y = F (x) vi−n = xi vi

= ϕi (vj )j≺i

ym−i = vl−i

i = 1...n

i = 1...l

i = m − 1...0

F ϕ v i ∈ V ≡ {1 − n, 2 − n, . . . , l − 1, l} G = (V , E ) j i (j, i) v v ∂ i

i

i

(j, i) ∈ E

⇐⇒

∂vj

ϕi ≡ 0

j

⇐⇒

j≺i .

j≺i

=⇒

j
G n m v = x j = 1 . . . n v = y i = 0 . . . m − 1 x = (x , . . . , x ) y = (y , . . . , y )

j−n

1

j≺i

=⇒

j

l−i

1

n

j ∈ 1 − n, . . . , l − m

∧

m−i

m

i ∈ 1, . . . , l

ni ≡ j : j ≺ i

1 2 ϕ

n i !

F (x) " # $ ϕ : Ê → Ê i

i

i

ni

Ê

Di ⊂ ni

cij ≡

∂ ϕi (ui ) j ≺ i ∂vj

ui ≡ vj j≺i ∈ Di F ϕi (i, j) cij = cij (x) x ∈ n

! "

Ê

v1 v−1 ∗ v0 v2 exp(v1 ) v3 sin(v2 ) v4 sqrt(v2 ) v5 atan(v2 ) v6 v1 − v2 = (v−1 , v0 ) (v3 , v4 , v5 , v6 ) = c1−1 v0 c10 v−1 c21 v2 c32 cos(v2 )

−1 c1

−1

c 10

c21

1

2

c61

c 32 c42 c52 c6 2

3 4 5

0 6 c42 c52 c62 c61

0.5/v4 1/(1 + v22 ) −1 +1

! "#

cij ! " # $ cij ϕi $ % cij ≡ 0 j ≺ i l × (l − m + n)

i=1,...,l C ≡ cij j=1−n,...,l−m (

&"'

⎡

c1−1 ⎢ 0 ⎢ ⎢ 0 C =⎢ ⎢ 0 ⎢ ⎣ 0 0

c10 0 0 0 0 0

0 c21 0 0 0 1

⎤ 0 0 ⎥ ⎥ c32 ⎥ ⎥ c42 ⎥ ⎥ c52 ⎦ −1

.

C ∈ Ê6×4 8

c62 = −1 c61 = 1 6 (x1 , x2 ) ≡ (v−1 , v0 ) C = C(x) Ê6×4 6 C 6 cij

F : Ên → Êm x ∈ Ên Ên Êm Êm×n F (x) ∈ Êm×n F x ej j = 1 . . . n

Ên ei i = 1 . . . m Êm F

F (x) aij ≡ eTi F (x)ej ≡

∂ T e F (x) ∂xj i

c˜ıj˜ P ≡ (i1 , i2 , . . . ik¯ ) (ik , ik+1 ) ∈ E cP ≡

#

cij =

¯ k−1 #

cik+1 ik

!"#

k=1

(j,i)⊂P

$% &'( am−i,j ≡

cP

.

!)#

P ∈[j−n → l−i]

* [j − n → l − i]

0 < j ≤ n 0 ≤ i < m $

aij

4 × 2 8

! " F (x) 1 # " ! F (x) $ %&' ( $ " 1 4×2 ! c1−1 c10 c21 c32 c42 c52 c62 c61 ) $ ! * c21 = 1 c62 = 0 5 ) X ⊂ n D = dom(F ) $ # C

Ê

Ê

C:X

−→

Ê l×(l−m+n)

.

C * {cij (x) : x ∈ X } X ⊂ n ( x # + , *

ϕi cij # cij

Ê

C ∈C≡

$i=1...l " cij (x) x∈X

j=1−n...l−m

⊃ C(X ) .

- C C(X ) .$ C X

# C dim(C) dim(C(X )) ≤ dim(C) ≤ l (l − m + n) .

#

ϕi (vj ) = γi ∗ vi γi / 0 γi

γi

G n m C ∈ C

B C= R

L ∈ S

Ê((l−m)+m)×(n+(l−m))

.

!"#

L det(I − L) ≡ 1

C m × n $ % & '() $ * + !,# !(#

i=1...m A = aij j=1...n

−1 A ≡ A(C) ≡ R + S I − L B

.

$ ',) - A

A:C

−→

Êm×n

det(I − L) ≡ 1

L $ A(C) C ∈ C

A=

"

$i=1...m aij (C) C∈ C ⊃ A(C)

.

j=1...n

$ dim(A) A(C) C ∈ C

. / % 0

F dim(A(C)) C

G

A(C) G

scarce(G) ≡ dim(A) − dim(A(C))

C

.

dim(A(C)) = dim(C)

1 C ∈ C

A ⊂ m×n 12 A(C) 3 C

Ê

dim(A(C)) = 5 < 6 = dim(C) " # A(C)

!

C

$

% & # ' ( y˙ = F (x) x˙ ∈ m ! x ¯ = y¯ F (x) ∈ n x˙ ∈ n y ¯ ∈ m % $ )

Ê

Ê

Ê

Ê

Ê

y˙ x ¯ $ x ∈ n ( x ˙ ∈ n y¯ ∈ m * + , ,

y˙

Ê

Ê

x ¯ #

' - .

1

−1 y˙

&

x ¯

$ (

"

6

$&

# "

$

/ -

v˙ i−n v˙ i y˙ m−i

= x˙ i

i = 1 . . . n P = cij v˙ j i = 1 . . . l j≺i = v˙ l−i

i = 0 . . . m − 1

v¯l−i v¯j x ¯i

= y¯m−i

i = 0 . . . m − 1 P = v¯i cij j = l − m . . . 1 − n ij = v¯i−n

i = 1 . . . n

'

˜ G

G

/ 0

/ -

$

2 (−1, 1) 1 %

( ! / , 1 . 2 3

3 c˜31

−1 1

c˜10 = c10 /c1−1

c˜41

1

c˜31 = c32 · c21 · c1−1

4

c˜51

c˜10

c˜41 = c42 · c21 · c1−1 5

c˜61

c˜51 = c52 · c21 · c1−1

0

c˜61 = (c61 + c62 · c21 ) · c1−1

6

G˜

C˜ = T (C) ∈

Ê

5×3

T : C → C˜

Ê6×4

˜ = A(T (C)) A(C) = A(C)

C ∈

.

T ˜ dim(C)

dim(A(C))

C˜ = T (C)

!

"

x˙

y˙

y¯

x ¯

C˜

l−m

#

dim(C)

$

i

(n + i)

"

"

l − m ≤ 2 ∗ dim(C) %

"

G C ˜ C˜ C

C ∈ C # A

C ∈ C C˜ = T (C) A l = m C˜ T C ⊂ C A(C) = A(T (C)) C ∈ C . C C T C˜ = T (C) C ! C˜ ≡ T (C) ˜ . dim(A(C)) = dim(A(T (C)) = dim(T (C)) ≤ dim(C)

C˜ " #

" F (x) $ y˙ = F (x)x˙ x¯ = y¯F (x) T dim(T (C)) = dim(A(C)) %

A : C → Êm×n " r ≤ dim(A) ≤ m n & X C '()

Ê

A(C) ⊂ m×n r T : C → C T (C) = C r T (C)

dim(A(C)) = dim(A(T (C)) = dim(T (C))

C∈C

.

! " A(C) r * +

! A , + C dim(A(C)) + %+

C ∈ C T : C → C˜ C˜ dim(A(C)) A(C) = A(T (C))

T

T (C) F (x)

T

A(C)

A(C)

A(T (C)

m n = 8 dim(A(C)) = 5

T (C) ≡ A(C) dim(C) = 6

! "

(j, i) ∈ E

#

(j, i) E chj += chi · cij h i

(j, i) E cik += cij · cjk k ≺ j

γ = cij = 0 chi ∗= γ h i cik /= γ k ≺ i

γ = cij = 0 cjk ∗= γ k ≺ j chj /= γ h i

(j, i) E (k, i) cik = 0 ckj += γ ≡ cij /cik chj −= chk · γ h k, h = i

(j, i) E (j, h) chi = 0 cih += γ ≡ cij /chj cik −= chk · γ k ≺ h, k = j

h

h

+ j

j

i

i

+ h k

k j

+ j +

i

k

h i h

k d iv 1 j

k

k

i

j

∗

v di

k

h

∗

k i

h

∗

h

k

i

k

k

j

h

∗

k

j

h

v di 1

i

div

h

h

i j

j

k

h

+

h

−

h

k

h

j

j i

− k

k

i

−

+ h

h k

i

−

k

(j, i)

G

˜

C G˜ C˜ A(C) = A(C) {−1, 0, 1} G˜

G + − cij ∗ div (j, i) ∈ E (i, k) ∈ E !

" #$% & #'%

( ) * + (j, i) ∈ E (k, i) ,

* ) m = n F (x)x˙ = y˙ x˙ ∈ Ên y˙ ∈ Ên & - " #.% &

" "

/ " 0 1 2 3 × 3 3 4 6 + 5

9 2 × 2 (j, i2 ) (k2 , j) i0

k0

i0

k0

j

j

k1

i1

k2

i2

=⇒

k1

i1

k2

i2

(j, i2 ) (k2 , j)

G G˜ ! 2 " # (−1, 1) # G˜ $ % −2

3

−2

1 −1

1 4

−1 −

2 0

3

4 2

5

0 −

5

(−2, 2) (−2, 1) (1, 2)

% (−2, 2) c1−2 = 0 & ' (1, 2)

1 " ( ( (2, 3) (1, 3) ' ) * (−2, 1)

−2

+ 1 + +

−1

−2

3

3 1

−1

4

−

2

2

0

5

4

−

0

5

(2, 3) (1, 4) (2, 1)

10 8 (−2, 1) (2, 4) 1 Ê3×3 8 ! " ! ! −2 −1

−2

3 + +

1 −1

4

1 div

∗

1

v di ∗

2 0

5

0

∗

3 ∗

1

4

2

∗

div

5

(−2, 1) (2, 4)

# v(t, u, w) (u, w) $ t % t u w h = 1/˜n n˜ > 0 tk = k h ,

uj = j h ,

wi = i h

0 ≤ i, j, k ≤ n ˜ .

" v(t, un , w) = v(t, 1, w) = v(t, 0, w) = v(t, u0 , w)

v(t, u, wn ) = v(t, u, 1) = v(t, u, 0) = v(t, u, w0 ) .

& $ !

vk,j,i ≈ v(tk , uj , wi )

uk+1,j,i = fh (tk , uj , wi , vk,j,i , vk,j−1,i , vk,j,i−1 ) vk−1,i ≡ vk,˜n−1,i vk+1,j,i

vk,j−1 = vk,j,˜n−1 v

.

!

n ˜ = 3

"

v0,j,i j=1...˜n,i=1...˜n

p = (i − 1) ∗ 3 + j

# m = n ˜ vn˜ −1,j,i j=1...˜n,i=1...˜n n ˜ 3 dim(C) = 4˜ n2 (˜ n − 1) 2

$

$ n ˜ 4 % &

dim(A) = n ˜ 4

dim(C) ≥ dim(A(C))

1 2 ˜ /(˜ n 4n

− 1)

) ˜ − 2]2 scarce(G) ≥ n ˜ 4 − 4˜ n2 (˜ n − 1) = n ˜2 n %

n ˜ ≥ 3

.

' $

#

7

8

9

4

5

6

1

2

3

( ) $ & *

$

v

(uj , wi ) 3˜ n2 (˜ n − 1)

n ˜ 2

n ˜ /3 ≈

√ n/3

F

!

"#$ %& ' (

( )* +

½½ ,#-./0( 1.2-3

* '7 )* #--3( #/3 2 #/- 9 (

( + #- % * )*(

"4$ % 5 6(

"8$

( 4::: "/$ 9 (

! "# (

* ( 4::8

$ % ( 5=>:4( 3 5

";$ 9 < +

(

= 5 ) ( 4::4 "3$

& $

(

".$ * (

( ? @ A ( (

, 9 9 % 5 ( 0( )*( (

( #--#( #2#3

% '# ( ? ( ) 5 (

"1$ < (

? < ( 9 ( #--3

1 2 3

1 1 2 3

! "# $ % &' ( ) *+, --. / 0 1 .2 3 0 "# $ &10 % &' ( ) 4 .5 &10

" 6 0 0 0 6 7 8 9 0 6 & ( 7 04 ( 7 0 7 7 0 7 & 7 0 04

! !

" #

# $ %

%

% & '

(

% %

# % )*+ , -.

# % )/+ # % %

Tc

† † † HHHM = −t ciσ cjσ +U ni↑ ni↓ +gω0 (bi +bi )niσ +ω0 bi bi . ! i,j ,σ

i

i,σ

i

" t # U $ c†iσ σ % i niσ = c†iσ ciσ & ' g = εp /ω0 εp (†] ! ω0 bi ! ( )*+ , -! & ).+ )/+ 0 " 1 #'%! '%! U ! )2+

0 3 ! " 4 5'! 4 '-67! - # 4 '

"

5'

{|Φi } Hi,j = !Φi |HHHM |Φj .

Hi,j ! !

" ! #$% & '() * +,+ HHHM !

! ! -

! N = 8 16 * & ! ! . % % i ni↑ +ni↓ z 12 i ni↑ −ni↓ ! {|Φu,v = |u

el

⊗ |v

ph ;

u = 1, . . . , Del ; v = 1, . . . , Dph },

/

Dtot = Del ×Dph |v

ph

=

" $mi,v 1 b†i |0 m ! i,v i=1 N #

ph ,

mi,v ∈ [0, ∞]

% M i mi,v ≤ M M Dph = (M + N )!/M !N ! ! M ω0 "! # $ HHHM E M ω0 M % {t, U, g, ω0} & % "' N = 8 M M Dtot = Del × Dph ∼ 1010 ! ( "' $ "' % ) ) *+* ) *+* "! , -. /0

'*12 -#3 ## #40 "' H 56 7 % '*12 '*12 8 4

! " # $ % & !'!

ρii =

∗ ψij ψi j ,

j

i j ⏐ : ⏐ψ ⏐ : ⏐ :⏐ : ⏐ψ = ψij ⏐i ⏐j .

ij

!" ρ # $$ % • & ' & l ' m • % & l + 1

' • % & l − 1 % ( $ ) & $ * +,+ $ #$ $ $

- ' $ & . /0 !"$ ¯ l+1 H

|

|

¯ R H l −1

!" H¯ l+1 # H¯ lR−1 $ % # & $ % ' ## # ( $"

m ' $ m $ 1 m m

$ 2 m

m = 500 !! "#

$ %

& '( ) !

!" # $ # $% # &

&% '# (") '% % & *# '# +,-! # ,--. %% / '#

0 % !0 / % '% 12 &%3 % !0

4"- % %' /* 5 # ' % 6 1

0+" 78/ 9 : ; $<

61

;1

4"-

670%'1 : $<

9

9

"-

0% = ( :19 $<

92

1

"-

!' ;2 :8 "$<

2

19

4"-

!0 ;9 :6 "$<

12 :6 1<

4"-

! * + , , '-) . /0 1 1 * '2) . /0 $ "3 4 -1- ! U = 4 m = 2000# $ - $5 6 4

/ * 7 68 * +

& . "#

Performance [MFlop/s]

2500

2000

1500

1000

500

0

90

M

IB

p6

(

HP

0

67

G (1

el

Int

P

Xe

D on

G 2.4

(

ir

nF

Su

00

8 e3

I SG

)

Hz

Hz

Hz

rx5

)

)

)

)

Hz

G 1.3

(9

M 00

00

Hz

M 00

(5

4 n3

igi

Or

!"

! " # $

!

$ %&

' ( ( )*

($ " + , - %. /. , &

)0 1* , U = 0 & )&#*

gc (ω0 ) $ ω0 → 0 2 &# " 3 P = +1 4 #

3

%. 5 g = 0

!

P = −1

!"

##

Peierls Insulator

Mott Insulator

u/λ >> 1

∆c = ∆s > 0; P = +1

∆c > ∆s = 0; P = −1 u/λ << 1

CDW

SDW

u = U/4t λ = εp /2t = 2αg 2 α = ω0 /t $ %

&

'( )(* ) + # *

# , & # # - . + , # , . / , # # , , // , # & % † , . Ekin = −t! i,j σ ciσ cjσ 0 #

U

+ 1

0!"

. * #

Ekin

!"

###

, # & # $

# # u/λ* * , # $ #

2

# , 3! 4 * *

5 " # 5

# . 5 ,

Ekin + , &

, 6

Sc (π) =

1 1 1 (−1)|i−j| !(niσ − )(njσ − ) 2 N i,j 2 2

,

7

σσ

1 Ss (π) = 2 (−1)|i−j| !Siz Sjz N i,j 8 ,

Sc (π) Ss (π)

,

Siz =

1 (ni↑ − ni↓ ) . 2

+ 7 , /

4, * , 0!" . !"

& U g ω0 * &

ED (M=24) DMRG (np=7;m=600)

0.5 -5.0

ED: U/t=2 ED: U/t=4 ED: U/t=6

-6.0

DMRG

Ekin(ep=2,U)

Ekin(εp=2,U/t)/Ekin(0,0)

1.0

-7.0 -8.0 4

0.0 0.0

2.0

8

12

16

4.0 U/t

M

20

24

6.0

8.0

U/t g 2 = 2 ω0 /t = 1 ! "# #$%& Ekin ' U g2 = 2 ω0 /t = 1( ) "# * ' ! M ! #$%& np = 6 ! * m = 1000 ! ) + ,- ! ./.

! "# # $ "# Sc(π)

% Sc (π) N → ∞ & # # % % % ' ( ) * % ' N = 128( * + * ,- . ,- "/% ! 0 % + + $ 1 0 "/ 0 # # 2 -& . ,- * %

$ 3 4 5 -& "/

0.04

1

U/t=6

U/t=2 0.8 0.03 0.6

Sc(π)

Sc(π)

-2

1.0×10

0.1

0.02 0.08

Ss(π)

Ss(π)

-3

0.4

7.5×10

-3

5.0×10

-3

2.5×10

0.0 0

0.2

0.05 0.03

0.01 0.1

0.05

0

-1

N

0 0

0.05

0.1

N

0.15

0

0.1

0.05 -1

N

0.2

-1

0 0

0.05

0.1

N

0.15

0.2

-1

!" # $ %& ! " & ! " m = 1000 5 ' (

( ) N = 128 ( m = 800 ' ( *#

T = 0 σ reg (ω) =

π |!ψ0 |ˆj|ψm |2 δ(ω −Em +E0 ) . N Em − E0

m=0

|ψ0 |ψm % † E0 Em ˆj = −iet iσ (ciσ ci+1 σ − c†i+1 σ ciσ ) ! " # $% σreg (ω) % &$'( &&$'( )*+ &&$'( , - η ! η . η &&$'( /& η /& 0 % &&$'( " ,- &$'( 1223 4 5 ω ω0 &&$'( " #

η ! "# $ %%$& ' (( $) &'$

reg

σ (ω) [arbitrary units]

3 ED DDMRG η=1 DDMRG η=0.2 DDMRG η=0.1

2

1

0 0

1

2

3

4 ω

5

6

7

8

N = 8 U/t = 6 g 2 = 2 ω0 = 1 ! " m = 200 # $ % # η = 1 η = 0.2 η = 0.1 & η " "

" &'() $ % *# % # + # ,

* + , -. %! -%# / . 0# %$& / / / ! 1 0 -% 2 %$& 3 / 4$ + -% % %$& %%$&#/ /

! ! " # $ " #

% &&' &'( )* * ( + ,&' ), - + !.&$ )! " * * $ + / &&'0 )& & 0 + '12 )' 1 2 + !.&0 / 2 * 0 ! " * )( !+ )! 3+

! " #$%%%& $ ' ( ) ) * ! + , - ,. / ' ! 00 # 110& 0 233 4 + ) 5 ( ! / $067$8 # 10&9 : 4 ; < , 5 < 5 $87$61 # 10& = < !2 5 , < , 2 4 , ' 5 #+>& 0$870=$ # 181&9 !2 5 , $ < ! 5 ' 5 #+>& 0=07061 # 181& 8 !; 3 ? 5 ( ) $%67$ 6 # 116& , ' @ / A 5 ; ' ! ! 5 $08;$=1 #$%%%& ) )B2 C 3 ; ? ; ; 5 ( ) 00; 0 # 116&

! " #$ % & % ' (" " ) # *+++(*+, -**+. * ! / $ ( $ $ 0 '1 % 2 % ($ & ' 3 45% -'.0 % " 6 $ % '%% *** 7(7* $%(8% # 9% -7:::. : ; < 1 2 $ " ) =

7+>(7++ -**7. ; < 1 ?% 2 $ " ) # :>@,(:>,+ -**>. 7 < ! ; < 1 2 $ & & " A % < 3! 3 9% -'.0 ; (< 1 2 0 ? / < " = / ! $ <1("!(& B " ! 3 $1 ; ** = " 8 ,7 $% # 9% -***. > ' 4! ; (< 1 2 $ " " 9 < " ) # +>+(+>, -**. @ % ' 4! ! "2 ( % ; < 1 2 $ ?% (< , ! ? " C$ 2 & ( ) 6 &?< " $ -7::. + ' 4! ; ( 1 2 (% $ " ) # :@,@ -7::7. ! ? D #B

? # $ " <

( ; " # >7(>> ,+7(,+> -7::7. ! ? " 3$ < ! "( <

( ; ' " 4 # (+ -7::>. * ! ? " 3$ < ! % ? D #B (

? # $ C( ($ 0 <

( )( "( & % -'.0 % " 6 $ % '%% < 7::7 >>*(>@* $%(8% # ( 9% -7::>. 7: < 92 ? C % ? ? # & <

& C ; " ) =

7:@(7: -***. 7 " # & 4$2 ? " 3$ < ! ( % $ (E 99 -7::>. 77 3 9% ; ( 1 % $ $( /( " ) # *7(*>: -**,.

1 1 2 1 2

λn · T (X, t) = qn (X, t)) ! "# $ "$ !

%" ! & ! '(') *(*+ ,(,-! ,(% !

! " # " " $ %&' #

$ ( ) * ) " "+

C0 ρ0

∂T (X, t) ∂T (X, t) ∂T (X, t) ∂T (X, t) ∂ ∂ +u = λ0 + λ0 ∂t ∂z ∂x ∂x ∂y ∂y

∂ ∂T (X, t) + λ0 ∂z ∂z

X = {x, y, z} u

z

nd rd

!

Ω0

!

" qn # $

%

%

&

'

% !

( %

( ! %

!

!

)

(

&

!

!

X ∈ Ω0

TL

TS

!

" #

C0 ρ0

$

∂T (X, t) ∂T (X, t) ∂T (X, t) ∂T (X, t) ( ∂ ∂ +u = λ0 + λ0 ∂t ∂z ∂x ∂x ∂y ∂y

!

%

u

(λ T )

&

! ' (

q = qI (z)

' $

)

α = αII (z) Tp > Tcr

$

X ∈ Γ0 : T (r, z) = Tr ∂T (r, z) =0 X ∈ Γ∞ : ∂r ∂T2 (r, z) X ∈ Γ1 : −λ2 = q1 ∂z ∂T (r, z) − αn (z)[T (r, z) − Tmt ] X ∈ Γ2 : −λ2 ∂r ∂T (r, z) =0 X ∈ Γs : ∂r F m=1,2 ∂T1 (r, z) ∂T2 (r, z) X ∈ Γ1,2 : λ1 − λ2 = U · ρ2 · L · ∂r ∂r ∂z

1 ∂ ∂T (r, z) ∂T (r, z) = Ωm : Cm ρm · U · r · λm ∂z z ∂z ∂z m=1,2

' (

∂y∂ λ0 ∂T∂y(Y ) ' ( ∂T (Z) ∂ λ 0 ∂z ∂z

C0 ρ0

∂T (X, t) ∂T (X, t) ∂T (X, t) ∂ +u = λ0 ∂t ∂z ∂x ∂x

X∈ X∈

Γ0 : T (X, t) = Tp ∂T (X, t) = qk (X, t) Γk : −λ0 ∂n

k = I, II, . . . , n = x

n = y

Γ0

Γk k k = I, II, . . . , qk !" k Co ρo

# " $%

∂T ∂T +w ∂t ∂z

T f +1 − T0f +1 T0f +1 − T0f +w 4 ∆t ∆z

= 0

z e = 4 ∆z = u∆t

z +

∆

∂T ∂T +w ∂t ∂z

Φe

0

T f +1 − T0f +1 T f +1 − T0f T0f +1 − T0f + 4 = 4 ∆t ∆t ∆t

2 2 T0f +1 − T0f Tef − T0f Φ Ψ + qof Φe (Ψe − 1) = e e f ∆t R oe e=1 e=1

∈ Ω0

$ $ %0

=

=

T0f

+

∆t C0f ρf0

1 2 Tf − Tf e 0 e=1

f R0e

Φe Ψe +

2

2 q Φe (Ψe − 1) f

(

e=1

$

$

Ψe

Ψe = 1, e = 1, 2 $ Ψe = 1 $

) * $

$

(z = 0)

$ $ $

Tp

'

&

$

!" # $ %0 $ $ &

C0f ρf0

T4f +1

T (X0 , 0) = Tp

+ $ #

( $ $

$

Γ 1

Γ1 ! " # $

% & ' ( Γ1

% $ )* "

q= r1

t1 − t2 1 λ1

ln dd21 +

1 λ2

ln dd32

$

+

% T1 $ '$ 2 $ $ 1 1

1 2 3 T2 λ1 λ2

Ì1 Ì2 Ì1 Ì2 Ì1 Ì2

& # % $

!! #! #" #&$ # % ## # #%

" ! % % %& % %# %

#$ #$! #"" #!% # &&

%% %$ %" %! $ $

# #! " %% $ "#

' (

) %% * ) +, * & - .

# $ $% ' % #(% )%

λ

# ρ

L S

&

&

& & &

!" !" !"3 0 0

*'+ % , $ %- $ ./ $ $ & % %0 $ ./ " % $ 1% %$ % - & 2 % " % $ 3 % $ % % $ %/%- 2 '/- % - " " % 2 4.) .# %% % - , $ %- ,% " % 5% %0 6 2 (%/ %% % *" %2% % $ % %7 ,%

2 " & % 5 2 % %6 % 2 " 8& % 5 %6 % $ $ %- - -! 2 2 " "

%%/% $ %" 2 "" $

%0 Scrit % %%$ $2" 7 Scr = Pp /σT ,

P σT σT = 1, 2 1, 5

!" #$ %&'( )crit = 163.636 × 10−4 m2 ( Sliq ( * Sliq = Scast − Scrit ( Scast ( 176.625 × 10−4 m2 ( Sliq = 12.989×10−4 m2 ( Rliq ( 2.03.10×−2 m. $ + # ( Lcrit ( * Lcrit = Rcast − Rliq = 75 − 20, 3 = 54, 7mm.

,( + # ( Lsol ( !(&, - ,. +

(

( + # / 0 ( 1 2

!"#$ %

&"' ( )* + + , - "$$# ./ 0 &1' 2 - - 2 3 % &4' 5 + , 2 , 0 . "$67 &7' (8

( 22 . 9

: 1" ( "$$; &#' 0

5 8 0

5 "$$6

1

1

2

2

! " #

$ % & '( ) *! !* $ " #

+ , , + , , ,,+ - , , ,

+ % , # ( - , ++ ,. # , , ,+ / , + ,- , # # ( 0# , +, , , , , , , # , +( -/ + ,+ , , , 1 1 ,2 -# (

⎧ y − ν∆y + (y · ∇)y + ∇p = Bu (0, T ) × Ω, ⎪ ⎪ ⎨ − y = 0 (0, T ) × Ω, ! (0, T ) × ∂Ω, y = 0 ⎪ ⎪ ⎩ y(0) = ϕ Ω. z L " Bu # $ B $ Ω ⊂ R # % % % t

2

2

Bu = K(y)

% % %

$ %

!"

# $ %

z

&

' & &

K

A

'

& '

b(y)

yt + Ay + b(y) = K(y). (

& & ) *

K z

|y(t) − z(t)|H 1 ≤ ce−κt

c

κ

+ ,," &

- & & , . / 0 ,1 ,! ,2 ,+" &

+ ,," -

& & 2" & &

& & &

& ,3 ,4" ' & .

* / - 4 )

'

c

C

' '

T > 0 Q = (0, T ) × Ω Ω ⊂ R2 V = {v ∈ H01 (Ω)2 , v = 0} H = L2 (Ω)2 {v ∈ C0∞ (Ω)2 , v = 0} & 5 H H & V ) ' $& V → H → V 6 H V - '

(ϕ, ψ)V = (ϕ , ψ )H

&

ϕ, ψ ∈ V.

Lp (Z) (1 ≤ p ≤ ∞) & & ϕ : (0, T ) → Z p ' (1 ≤ p < ∞) (0, T ) (p = ∞) 7

Z

' 5

L2 (U) U U

B : U → V

Uad ⊆ U ! " #$ W := W (V ) = {ϕ ∈ L2 (V ) : ϕt ∈ L2 (V )}

H 2,1 (Q) := {ϕ ∈ L2 (V ∩ H 2 (Ω)), ϕt ∈ L2 (H)}.

#

(u · ∇)v w dx.

b(u, v, w) := Ω

" y ∈ L2 (V ) b(y) !b(y), v V ,V := −b(y, y, v) v ∈ V % V t ∈ (0, T ) b(y) ∈ L1 (V ) &' ( )* ! y ∈ L∞ (H) b(y) L2 (V ) y ∈ W W L∞ (H) &+* u ∈ L2 (U) " #$ , y ∈ W

d (y(t), ϕ)H + ν(y(t), ϕ)V dt = !b(y) + Bu(t), ϕ

V ,V

ϕ ∈ V t ∈ [0, T ]

)

χ ∈ H. ) - . / . 0. =: ν > 0 #

/ # &'* Ì ϕ ∈ H u ∈ L2(U) ) (y(0), χ)H = (ϕ, χ)H

y ∈ W

A : V !Ay, v

V ,V

→ V

:= ν(y, v)V .

u = 0 V ! yt + Ay = b(y), y(0) = ϕ,

b(y) "# m ∈ N $ (0, T ) h = mT tk = kh! k = 0, 1, . . . , m# z ∈ W → C([0, T ], H) # % J k : V × U → R,

(y, u) →

1 z = h k

tk + h 2

tk − h 2

1 γ 2 2 |y − z k |H + |u|U , 2 2

&

z(s, ·) ds

z(t, ·) = 0 t > T # ! k = 1, . . . , m i = 1, 2 ek : V × U → V ek (y, u) = (I + hA)y − hb(y k−1 ) − y k−1 − Bu,

y k−1 # ' $ J k (y, u) ( ek (y, u) = 0 V ,

u ∈ Uad ,

Pk

y 0 = ϕ# ' ϕ V # y k−1 (y k , uk )

ek (y, u) = 0 V (y k , v)H + νh (y k , v)V = (y k−1 , v)H + !Buk + hb(y k−1 ), v )V ,V

∀ v ∈ V.

)

ϕ ∈ V ! V # ' ! uk ∈ U *# ) y k ∈ V

|y k |V ≤

C k−1 |y |H + h|y k−1 |2 V + |uk |U . νh

J k ! ek Uad + (Pk )! k = 1, . . . , m! (y∗k , uk∗ ) ∈ V × U # !

λk∗ ∈ V

A

(y∗k , uk∗ )

(I + hA)y = Bu + y k−1 + hb(y k−1 ), (I + hA)λ = −(y − z k ), (γu − B λ, v − u) ≥ 0

(y∗k , uk∗ )

!

(Pk )

v ∈ Uad ,

(y, u, λ) = (y∗k , uk∗ , λk∗ )

(Pk )

V ×U ×V

"

#

Jˆk (u) = J k (y(u), u)

Uad

u∈U

$

y(u) ∈ V Jˆk u

% "

!

∇Jˆk (u) = γu − B λ,

u

λ

y λ B := (I + hA)−1 ek (y, u) = 0 y = B(y k−1 + hb(y k−1 ) +

& '

Bu)

(

yk − z k

!

Jˆk

# ) *

'

! '

! ' +

, &

y 0 = ϕ k = 0

t0

+ -

= 0 uk0

uk+1 = RECIP E(uk0 , y k , z k , tk )

RECIP E

(I + hA)y k+1 = y k + hb(y k ) + Buk+1 .

tk+1 = tk + h k = k + 1 tk < T RECIP E ! ! ˆ k) " uk0 −∇J(u 0

# $ % &'(

% )*+,* # &'( ! # u = RECIP E(v, y k , z, t) . (I + hA)y = y k + hb(y k ) + Bv

(I + hA)λ = −(y − z)

d = γv − B λ ρ > 0

/ RECIP E = v − ρd % # U = L2 (Ω)2 B ! # 0 !Bu, v

V ,V

= (u, v),

Bu = u.

1

&'( )*+,* .

uk0 = 0 .

h

(I + hA)y k+1 = y k + hb(y k ) − ρBB(y k − z k ) − hρBB(b(y k ) − Az k ),

y˙ + Ay = b(y) −

ρ BB(y − z) − ρBB(b(y) − Az), h

y(0) = ϕ.

y 0 = ϕ, '

.2

3 % ρ K(y) = − BB(y − z) − ρBB(b(y) − Az) h

..

.2 4 4 # 4 5 6 4 6 h ρ )*+,* . .2 # ! uk0 = 0 γ ' % K y z 5 .2 # z #

zt + Az − b(z) = −ρBB(b(z) − Az),

z(0) = ϕ.

ρ K(y) = − BB(y − z) − ρBB(b(y) − b(z)) + zt + Az − b(z). h

yt + Ay − b(y) = K(y) L2 (V ) y(0) = ϕ. ! ! "#$ %&$%&' ρ

|w(t)|2H,V ≤ Ce− h t

∀t ∈ [0, T ],

C ( w := y − z & ) ! ( uk0 * $ "#$ ' +& ) ! (I + hA)wj+1 = wj + h b(y j ) − b(z j ) − ρBBwj − ρhBB b(y j ) − b(z j ) , w0 = ϕ − z(0).

+ ) ! , -. )/. u ∈ Uad

& * ( ! uk+1 ∈ Uad ( 0 ( Uad & ) 1 ! -. )/. 2 -. )/. ,( ( ! RECIP E = PU (v − ρd)& ( (( * ( ( ( (Pk )& uk (Pk ) $ && Uad = U & uk ( ! !$ ( %$ ad

(I + hA)y k+1 = y k + hb(y k ) + uk (I + hA)λk = z k − y k+1 γuk − λk = 0.

3

) ! uk = −(BB + γI)−1 B(B(y k + hb(y k )) − z k )

&

% 2 ( L(H, H)& 1 ( ( S "'& .,( Bz k = BB(z k + hAz k ) S = γ(BB + γI)−1 BB,

(I + hA)y k+1 = y k + hb(y k ) −

1 S(y k − z k + hb(y k ) − hAz k ), γ

h

y

y 0 = ϕ,

yt + Ay − b(y) = −

1 S(y − z + hb(y) − hAz), γh

y(0) = ϕ.

yt +Ay−b(y) = −

1 S(y−z +hb(y)−hb(z))+zt +Az −b(z), γh

(I + hA)y k+1 = y k + hb(y k ) −

1 h S(y k − z k ) − S(b(y k ) − b(z k )) γ γ +z k+1 − z k + hAz k − hb(z k ).

"

u = K(y) = −

y(0) = ϕ,

K

!

1 S(y − z + hb(y) − hb(z)) + zt + Az − b(z) γh

# $

%&

|w(t)|2H ≤ C e−

α(γ) h t

|w(0)|2H

∀t ∈ [0, T ],

α(γ) =

γ (1+γ)2 .

"

K

min J(v k ) =

1 2

|wk+1 | + Ω

γ k2 |v | , 2

˜k P

'

(I + hA)wk+1 = wk + hb(wk + z k ) − hb(z k ) + v k .

(Pk )#

(

w = y−z

)

# *

(h, 1) u = RECIP E(v, y k , z k , z k+1 , t)

u

(I + hA)y = y k − z k + hb(y k ) − hb(z k ) + (I + hA)z k+1 + u (I + hA)λ = z k+1 − y γu − λ = 0.

RECIP E = u

(h, l)

l ∈N

! !

lh" # h" $ %

! !

&'()"

%

!

# *

' % ' ( +" $ ' &,)" ' ( %

ρ BB uk0 = 1 − ργ k+1 1 − z k + Az k+1 − b(z k ) + ρBB(b(z k ) − Az k ) . z 1 − ργ

I−

'( &,)" % -

z

1 J(y, u) = 2 %

t

L2 *

""

γ |y(x, t)) − z(x, t)| dx + 2 Ωo

|u(x, t)|2 dx,

2

Ωo

Ωc

Ωc

% % -

Ω " / U = L2 (Ω)"

. 0 *- % B 1" / ! Bu = K(y) - % # ' .

% ' ( +"

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T = 1 T = 2 Ω = [0, 1]2

y(x, 0) = ϕ(x) = e

(cos 2πx1 − 1) sin 2πx2 −(cos 2πx2 − 1) sin 2πx1

,

e z(t, x) =

ψx2 (t, x1 , x2 ) −ψx1 (t, x1 , x2 )

,

ψ ψ(t, x1 , x2 ) = θ(t, x1 )θ(t, x2 )

θ(t, y) = (1 − y)2 (1 − cos 2πyt).

10 ν = 1/10 T = 2 ! ! h = 0.01 ! " #$%& %##' (&( ) *+, # % ! ρ = 0.1 % - |u(x, t)| ≤ 103 ) . - |y(t) − z(t)|H %

−3

2

10

10

|u(x)|<=1e−3 unconstraint

|u(x)|<=1e−3 unconstraint

0

10

−4

10 −2

10

−4

10

−5

10 −6

10

−6

−8

10

10

−10

10

−7

10

−12

10

−14

10

−8

10 −16

10

−9

0

0.2

0.4

0.6

0.8

1

1.2

1.4

|y(t) −

1.6

1.8

z(t)|2H

2

10

0

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4

|u(t)|2L2 (Ω)

1.2

−0.2

0.2

−0.2

0

0.1

0.2

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T = 2

t ≥ 1.4

!"# " $ L2 ! |y(t) − z(t)|H % γ & '

" !"# ' " ( ' 104 108 ) *

2

2

10

10 γ=0 −4 γ=10 −2 γ=10 0 γ=10

0

10

−2

γ=100 1 γ=10 2 γ=10 4 γ=10

0

10

10

−2

10

−4

10

−4

10 −6

10

−6

10 −8

10

−8

10 −10

10

−10

10

−12

10

−12

−14

10

−16

10

10

−14

10

−18

10

−16

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

|y(t) − y¯(t)|2H γ

! "# $ ! %$ & ' ! % (# )

( * *

%$ & % + ,--. ! " "/ * 0 0# ( $ 1 2 * #3345 ,---. 3 ! " % 6 * ) 0 +# &

$ 1 7 + & * #-483 ,--3. 8 ! ) & !#

9 % ' ! )

,-55. : 6 ; + ;# 9 $ ( * 7 ,--. < = > !9 ) :* * *# * # % 2 #334385 ,-5-. ? =/$ *: * # 2 @

$

1 ! * 2 * 9 #53453 ,. 5 " :!# :

!%6 2 6

7 ! %$ 6

' A22 2

6

! 38 ,--3.

!" #$##$ %

" & ' ( ) *$##$+ ,# - - . / 0 01 2 )3 4& !" 5 / 67,8679 : ; / *,5+ ,, <1 3/ =

./ ! " !

,8$> *$##$+

,$ : ( . /

. ? ? ,,$##6 %

" & ' 3 *$##6+ ,6 : ( . /

2 !

,@ A B B ( C / 2 01 1 1 2 ! D . E

,9@58,,97 *,5+

,7 A B B ( 1 2 ! D . E

>7@8>55 *,5+

,> A . - D . 2 2 01 / D F

$@78$79 *,9+

,5 . . 2 2 C / 1 ? 2 D F

,$6 8 ,7> *,+

,9 " ? 4= *,5+ , "G F &/ ! F 2 / H!

@@58@7> *$##,+

1 2 3 1

!"# $

2

!"# $

3

!"# $

%

&& %'( ) & * % ( & ( ) &

+ % (

% +

% (

! " ! # $% & ' ! # & ()* ! +

& !! , - . k / 0 10 ! 2 1, ! 3

k

k

!

"# $%% & %'(

)

* + ) , - . ' . / 0

, . &

.

1

2

2

min cT x

Ax = 1 x ∈ {0, 1}n. 3

A

2 2 i

xj

aij = 0

aij = 1 j

j cj

cj 4 - * + * + 5 6 0

3 6 0

/77 2

! "

# $ % &

$ ' #

$ $ (

$ ) %

*

+

$ % ! $

,"" $ $ $ .

# $ "

!" #"$ !! "### ""#$ # !!$ !

) $ /#

$ /# $ ' $ '

!"#

$

& '(#)

%

* ** * *

'

*

'(#) &

+*

&

, - & * ) ./0 & '(#)

&

-

min cT x

Ax = 1 x≥0

(P )

max uT 1 T

A u ≤ c.

(D)

) & & & 1& &

3

2 *

- &

- *

-

) /4 + ) 54 )&

A0

3 &

)

k := 0

k

(P ) T

min ck xk k k

A x = 1 k x ≥ 0, &

xk

uk uk (Dk )

T

rk := min cj − uk A.j j∈J

J (P ) rk < 0

Ak Ak+1 k = k + 1

(P k ) ! " # $ ! V % & ! ! fi fj

fi fj '( fj ! fi ) (fi , fj ) cij − uki " * ) % cij ) + ',

% %

' ) " * ' -

fi 0

super source

cij − uki

fj −ukj super sink

! " # ! ! uk+1

! uk $ ! % ! & ' !( $ ! c1 u1 u0 c2 c3 ! ) c1 ! * ! ! " u3 u0 u1 0 11 00 00 u 11

c2

u1

2

u 1

c

1 0 0 1

11 00 00 11 3

11 00

Optimal

u

c3

+ ! & ! ! ! " !) ' ( ! $ ! , ! -./

% ! ! ! P˜ ! D˜

01 (P˜ )

T T ˜ − δ− y − + δ+ y+ min cT x A˜ x − y− + y+ y− y+ x ˜ , y− , y+

=1 ≤ ε− ≤ ε+ ≥ 0,

˜ (D)

max u ˜ − εT− v− − εT+ v+ AT u˜ −˜ u − v− u˜ − v+ v− , v+

≤c ≤ −δ− ≤ δ+ ≥ 0.

[δ− , δ+ ] ∗

x

ε− = ε+ = 0

∗

∗

x ˜ u ˜

,u

∗

(P, D)

˜ (P˜ , D)

∗

˜ < δ+ ε− ≥ 0, ε+ ≥ 0 δ− < u

˜ D

ε−

ε+

P˜

! k

(P˜ )

T

k k y − + δ+ y+ min ck xk − δ− k k A x − y− + y+ = 1 xk , y− , y+ ≥ 0,

k

˜ ) (D

max uk 1 uk Ak ≤ ck k k δ− ≤ u k ≤ δ+ .

"

#

$ %

&'() δ+ δ−

#

* &+)

,

δ+

δ−

δ % k+1 k+1 k k δ− = δ− − ∆δ− , δ+ = δ+ + ∆δ+ .

δ+ δ− % # ∆δ

-

,

T

cj −uk A.j uk

!

" !

#

!

$% $% $%* $%' $%, $%& $%( $%+

& )& *' ) + ( &

'(() ')* *(++ ''*& '('& ',(& ''& * '

)(' )&) )&, )(' )&' )&) ) )+ )&'

(('& * ,& * ,&+ )+'( +')) ,+& *(*

-

((

' *'

),

&,

! ||u − u∗||2 u u∗ "# #

||u − u∗ ||2

Standard method

2.5e+06

2e+06

1.5e+06

1e+06

500000

0

0

20

40

60

80

100

120

140

160

180

20

(iters)

- -

%

√ ni=1 ui ≤ cj cj / n ! %& %' % %) %( %. %, %+

∆δ = 700 " " # $ &'( &)(* +*), &( &)*++.&. &+& +**' &*&),( *).&.. &),& *).& &), &(.. *)-. &. &,) *).* &( &)+ *),* &(& &))( *'

')&) ()( )(( (*.) )*-& ','+ )'*) )& *'*

/0

1 ! 0 )

2

!

0 ! / 3!

!

1 0 45

6 &----- ! .(---

3 ! ∆δ / 0

( % ∆δ 7 0 ∆δ 1 6 ! / 7

||u − u∗ ||2

Stationary BoxStep method

80000 70000 60000 50000 40000 30000 20000 10000 0

0

20

40

60

80

100

120

14

(iters)

#Var(s) 5000

FS 1 FS 2 FS 3 FS 4 FS 5 FS 6 FS 7 FS 8

4500 4000 3500 3000 2500 2000 1500 1000 500

128

256

512

∆δ

1024

2048

4096

819

#LP(s) 400

FS 1 FS 2 FS 3 FS 4 FS 5 FS 6 FS 7 FS 8

350

300

250

200

150

100

128

256

512

∆δ

1024

2048

4096

819

! " # $ ∆δ

∆δ

√ 10080/ 2 # 7128

∆δ

! " [31, 7128] [512, 1024#

$ % & ' $ ∆δ

$

Time (secs) 20000

FS 1 FS 2 FS 3 FS 4 FS 5 FS 6 FS 7 FS 8

18000 16000 14000 12000 10000 8000 6000 4000 2000 0

128

256

512

∆δ

1024

2048

4096

819

∆δ ( $

) * + , - " ) " - $

.# / 0 1 2 3 / 4 ( " / 1 4 /05+4

66

! ! ""#$!% $ & $

'

()%*+),*- +--.

* /$ 0 / 1 $

. ! # 2$

$ 3 $ ! $ # *4,+*-

"

+--* ( 5 /

$ /$

$ 4 $$8

$$6

+-(%-,*7 +---

$ 9 ! $ :

$ $ '

# (.+,(-. +--4

) 2 &

/ $ ;$$ : & 2! : +---

7 ' '! $ <$ 1 $ /$

= /:$! "

$ & 9$ 2$ $9 $

*+ +--7

. 1 2

= 29" 3! "

$ =$ & =!:!$! # 2! $! #

$ 3=/ +--- - >:

2

$ & $ ?

! "

.%7@*,7+ +-)@

+@ / > A > ! 9 =$$ 9 $$ $ $ 2! $! # 2 $ $! ' $ $ $ $ & 2! " : +--- ++ 3 8 $# & "! 5# $$6 $ (@4 +-74 +

2 ? #

*B*C%*.-,

"

/

$ 2 ;$$ : & +--- +*

! = = D8

= >$ / $ 3 3 = $ $! ! ""$! =##! & =$$ 9 $$ +--7 +( 3 $

()

#$ % & ' #

E' +--7

+4 2 0 =

(

6

* (

+74* &

! # = :$ =##! & $ ! 9 ! "

$ # -*,*@7 #$ @@@

1 2 1 1 1 1

2

! " # $ % & ' '( 104 4.105 " # #

)*%*+ ,nm3 )-%./ ,nm3 % "

# /%///.'/%//0* '( # %

! " # $ % ! & "

' # % ( " ! ' ! ) * " ' # ' " % " $ ! # "% + & +

,- ! ) ' , + & ' .!/ %! 0 '

! " # " $ # % # # & # "' "( ) * "+ ", -# # # . / ! 0 # 1 104 2.104 4.104 4.105

1 2.104 4.105 * ! 1 2 ,++ 3nm3 4 56 7( 8) 9 * 2:914 , ; ϕ(r) = −0.18892(r − 1.82709)4 + 1.70192(r − 2.50849)2 − 0.79829

2"4

< ϕ(r) A0 8 # = >((A0 < # 6 ? 9 ,++ 3nm3 = 6 9 # ; 0 # ''"A0

# ? 6 * ''+A0 ''"A0 5 2''+A0 4 # &

& *

''"A0 2.104 ,>$ 3nm3 # "

ith ith

A0 ith

ith A0 ! ith " ### A0 $ Nv

ith (2000 − Nv )4πR3 Vf = %& 3×2000 '

Nv &(###

Vf

%###'

!"# $ !%# & ' ()% & 1st

! " ! # $ & 0

1 ! # $ % A 1st ! # $ % 2st ! # $ % A0 2st ! # $ % st

A0

%

& & & &

'(!)*+

, ./ 1/ ,1

, ./ 1/ ,,

,.0.1 1 ,.,

, .0 1 1 ,0

,.,1 1 ,-1

* + , + , - ' '

- * (- A0 (. A0 !# (%/ A0 ' ' & 0 '1

!" # $

∼ %% # & ' # ( (

$ )

# * +

,

- - - -

. / 5 6 A0 7 8nm3 , * + A0 * + , % * + A0 % * + 7 ≥ A0 , *)+ *)+ # , *)+ *)+ , *Vc + *Vc + , *Vf + *Vf +

01!% 02 0202 034 33 33 3% 31"1 2.104

2.104

2.104

4.105

243 0% 003 4% 032 %1 % """ % 42 ! !00 !% ! 4 0% %1 202

243 0% 0!04 4% 02 " % "31 % 3%!" ! !%1! !%%24 4 0%4 %1 "3

20" 0! 00! 4% !!4 !" % % ! % %%" ! !%01 ! 0 4 0 " %2 201

243 0% 0% 4% 013 14 % "%4 % 02! ! ! 20 !"% 4 %"3% %1 10!

! " A0

Vf

A0 !"#$ !"#% !"&'$ !"#% ( ) *

"#+"$ ( ( (

A0 (

Vf ! " Vf " #$ $ " %$ &$ ' ($ ! ") ! " Vf * +$ $ * " %$ &$ ' ($ ! ") ,!")$ " ! " Vf

- -$ ) , ")$ "" ! " Vf

" - . -

" ) - . - - - Vc "" Vc =

Vt Va

/ Vt " !" "" !

$ Va !" ∼ π×1.263 A0 % " " !)

" A0 ! ! " !)

$ ! ") !)

fc (z) fa (z)$ fc (z) * ! 0 "1 2 "" " " 3 4 104 4.104 $ ! ") 5 ) " ! " " % ( " " 0 " % ($ 3 "")$ ! )"6 "

7

8 " 3 8 " % 8 " 4 8 " (

3 ) $ 9 : " ( )$ ;nm3 7 6 $A0 < 6 : ! ≥ A0 ! ! '

** 104 *# # * *

2.104 *# # *

#* 4.104 *# + #

# 4.105 *# * +

1

! A "#$ %

" A & "'$!

( ( $)! *nm

+"+,,

++,, & ( - st

0

0

3

! " # !

4.10 . &

(/ """"0/""" ) ( (/

1 1 1 1 (

104

5

! " # $

! %

&' ((( )*

+ ,& - . + . /% ( 0 - ' ' 0

- + 1%" 2 # 3

((( 3 ) 0 - ' ' ' 4 0

- + ""5 # %"

"% 6 3 )) 3 0 - ' ' ' 4 + " # %" "% (( 6 ) 0

- -

.76 8#

9

" 3 )) * &" 6 # &" 6# ' 3( 3*) (3 0 - ''-

00-

048 + ""5 # %

" "%' * (( ( 9 1& 4 + " 1 &

"% 1 && " *

((( ) 8 % : " .::: 1 &

"" 2 #

: - % - ) * 3* '" 9 " 1 &

"" "% 2

*( 8 9 '" 1 &

"% 1 && "

( 9 8 ;% < , '" # , < 2 = 2

% &

"% ) >

) '" % : " % -% ! 4 %" 2

& " * ((( )

3 '" 9 8 1 &

" - 2 # : -

% - 3 )3 '" 8 9 :55?@ . "% 2 # : - % - 3 )3 * '" 9 " 1

%% " @ 2 # 2"" # "%

2 3 *) A, %" " ! " # $

! %

&' ((( *)(*3(

( . < 4 2""2 8 (

! " # $ % & ! " ' ( % ) " * + % * %

! "#$%& ' (

! ' ) ( * ( ( "#$+ "#$,& -. '

/ 0 1 ( 2

"#3#&

' ( 2

"#$4& "#$3& - ' . (

-

ut + u.ux + v.uy + ϕx − f.v + T x = 0 vt + u.vx + v.vy + ϕy + f.u + T y = 0 ϕt + (ϕu)x + (ϕu)y − r = 0

!"#

$ u v x y f % r ϕ = gh h g

T x = (τbx − τsx )/ρh , T y = (τby − τsy )/ρh

c=

1 1/6 nR

, τby = ρgv

!#

u2 + v 2 /c2 , τbx = ρgu u2 + v 2 /c2

τsx = ρa Cz Wz2 cosϕx , τsy = ρa Cz Wz2 cosϕy , τbx τby x y & n '

R τsx τsy

x y ρ

ρa

Cz ( Wz ϕx ϕy x, y

) *+ !"#

m

!f (x, y), Vi =

f (x, y) · Vi dxdy

elements

f (x, y) · Vi dxdy , i = i, j, k

=

global

ue =

uj (t) · Vj (x, y)

j=i,j,k

uj

Vj

u

Vi =

A

1 (ai y + bi x + ci ) 2A

bi = yj − yk , ai = xk − xj , ci = xj · yk − xk · yj (6) i, j, k

! " # $ %

&i

bi ∂Vi ai ∂Vi = ; = , (i = 1, 2, 3) ∂x 2A ∂y 2A

'

( ) "*+ , , - . " -

+., [M ] ϕ + [K1 ] {ϕ} = [M ] {r}

/

[M ] =

m 1

[M e ] ; [K1 ] =

m 1

[K1e ] ;

m + . , m +., ϕ = ϕe ; {ϕ} = {ϕe } 1

1

0

M e = ⎡ ⎤ Vi Vi dxdy Vi Vj dxdy Vi Vk dxdy ⎡ ⎤ A A A 211 ⎢ ⎥ A ⎢ ⎥ e Vj Vi dxdy Vj Vj dxdy Vj Vk dxdy ⎥ = ⎣1 2 1⎦ [M ] = ⎢ A A A ⎣ ⎦ 12 1 1 2 Vk Vi dxdy Vk Vj dxdy Vk Vk dxdy A

A

A

[K1e ] = ⎛ ⎞ ∂Vj ⎝Vi · ϕj ⎠dxdy [K1e ] = (Vj · u∗j ) · ∂x j=i,j,k j=i,j,k A ⎞ ⎛ ∂Vj ⎝Vi · ϕj ⎠dxdy Vj · vj∗ · + ∂y [K1e ]

=

A e [K11 ]

e [K11 ]=

1 24

e ]= [K12

1 24

j=i,j,k

e + [K12 ] ⎡ (2u∗i + u∗j + u∗k )bi ⎣ (u∗i + 2u∗j + u∗k )bi (u∗i + u∗j + 2u∗k )bi ⎡ (2vi∗ + vj∗ + vk∗ )ai ⎣ (vi∗ + 2vj∗ + vk∗ )ai (vi∗ + vj∗ + 2vk∗ )ai

j=i,j,k

⎤ (2u∗i + u∗j + u∗k )bj (2u∗i + u∗j + u∗k )bk (u∗i + 2u∗j + u∗k )bj (u∗i + 2u∗j + u∗k )bk ⎦ (u∗i + u∗j + 2u∗k )bj (u∗i + u∗j + 2u∗k )bk ⎤ (2vi∗ + vj∗ + vk∗ )aj (2vi∗ + vj∗ + vk∗ )ak (vi∗ + 2vj∗ + vk∗ )aj (vi∗ + 2vj∗ + vk∗ )ak ⎦ (vi∗ + vj∗ + 2vk∗ )aj (vi∗ + vj∗ + 2vk∗ )ak

u∗ v ∗

u∗ = un+1/2 = 3un /2 − un−1 /2 v ∗ = v n+1/2 = 3v n /2 − v n−1 /2

! n+1

n

n+1

n

− {ϕ} + {ϕ} {ϕ} n+1/2 + [K1 ] = [M ] {r } ∆t 2 # n n + 1 tn+1 = tn + ∆t, tn = n∆t [M ]

{ϕ}

" $ " $ 2 [M ] {ϕ}n+1 − {ϕ}n = 2 [M ]{r}n+1/2 − [K1 ] {ϕ}n + [K1 ] ∆t

"

$

% & ' ( ) *+ ∂v u ∂x . L(u, v) = u∂v/∂x

, ( ) *+ ! & ' & ' -

ξ = kh η = lh h

u(∂v/∂x) u = exp(ikx) , v = exp(ilx)

|T.E.|

∼

[4η 4 +8η 2 +7η 2 ξ 2 −2ηξ3 ] 720

ξ = η || ∼ 17η 720 4

!" # Z :=

∂v/∂x)

⎫ ⎧ ⎤⎧ Z1 ⎪ v2 − v1 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢1 4 1 ⎥⎪ ⎪ Z v ⎪ ⎪ ⎪ 2 ⎪ 3 − v1 ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ... ... ⎥⎪ ⎪ ... ... ⎨ ⎬ ⎨ ⎥ 1 h⎢ ⎢ 1 4 1 ⎥ Z v = j j+1 − vj−1 ⎥⎪ ⎪ 6⎢ 2⎪ ⎢ ⎥⎪ ⎪ ⎪ ... ... ... ... ⎪ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 1 4 2⎦⎪ Z v ⎪ ⎪ ⎪ m ⎪ m+1 − vm−1 ⎪ ⎪ ⎩ ⎭ ⎩ 21 Zm+1 vm+1 − vm ⎡

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

$

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

% & '!" w = u∂v/∂x ⎧ 3u1 Z1 + u1 Z2 + u2 Z2 + u2 Z2 ⎪ ⎪ ⎪ ⎪ u 1 Z1 + u 1 Z2 + u 2 Z1 + u 2 Z3 ⎪ ⎪ ⎪ ⎪ +u3 Z2 + u3 Z3 + 6u2 Z2 ⎪ ⎧ ⎪ ⎫ ⎪ ⎪ ⎤ ⎪ w1 ⎪ ⎡ M ⎪ ⎪ ⎪ ⎪ 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w u Z + u ⎪ ⎪ ⎪ 2 j−1 j−1 j Zj−1 + uj−1 Zj ⎪ ⎥⎪ ⎪ ⎢1 4 1 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎢ +u Z + uj Zj+1 ⎪ ⎪ ⎪ j+1 j ⎪ ⎥⎪ ⎪ ⎢ ... ... ⎨ ⎥⎨ ⎬ ⎢ 1 +u Z + 6uj Zj j+1 j+1 ⎥ ⎢ 1 4 1 ⎥ ⎪ wj ⎪ = 12 ⎪ ⎢ M ⎪ ⎥⎪ ⎢ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ um−1 Zm−1 + um Zm−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 1 4 1⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +um−1 Zm + um+1 Zm ⎪ ⎪ ⎪ w m ⎪ ⎪ 12 ⎪ ⎩ ⎪ ⎭ ⎪ +u ⎪ wm+1 m Zm+1 + um+1 Zm+1 ⎪ ⎪ ⎪ +6um Zm ⎪ ⎪ ⎪ ⎪ u Z ⎪ m m + um Zm+1 ⎪ ⎩ +um+1 Zm + 3um+1 Zm+1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(

|T.E.|

∼

[2ξ 3 η+3η 2 ξ 2 +2η 3 ξ−4η 4 ] 720

)*

3 4 η

ξ = η | | ∼ 720

u v !

"

∂u/∂x = Zxu , ∂v/∂x = Zxv

## # ∂u/∂x ∂v/∂x # Zyu Zyv y

# $ $ u 1

{u}n+1 − {u}n + {(u.Zxu )} + {(v.Zyu )} − {f.v} + {T x} [M ] . ∆t

2

+ [K2 ] . {ϕ} = 0

""

$ % v 1

[M ] .

{v}

n+1

n

− {v} + {(u.Zxv )} + {(v.Zyv )} + {f.u} + {T y } ∆t

2

+ [K3 ] . {ϕ} = 0

$ [K2 ] =

m

[K2e ] ; [K3 ] =

m

1

"&

"'

[K3e ]

1

$ ⎡

bi

Vi dxdy bj

Vi dxdy bk

Vi dxdy

⎤

⎡ ⎤ A A A bi bj bk ⎢ ⎥ 1 ⎢ bi 1 Vj dxdy bj Vj dxdy bk Vj dxdy ⎥ [K2e ] = ⎢ ⎥ = ⎣ bi bj bk ⎦ A A A 2A ⎣ 6 ⎦ bi bj bk Vk dxdy bj Vk dxdy bk Vk dxdy bi A

⎡

A

A

"( ⎤

⎡ ⎤ A A A ai aj ak ⎢ ⎥ 1 ⎢ ai 1 Vj dxdy aj Vj dxdy ak Vj dxdy ⎥ [K3e ] = ⎢ ⎥ = ⎣ ai aj ak ⎦ A A A 2A ⎣ 6 ⎦ ai aj ak Vk dxdy aj Vk dxdy ak Vk dxdy ai ai

A

Vi dxdy aj

A

Vi dxdy ak

A

Vi dxdy

")

∆x = 100 m , ∆y = 50 m , ∆t = 60 sec ,

f = 2w sin ϕ

w

= 7.29x10−5 sec−1 ϕ = 150 τsx = τsy = 0 n = 0.035 g = 10 ms−2 r = 0

u = u(0, y, t) = 0.500 − 1.000 m/s v = v(0, y, t) = 0.0 m/s, h = 4.00 + 0.5 sin(πt/21600) m, v = v(x, 100, t) = 0.0 m/s, v = v(x, 300, t) = 0.0 m/s

! " # " # $

(k) − hi |) = 0.001 m % εh = max(|h(k+1) i & εv = max(|vi(k+1) − vi(k) |) = 0.003 m

n∆t = 4320 ! "h#

∆

u !! !" # $

M

% & $ ' $&

() * & +& ,& -& -& . /% 0 %

1 +& *& , % & 2&"3 & !34'!5& (!)

& & ! / '

$ 0 ' - % 6& & 7 8 % 1 9& 6% , 6% & 4'5& (")

& & " / '

0 0 1 +& *& , % & 2&4! &! & ""'""& (3)

& & / % 7

0 1 * : 2&" & " & !44'!4&

!"# $%%& ' ( ) * + , -. ! 01) 2

$ (

,

/$./# $%% , $%!% 3 4

5 .1 5 6 4 ( (( 7 ,(

3((

-

$!8$# (

'!9.'##

# 01) 2 , * 1) $%#/ + : ((: 7 3 :. ( ( ; -

( 7 +

- <( ,8<8/#%8#/ = > ?& (

! "# $% & ' ! #% %( '! ' ( $ % '! ' ! $ ) ' '! ( * $

+ ! (

B div B = 0 B ! " #$ % & '

!

#! #$

! ()* & ! & #$ ! (+,* (+-* ! ! ! . / 0+12 ! 3

! " ! # $ % ! $ ! $ & '() $ * +

, - ./ 0 '() - # - 1

! $ + $ - 2 !

! / - 3

, ! ρ/ ρv/ E B + , ∂t ρ ∂t ρv ∂t E ∂t B

+ div ρv 2 I − BB + div ρvv + p + 12 B + div E + p + 12 B2 v − BB · v + div (B v − v B)

=0 =0 =0 =0

45

/ /

, , E=

1 1 p + ρv2 + 12 B2 γ−1 2

4.5

γ 6 45 4 7 5 '()

4 curl ∂t B+ curl (B × v) = 0 ⇒ div B = const

! " ## ## $%& # " ## ' () ( $*&

' # # #

# ˜ Ω div grad ψ = div B + ∂Ω ψ=0 ' ψ # ' B˜

˜ i,j − grad ψ| ' Bi,j = B i,j , # # - # # . # #

# ## # / B 0 1 , $2& # ( # $& 3 ! $& 4 # #

' # b(x) b(y)

i+ ,j i,j+ #

1 2

div

(0)

b

(x)

:= i,j

(x)

bi+ 1 ,j − bi− 1 ,j 2

2

∆x

(y)

+

1 2

(y)

bi,j+ 1 − bi,j− 1 2

2

∆y

%

1 # # #

!" #

$

div B = 0

%&!

' ! ( %) $ %&!

ψ

%&!

∂t B+ curl (B × v) + ∇ψ = 0 D(ψ) + ∇ · B = 0

D

!+

*

D

$, + &,

$ # %&! - .

/

& / ,

,

,

, )

0 , / 1 0 ' #

,

) %&! 2

, ' /

B (x)

B (y)

/

&,

B (z)

IRD D

∂t u + F (u) = 0

F

u∈Ω⊆

Ω

!

C

" #

C (F (u)) ≡ 0

$

# %

C (u) = const

&

% # % # '() ( # % % (# "%

u=B ˆ #

C (u) = ˆ div B

! "%

Ω "% (xi , yj ) * ∆x × ∆y % h = max (∆x, ∆y) + ˜ n : T → IRD * "% ∆t

B

, B n "%

-

B n "% Bni,j .

% 4 !

T

"%

K = (i, j)

" # /

! %

01, "0 1, "

V = g : T → IRD ΦK : V → V (i, j) ˜ B

˜ : T → IRD . Φi,j (B) " $

Φi,j B˜ (i, j)

k,l

2

B˜ (k, l)

B˜ B˜ ˜ Φi,j (B) B˜

n Bn+1 i,j = Bi,j +

" $

˜ n

Φk,l B

(k,l) (i,j)

. i,j

!""#

B˜ (i, j) $ !""# % > i,j · B ˜ + O (hm ) div B|i,j = div !"&# > i,j

B˜ ' div ( !""# ) *"+,

ˆ (g) Φ i,j

> k,l · Φ ˆ (g) = 0 div i,j

∀ (i, j), (k, l)

!"+#

"" !"+#

{Φˆ (g) i,j } g = 1, 2, ... (g) " $ (g) ˜ Φ ˆ B Φi,j (˜ u) = ϕ !"-# i,j g i,j

. ) ϕ(g) i,j $ ϕ(g) i,j /

Φi,j (g)

ϕi,j > k,l div !

"

# $ ! %& (x)

(0) ˜ divi,j B :=

(x)

(y)

(y)

Bi,j+1 − Bi,j−1 Bi+1,j − Bi−1,j + . 2∆x 2∆y

'

# ( div() (x)

() ˜ := divi,j B

(x)

(y)

(y)

{Bi,j+1 }x − {Bi,j−1 }x {Bi+1,j }y − {Bi−1,j }y + 2∆x 2∆y

(

{ψi,j }y = {ψi,j }x =

1 4 1 4

)

*

(ψi,j+1 + 2ψi,j + ψi,j−1 ) (ψi+1,j + 2ψi,j + ψi−1,j )

+

! ,* () ˜ divi,j B

(i, j) × -

.

( O(h2 ) #

* / . - - O(1) ( $

0 O(1) * . O(1)

K = (i, j) div() B B

! div() " #! (i, j)

(i, j) $"% & ! # !

ˆ (1)

ˆ (1)

Φ = (−∆x, ∆y), Φ = (−∆x, −∆y), i,j i,j

i+1,j+1

i,j+1 ˆ (1)

= (∆x, −∆y), ˆ (1)

Φ Φ = (∆x, ∆y) i,j i,j i,j

'

i+1,j

( ( !

!

( ! ) ! K ! ! *

ˆ (g) ' ! Φ i,j + div() !

,- # $.% /01

n Bn+1 i,j = Bi,j +

∆t ∆t 1 − G (F 1 − Fi+ 12 ,j ) + (G i,j+ 12 ) ∆x i− 2 ,j ∆y i,j− 2

F G !!"# # $#

% & ' # ( )# ) ) * ) Φ( ) ) & + , - F + + G . f # x+ y + , # Fi+ 12 ,j

1 = −fi+ 12 ,j 0

Gi,j+ 12 = fi,j+ 12

0 . 1

/0

) )+

( )

Φi+ 1 ,j

2 i,j

( )

Φi,j+ 1

2

i,j

= fi+ 12 ,j = fi,j+ 12

0 ∆y

−∆x 0

( )

Φi+ 1 ,j

2 i+1,j

( )

Φi,j+ 1

2

= fi+ 12 ,j

i,j+1

= fi,j+ 12

0 −∆y ∆x 0

/ //

, n Bn+1 i,j = Bi,j +

$

∆t " ( ) ( ) ( ) ( )

Φi+ 1 ,j + Φi− 1 ,j + Φi,j+ 1 + Φi,j− 1

2 2 2 2 ∆x∆y i,j

/-

( ) * . / 1 div(0) div() # # * (g) Φi,j

2 " $ () (1) (2) Φi,j+ 1 = − 18 fi,j+ 12 Φi,j + Φi,j 2

/3

! " # ( ) Φi+ 1 ,j 2

Bn+1 i,j $

∆t " ( ) ( ) ( ) ( )

n Φi+ 1 ,j + Φi− 1 ,j + Φi,j+ 1 + Φi,j− 1

Bn+1 i,j = Bi,j + 2 2 2 2 ∆x∆y i,j $

∆t " ( )

( ) ( ) ( ) Φi+ 1 ,j+1 + Φi− 1 ,j+1 + Φi+1,j+ 1 + Φi+1,j− 1

+ ! 2 2 2 2 ∆x∆y i,j $

∆t " ( )

( ) ( ) ( ) Φi+ 1 ,j−1 + Φi− 1 ,j−1 + Φi−1,j+ 1 + Φi−1,j− 1 . + 2 2 2 2 ∆x∆y i,j " # $ % &

! & 1 1 1 n+1 n ∆y (!fi,j− 2 − !fi,j+ 2 ) Bi,j = Bi,j + ∆t '! 1 1 1 ∆x (!fi+ 2 ,j − !fi− 2 ,j ) ()*+ , fi+ 12 ,j

1 1 ∆y (!fi,j− 2 1 1 (!f i+ 2 ,j ∆x

− !fi,j+ 12 ) −∂y f + O(h2 ) = ∂x f i,j − !fi− 12 ,j )

f

(v × B)(z)

! "

# $ %

div B = 0

& '(

" )%

*+

$

% #

% %

,

-

B

"

v = v0

B"

sin ϕ cos ϕ

.

ϕ %

*+

B (x, t = 0) =

B0 B1

x < −y tan ϕ x > −y tan ϕ

/

B0,1

!

( ϕ, sin ϕ)T

! +

T

R(ϕ)(1, 2)

B0 = R(ϕ)(1, 1)T

B1 =

+

R (ϕ) =

"

cos ϕ − sin ϕ sin ϕ cos ϕ

0

$ % #

(

B2 div() B tan ϕ = 1 div() B ±∞ 2

ϕ = 0 ϕ = π4

() div

1 2

ϕ

tan ϕ

=

! " 1 B2 # tan ϕ = 2 $

B % #

() div ! " &

! ' '('( [−1, 1]2 )( * 2 B B

+&

tan ϕ = 12

! " # $ %& ' " $ ( "

!" # ψ B˜ Bi,j = B˜ i,j − grad ψ|i,j $ div grad % # &'"

# ( )

) div() ψ

) div()

( ( % !" %

% %

&* #

+ %

, (

%%

- # ." %

% % (x) (y) bi+ ,j bi,j+ 1 2

1 2

div(0) b

=0 i,j

∀ i, j

Bi,j

⎞ ⎛ (x) (x) 1 ⎝ bi+ 12 ,j + bi− 12 ,j ⎠ = 2 b(y) 1 + b(y) 1 i,j+ i,j− 2

∀ i, j

⇒

(x)

div() B = 0

2

2

(y)

bi,j+ 1 2 Bi,j

bi+ 1 ,j

!

B (x, y) =

ϕ(x, y)

∂y ϕ (x, y) −∂x ϕ (x, y)

"

#

div B = 0

!

(x)

bi+ 1 ,j =

ϕi+ 12 ,j+ 12 − ϕi+ 12 ,j− 12

2

∆y

,

(y)

bi,j+ 1 =

ϕi+ 12 ,j+ 12 − ϕi− 12 ,j+ 12

Bi,j

∆x

2

$

ϕ(x, y)

ϕ

# $ % "&

ϕ (x, y) = B (x) y − B (y) x

'

B ϕ(x, y) B

(

) $ '

# *

+ , # $ - . / Ms

x = −0.6 # = 8

8 x = −0.6 ! r0 = 0.4 "

! r0 " # $

B

eϕ = (−y/r, x/r)T

B

ϕ

p

!

1 p = p − B2 2

*

"

%&'(

B

r = Bmax eϕ r0

%&)(

+

B

"

ϕ (x, y) =

Bmax = 1.3

Bmax 0

r02 −(x2 +y 2 ) 2r0

%&'( !

r0 = 0.4

x2 + y 2 < r02

%&,(

t = 0.3

! "# $ ! % & '() )* t = 0.5

y t = 0.5

x = 4.4

[−0.8, 4.2] × [0, 2]

1000 × 2500 0.9

! " #$$% &

" '()* '(+* ,%-. '/* # 1

0

" "

" 234 + 3 ) 56#

371

1

588

9 4 + :

t = 0.3

0

:

B

!

:

4

2

"

" " 4

B ! t = 0.5 B

t = 0.5 ! " # $ $ 3.5 " 6.5 % $ # & '( )*

"

# $ " %& " ' # $

( %&

" %& & B

div()

'

B

)*+, )**,

div() B = 0

!"#$ !"%$

!"$ &' ( (' (' ' ) * + ½ ' ,"---. /01 !/$ &' )2 &' *' ∇ · B

' ) * + ' ,"-31. %/4 !#$ '

' +5' ' 6 ) ' ,"--3. #"0 !%$ ' 6' 7 ' 8' 79' ' ' *' ( ' ' ' ' ! " #! $ ' ) * + ,/.' ,/11/. 4%: !:$ (' ' # ! % " % &

' # ' 6;66 + /11"/4/#' ,/11". !4$ <' *5 ' )8' & # ( ) " # ' 6 ) ' ,"-33. 4:!0$ = ' < 5' +6' * & " +' (' > ? ,"--4. !3$ )@' 6 ' ' * ' , ' 6 +' > ? ,"-4%. !-$ )' =( ( ' * ' $- . $* ' )* + ' ,"--4. /1/ !"1$ ' *' A ' +' (' 5' ( B' <' C ' 2' ! " #

& / # ' )* + ,/.' ,/111.' %3%

! "#$ % ! & &'(

" )*$

½ + ,

- )

. %

! /* .

$(#"! . ("

%

00-

! " # ∇ · B " $ %! " " # &

)

.

$(#"! . 1 " 00-

2 )

" )*$

000

3 . 4$ #5

%$ 4$ (" .%)

$$ 00

!"#$% &

'

(!

) * + !

) *

' ! !

' ' (

!

" # $

%

& ! # '

!

+ ), ) & - . & 0 1 - .202/*

&/

½

! "

# $ %&&'( ) %&' %**' + ,

! , - %&.'( / - 0 %*1' )2( / ) %&3' /( + 4( +( + 5 6 %*.'( 7 ( ! 8 %*9' : ( -%*&' 6 ,

( + 5 0 %&;' 6 , " ( < + ( 5

( 5 / %;' < = (

(

x

Ω τ ∈ Ω τ = t z (t, z) x : Ω → Ênx : τ → x(τ ) F [x, p, q] = 0

! "! # !

• •

p ∈ Ênp q : Ω → Ênq $

% & ' D[x, p, q] = 0 (

D[x, p, q] ∈ ÊnD ' p q x )( $

" $ ) ' $

N1 $ q j j = 1, . . . , N1 & $) * $ j ∈ {1, . . . , N1 } $) ηij τij ∈ Ω

σij i = 1, . . . , M j

F j [xj , p, q j ] = 0,

Dj [xj , p, q j ] = 0

+

xj : Ω j → Ên qj : Ω j → Ên j = 1, . . . , N1

p j x

j q

, $ + , $ j ∈ {1, . . . , N1 } xj * xji := x(τij ) τij i = 1, . . . , M j

xji hji (xji , p, qj )

ηij = hji (xji , p, q j ) + ji ,

i = 1, . . . , M j

ji 2

ji ∼ N (0, σij ),

i = 1, . . . , M j .

(ηij − hji (xji , p, qj ))/σij 2

j

min x,p

N1 M (ηij − hji (xji , p, q j ))2 j=1 i=1

F j [xj , p, q j ] = 0, j

j

j

D [x , p, q ] = 0,

σij

!"#

2

!$# !%#

j = 1, . . . , N1 j = 1, . . . , N1

& x N1 x = xj , j = 1, . . . , N1 .

!"#!%# j F j [xj , p, qj ] = 0 '

sj ∈ Ên

ξ j xj F j [xj , p, q j ] = 0 xj = ξ j (p, q j , sj ) xj = xj (p, q j , sj ) xj (τ ) = xj (τ, p, q j , sj ) ( !"#!%# sj

j

M (ηij − hji (xj (τij , p, q j , sj ), p, q j ))2

σij

i=1

2

Dj [xj (p, q j , sj ), p, q j ] = 0 ˆ j [xj (p, q j , sj ), p, q j ] = 0 D

v = s1 , . . . , sN1 , p ∈ n r1j (v)22 r2j (v) = 0 r1 : n → n1 r2 : n → n2 n2 < n ≤ n1 + n2

Ê

Ê

Ê

Ê

Ê

min v

N1

r1j (v)22

j=1

r2j (v)

= 0,

j = 1, . . . , N1

" " $ $ r1 = r11 , . . . , r1N1 r2 = r21 , . . . , r2N1 ! r1 J1 J= = J2 v r2

J1 J2

v " #

J1 J2 $ ⎛ r1 ⎛ r1 ⎞ ⎞ r11 r21 1 2 0 0 s1 0 · · · p s1 0 · · · p ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ J1 = ⎜ ⎠ , J2 = ⎝ ⎠ ⎝ N1 N1 N1 N1 r r r r 0 · · · 0 s1N1 1p 0 · · · 0 s2N1 2p J2 % n2 &' J % n () % *% +,- . $/ J1T J1 J2T −1 J1T 0 + J = I0 . J2 0 0 I

/ vˆ . $/ *

vˆ ∼ N (v∗ , C) v∗

C

C= I0

J1 T J1 J2 T J2 0

−1

J1 T J1 0 0 0

J1 T J1 J2 T J2 0

−T I , 0

vˆ

(100 · α)%

α ∈ [0; 1]

v∗

G(α, v ∗ ) := {v ∈

Ên :

F2 (v) = 0, F1 (v)22 − F1 (v ∗ )22 ≤ γ 2 (α)}

γ 2 (α) := χ2n−n2 (1 − α) χ2 α n − n2 v ∗ v ˆ ! G(α, vˆ)

GL (α, vˆ) := {v ∈

Ên :

F2 (ˆ v ) + J2 (ˆ v )(v − vˆ) = 0,

F1 (ˆ v ) + J1 (ˆ v )(v − vˆ)22 − F1 (ˆ v )22 ≤ γ 2 (α)}. "

GL (α, vˆ) = {v ∈

Ê

n

: v − vˆ = −J (ˆ v) +

δω 0

, δω ∈

Ên , δω2 ≤ γ(α)}. 1

N2 ! j = N1 + 1, . . . N1 + N2 " • •

q j τij i = 1, . . . , M j wij ∈ {0; 1} i = 1, . . . , M j #

$ " ! 2

ji ∼ N (0, σij /wij ),

i = 1, . . . , M j

a≤

wij ≤ b

i∈Ikj

cji wij ≤ cmax

i∈Jkj

Ikj , Jkj ⊆ 1, . . . , M j k = 1, . . . , K j wij ∈ {0; 1}

wij ∈ [0; 1] i = 1, . . . , M j

0 < wij < 1 wij

σij 2 /wij

cji wij

N1 + N2 • N1 η j j = 1, . . . , N1

• N2 η j j = N1 + 1, . . . , N1 + N2 ! "

" # ⎛

min x,p

N1 M (ηij − hji (xji , p, q j ))2 ⎝ j

j=1 i=1

F j [xj , p, q j ] = 0, j

2 σij

j

j

D [x , p, q ] = 0, F j [xj , p, q j ] = 0, Dj [xj , p, q j ] = 0,

+

N Mj 1 +N2

wij ·

⎞ (ηij − hji (xji , p, q j ))2 ⎠

j=N1 +1 i=1

σij

2

j = 1, . . . , N1 j = 1, . . . , N1 j = N1 + 1, . . . , N1 + N2 j = N1 + 1, . . . , N1 + N2 x = xj , j = 1, . . . , N1 + N2

min v

N1

r1j (v)22 +

j=1

N 1 +N2

r1j (v)22

j=N1 +1

r2j (v) = 0,

j = 1, . . . , N1

r2j (v)

j = N1 + 1, . . . , N1 + N2

= 0,

$%& $'& $&

v = (s1 , . . . , sN +N , p) J1 J2

J1 J2 J1 J2

1

2

!

" # •

j j ψL ≤ ψ j (xj (t), p, q j (t)) ≤ ψU ,

•

•

ϑjL ≤ ϑj (q j ) ≤ ϑjU ,

a≤

wij ≤ b,

i∈Ikj

i∈Jkj

Ikj , Jkj ⊆ 1, . . . , M j ,

•

cji wij ≤ cmax , k = 1, . . . , K j ,

wij ∈ {0; 1},

i = 1, . . . , M j ,

wij ∈ [0; 1],

i = 1, . . . , M j ,

j = N1 + 1, . . . , N1 + N2

$ j = N1 + 1, . . . , N1 + N2 qj wj % & '(

C ⎧1 · (C) ⎪ ⎪ ⎨ n T det(K CK) ϕ(C) = ⎪ max{λ : λ

C} ⎪ √ ⎩ max{ Cii , i = 1, . . . , n} K K T CK ξ = (q j , wj , j = N1 + 1, . . . , N1 + N2 ) ! x = xj , j = 1, . . . , N1 + N2 " min ϕ(C) #$$% ξ,x T −T J1 T J1 J2 T −1 J1 T J1 0 I J1 J1 J2 T C= I0 #$&% 0 0 0 J2 0 J2 0 #'%#$(% j = 1, . . . , N1 + N2 F j [xj , p, q j ] = 0 #$)% j j j D [x , p, q ] = 0 #$*% j = N1 + 1, . . . , N1 + N2 j j ψL ≤ ψ j (xj (t), p, q j (t)) ≤ ψU #$+% j j j j ϑL ≤ ϑ (q ) ≤ ϑU #$,% 1 n

a≤

wij ≤ b,

i∈Ikj

Ikj , Jkj ⊆ 1, . . . , M j , wij

∈ {0; 1},

cji wij ≤ cmax ,

i∈Jkj

k = 1, . . . , K j

i = 1, . . . , M j

#$-% #$'%

!"#

" . " /

! ξ v " C = C(ξ, v) # # # , + $ v : v − v0 22,Σ −1 := (v − v0 )T Σ −1 (v − v0 ) ≤ γ 2 % & ! ' min ξ,x

max

v−v0 2,Σ −1 ≤γ

ϕ(C(ξ, v))

" !$ !

( ) ' min ξ,x

max

v−v0 2,Σ −1 ≤γ

ϕ(C(ξ, v0 )) +

ϕ(C(ξ, v0 ))(v − v0 ), v

$ * ? ? ? ? ? min ϕ(C(ξ, v0 )) + γ ? ϕ(C(ξ, v0 ))? . +,? ξ,x v 2,Σ

γ ? ? ϕ(C(ξ, v0 )) ? v ϕ(C(ξ, v0 ))?2,Σ

. /

" 0 # 1 23 4 "

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1

' 3 "

! 0

3 45( $-6& 1 ! " ξ ! ϕ ! C $67& C ! J $-/& + ! ! J ! ξ J1 i ∈ {1, . . . , M j }

" j ∈ {N1 + 1, . . . , N1 + N2 } ! p j $ j wi " j j j j j j j r1i = · η − h (x (τ , p, q , s ), p, q ) i i i p p σij wij ∂hj ∂xj ∂hji i i =− j + ∂x ∂p ∂p σi ! hji := hji (xj (τij , p, q j , sj ), p, q j ) xji := xj (τij , p, q j , sj ) * ! q j !

j $ j wi " j j j j j j j r = · η − h (x (τ , p, q , s ), p, q ) i i i q j p 1i q j p σij

wij ∂ 2 hj ∂xj ∂xj ∂hji ∂ 2 xji ∂ 2 hji ∂xji i i i + =− j + ∂x∂x ∂p ∂q j ∂q j ∂x ∂p ∂x ∂q j ∂p σi +

∂ 2 hji ∂xji ∂ 2 hji + . ∂x∂p ∂q j ∂q j ∂p j i

j i j

∂x ∂x ∂p ∂q ∂q∂ x∂p ! "# $ % &'( ) # * + , ,

- . / - 0, % , "1 * , % , % w {0; 1} % / " % % ' -% 23( " 2

j

j i

w ! "

#$%& 4 -% 23( " , %

4 % % $& 5 ( % 1 4 % /%

A+B → C A+C D 3A → E

A B C D

E

L

!

n˙ C (t) = V (t) · (r1 (t) − r2 (t) + r3 (t)) n˙ D (t) = V (t) · (r2 (t) − r3 (t)) n˙ E (t) = V (t) · r4 (t) nA (t) = nA,0 + nA,e (t) − nC (t) − 2 · nD (t) − 3 · nE (t) nB (t) = nB,0 + nB,e (t) − nC (t) − nD (t) nL (t) = nL,0 + nL,e (t) nC (t0 ) = nD (t0 ) = nE (t0 ) = 0

! Ea,1 k1 = kref 1 · exp − n1 n2 R · r1 = k1 · V V Ea,2 n1 n3 k2 = kref 2 · exp − · r2 = k2 · R V V n4 Ea,4 r3 = k3 · k4 = kref 4 · exp − V R " n $2 1 r4 = k4 · ∆H2 V KC = KC2 · exp − R k3 = nA,e = nA,e1,0 · f eed1

1 · T 1 · T 1 · T 1 · T

−

Tref 1 1 − Tref 2 1 − Tref 4 1 − . TC2

k2 , KC

nB,e = nB,e2,0 · f eed2

nL,e = nL,e1,0 · f eed1 + nL,e2,0 · f eed2 ,

1

V =

nA · M A nB · M B nC · M C nD · M D nE · M E nL · M L + + + + + ρA ρB ρC ρD ρE ρL

t ∈ [t0 ; tend ] = [0 h, 80 h] nA nB nC nD nE nL

Ea,1 KC2

Ea,2 Ea,4

kref 1 kref 2 kref 4

∆H2 nA,0 nB,0 nL,0 nA,e1,0 nL,e1,0 nB,e2,0 nL,e2,0 f eed1 (t) f eed2 (t) T (t) • ! " # •

$

! "&'( #

"&'(

! %

•

!

# ) *

+ , % - * * * .%+/ & ! * 0

* - * 1 % * 234 * 5

kref 1 Ea,1 kref 2 Ea,2 kref 4 Ea,4 ∆H2 KC2

2.50 ± 0.02 0.835 ± 0.007 91.3 ± 0.7 0.834 ± 0.002 57.991 ± 0.009 0.65725 ± 0.00007 0.9 ± 0.3 1.1 ± 0.3

2.504 ± 0.006 0.836 ± 0.001 91.25 ± 0.02 0.83537 ± 0.00008 58.004 ± 0.006 0.6574 ± 0.0002 1.082 ± 0.006 1.29 ± 0.01

% * 6 274 /

nonrobust design

robust design

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

0.2

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

0.4

0.6 0.8 Parameter 2

1

1.2

1.8

1.4

1.6

1.4

0.6 0.8 1 1.2 Parameter 1

0.4

0.2

0.4

0.6 0.8 Parameter 2

1

1.2

1.4

1.6

1.8

1.4

0.6 0.8 1 1.2 Parameter 1

0.4

Ea,1 Ea,2 ! " # # $ $ ! "

#$ % $ $ " " &

"$

& " '( # !

ηkij (k

! &

1, . . . , m2 )

= ψ, θ, c) (i = 1, . . . , m1 ) (j = ψ, θ, c

ηkij = hk (ti , k(ti , zj ; p), p) + kij ,

2 kij ∼ N (0, σkij )

" )

min

m1 m2

p

"

−2 σkij (ηkij − hk (ti , k(ti , zj , p), p))

2

k=ψ,θ,c i=1 j=1

∂ψ = ∂t ∂θ = ∂t ∂(θc) −= ∂t +

∂ ∂ K(ψ; p) (ψ − z) + S(ψ; p) ∂z ∂z ∂ ¯ ∂θ ¯ p) + S(θ; ¯ p) D(θ; p) − K(θ; ∂z ∂z ∂(ρs) ∂ ∂c θDh (θ; p) − qc − + Q(c; p) ∂z ∂z ∂t

C(ψ; p)

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% 5% 2 % 5% 7 57 8 57 % 9

s, s = 1, . . . , n z, z = 1, . . . , m T

T (s, s ) := ρ(s = s |s = s) ρ(z = z|s = s) z s O (s, s ) := ρ(z = z|s = s)δ , z = 1, ..., m

1 s = s δ = 0 . b (s) = ρ(s = s) t s ! " % b(s) = 1 0 ≤ b(s) ≤ 1 # $ % & $ $%&$ " a t ' ρ(s |s, a) ( a T (s, s ) ) * a z + 1 b (s) = b (s ) T (s , s ) O (s , s)

, N

t

t

t

z

t+1

t

s,s

t

s,s

t

t

s

a

t+1

t

a

z

s ,s

N :=

bt (s ) T a (s , s ) Oz (s , s) .

s,s ,s

$%&$ a = π(b ) * * " $%&$ a = π(z ) * * & - . a r(s, a) V (b ) := E γ r(b , a ) r(b , a ) = r(s, a )b (s) / t

t

t

t

T

π

i

t

i=0

t+i

t+i

t

t

t

s

t

π γ ∈ [0, 1]

bt

Qπ (bt , at ) := r(bt , at ) + γ

π

ρ(zt+1 |bt , at ) V π (bt+1 ) ,

zt+1

π

b a ! ρ(zt+1 |bt , at ) = bt (s)T at (s, s )Ozt+1 (s , s )

" #

s,s ,s

π∗

zt+1

!

bt

at bt+1

$ %

& ' &

()*

∗

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∗

ρ(zt+1 |bt , at ) V π (bt+1 )

+

zt+1

∗

∗

V π (bt ) = arg max Qπ (bt , at ).

,

at

- & .

($* (//* - 0 1 1 (/2* 1 1 (3* 1 1 (+* % 1 1

4

/,,# (/*

4 '

& & 5

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(6*

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8 (/2* "

5

9

. &

π

1

! " # $

% & '()

* * +

*

$

T

a=1

:=

0.5 ct 1 − 0.5 ct 10 a=3 := := , T , T 01 1 − 0.5 ct 0.5 ct co 0 1 − co 0 z=2 := := , O , 0 1 − co 0 co 10 −1 −100 cr . r(s, a) := 10 −100 cr −1

ct 1 − ct 1 − ct ct Oz=1

a=2

'() co = 0.85 cr = 1 ct = 0.5 γ = 0.75

, $

2 π1 (z) = 1

z = 1

π2 (z) = 3 ∀z .

z = 2

, #

# " , &$

T

a=1

:=

0.995 0.005 0.02 0.98

, T

a=2

:=

0.97 0.03 0.005 0.995

0.9 0 0.1 0 , Oz=2 := , 0 0.1 0 0.9 −1 −1 −1 −2 r(s, a) := 20 + . 0 0 −1 −2

Oz=1 :=

-. s = 1

s = 2 /

! " # $

%

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%

&

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,

- ,

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! ! ! ! "! " # cr $ " cr " !$ ! !% ! % # " "! !& !$ ' cr % 0.2# " ! # $

V π (s) ρ(s|π) π !V π !Rπ := % t tγ

!V π =

γ t rt0 +t ,

t

t0 ρ(s|π) !" # $ % !R co & ' π1 (z) π2 (z) = 3 & ( & ) *

$ +

! "c0 = 1 # "c0 = 0.5 # $

% % & & % " cr # %

% ! ct % ' & ( & T a=1,2 1

| det(T a)| 0 ≤ | det(T a )| ≤ 1

T =

0.5 0.5 0.5 0.5

,

det(T ) | det(T a)| a | det(T )| 1 ! " # $%&% ' ( ( %)*%

+ , $%&%, $%&% $ % & % , " # , - *" " ' . / 0

a = 1 a = 2 a = 3 1 & s = 1 s = 2-

( 23) , , s = 3, 4 )- z = 1, 2, 3 4 5 678 ( " 9 + ' ( 1

⎛

TSa=1

a=1 TH

T a=2

T a=3

⎞ 1.0000 0 0 0 ⎜ 0.0500 0.9492 0.0008 0 ⎟ ⎟ := ⎜ ⎝ 0.0333 0.0333 0.9308 0.0025 ⎠ 0.0033 0.0033 0.0025 0.9908 ⎛ ⎞ 1.0000 0 0 0 ⎜ 0.0500 0.9492 0.0008 0 ⎟ ⎟ := ⎜ ⎝ 1.0000 0 0 0 ⎠ 0.1000 0 0 0.9000 ⎛ ⎞ 1.0000 0 0 0 ⎜ 0.0250 0.9733 0.0017 0 ⎟ ⎟ := ⎜ ⎝ 0.0017 0.0017 0.9942 0.0025 ⎠ 0.0002 0.0002 0.0025 0.9972 ⎛ ⎞ 0 0 1.0000 0 ⎜ 0 0 1.0000 0 ⎟ ⎟ := ⎜ ⎝ 0.0008 0.0008 0.9958 0.0025 ⎠ 0.0033 0.0001 0.0025 0.9941

d a=1 Tda=1 := d TH + (1 − d) TSa=1 , d ∈ [0, 1] .

⎛

Oz=1

⎛ ⎞ ⎞ 0.9 0 0 0 0.08 0 0 0 ⎜ 0 0.1 0 0 ⎟ z=2 ⎜ 0 0.6 0 0 ⎟ ⎟ ⎟ := ⎜ := ⎜ ⎝ 0 0 0.1 0 ⎠ O ⎝ 0 0 0.2 0 ⎠ 0 0 0 0.1 0 0 0 0.2 ⎛ ⎞ 0.02 0 0 0 ⎜ 0 0.3 0 0 ⎟ ⎟ Oz=3 := ⎜ ⎝ 0 0 0.7 0 ⎠ . 0 0 0 0.7

r(s, a) r(s) r(a) r(s, a) = r(s) + r(a) . !"# a $ % !& r(s) 5 r(s, a) = r(a) + 5 r(s)

⎛

⎞ ⎛ ⎞ ⎛ ⎞ −6 −8 −10 −1 −1 −1 −11 −13 −15 ⎜ −6 −8 −10 ⎟ ⎜ 0 0 0 ⎟ ⎜ −6 −8 −10 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ r(a) := ⎜ ⎝ −6 −8 −10 ⎠ , r(s) := ⎝ 0 0 0 ⎠ , r(s, a) = ⎝ −6 −8 −10 ⎠ −6 −8 −10 0 0 0 −6 −8 −10

T → ∞ γ = 0.9 γ = 1 γ < 1

! " #$% & & '( ) * + ! " # $" "% &'"

, ⎧ ⎪ ⎨3 z π(z ) = 1 z ⎪ ⎩ 1 z t

t t t

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-& , ⎧ ⎪ ⎨3 z π(z ) = 2 z ⎪ ⎩ 2 z t

t t t

=1 =2 =3

( ," . #& a = 1 a = 3 & , ( & ," & , ( & , a = 1

% t |at − at−1 | !"#$!

d ! R

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( ( ( #= ## ### & >) 2# & # ? $ :+333; *4, ! < "( @ @. " !# >(% # /# )-56A8 B/! B % # ?$# $ C57D4 :+33D; *C, " B $ < ! 2. 1/!-(+. ! $# #( (% # # ! # 2 3C473DC :5AAA; *D, % &. # " . ! 2%#:; ># ## $ > # ## # <# !( # 5637599 :+33A; *6, % % !. # E # " ## #($ > ! +A6A7+A85 :+338; *8, < <$# ! $ # " <# . !"!(/!-. ! " ! ( /! - ) # ### $ E !

# ++937++3D :+333; *9, <$# !. !- 7 ! )(6C "$ B/! ( .FF(#D ( F $F=F :5AA4; *3, @ / <$# ! . $ B# $( ! B/! !# ? >(% # BGA5 +957+3+ :5AA5; *+A, % . ! !#$ " ( $ 1' :5AAA;

! " # ! "

$ %&' ! " ! ( ) !! # ! ! * + , -

! . / 0 0 !

!"#$% & # !"% $' (

) *

&

&

! !

p

"

G(z) z ! u(k) U (z) y0 (k) Y0 (z) #

!

d(k)

D(z)

$

% #

G(z) y(k) = y0 (k) + d(k) &! !

#

yd (k)

e(k) = yd (k)−y(k) # u(k)

!

# # !

! ' ! ! !

ϕ

! !

!

u(k) = u(k − p) + ϕe(k − p + 1)

(

#

z

G(z)

#

! !

U (z) = z −p [U (z) + ϕF (z)E(z)]

)

! ! ! ( # !

!

! * ) !

z

F (z)

#

+ !

$ ,-. !

F (z) #

1 − z −p [1 − ϕG(z)F (z)] E(z) = 1 − z −p [yd (z) − D(z)]

!

p

!

"

# $

%

&

P (z) = 1 − z −p [1 − ϕG(z)F (z)] = 0

%

'

P (z) 2π 2π &

G(z) F (z) 2π '

P ∗ (z) = z −p [1 − ϕG(z)F (z)] z () ' F (z) z −p G(z)F (z) ! p

! * +

P ∗ (z)

z

()

|1 − ϕG(eiωT )F (eiωT )| < 1

ω

#

T

,

' -./

0

,

1

%

ϕG(eiωT )F (eiωT ) ω

() " #

[1 − ϕG(z)F (z)]

F (z) z p E(z) = [1 − ϕG(z)F (z)] E(z)

% ,

ω ϕG(eiωT )F (eiωT )

G−1 (z) F (z) = G−1 (z) ϕ = 1

z p = 0 !

0 < ϕ < 1 1−ϕ ϕF (z)G(z)/(z p −1) = −1 p F (z)G(z) = 1 ϕ p F (z) = G−1 (z) "

# G(z) $

% & % ' ( %

' )*+ % ,

% % - -.-, /,01 % 2 *1** // /// % ' ,-,/2 ,-,, /2- //2- 3 4

,-,/2 /// 5 & (−23.204)1000 % & ' 6 ' % G−1 (z) % 7

G(z)

(z − z1 ) z1

(z −1 − z1 ) z

ϕG(eiωT )F (eiω T )

!

" +ϕ ϕ ) *

#

G(s) = aω02 / (s + a)(s2 + 2ζω0 s + ω02 ) $%% #

& ' ' ( ) a = 8.8 ω0 = 37 # * ϕG(z)F (z) = ϕ∗ (z − z1 )(z −1 − z1 ) ϕ∗ + ϕ & ,( # +

-

- . +

- . +

p

$ p = 10

&

' +

-

0

4

2

−4

Imaginary Axis

Magnitude (dB)

−2

−6

−8

−2

−10

−12 0 10

0

1

2

10 10 Frequency (rad/sec)

3

10

3rd

−4

−4

−2

0 Real Axis

2

4

3rd

y0 (k +3)+a2 y0 (k +2)+a1 y0 (k +1)+a0 y0 (k) = b2 u(k +2)+b1 u(k +1)+b0 u(k)

u(k) G(z) ! "#$ % & d(k) u(k) y0 (k) = −d(k) & d(100) = 1 & k = 1, 2, 3, . . . , 198 % ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ s(1) a0 a1 a2 1 0 . . . 0 ⎢ ⎥ b0 b1 b2 0 . . . 0 0⎥ ⎢ 0 b0 b1 b2 . . . 0 ⎥ ⎢ s(2) ⎥ ⎢ 0 a0 a1 a2 1 . . . 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ 1 ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎥ ⎣0⎦ 0 . . . 0 b0 b1 b2 s(200) 0 . . . 0 a0 a1 a2 1

'

u(k) ! s(k) k ( )** % ( 100th (+, ! % 198 × 200 200 × 1 % 198 × 201 201 × 1 & -. /.0 s(1) = s(200) = 0 1 -!2 " 3 # % " 4 5* ,* ! ,* +* !*() 6**4 . ((* ()* ! 7 !(* 64 10−3 "

−7

5000

2

x 10

4000 3000

1

2000 0

0

s(k)

s(k)

1000

−1

−1000 −2000

−2

−3000 −3

−4000 −5000 0

20

40

60

80 100 120 Time step k

140

160

180

200

s(k)

−4 70

71

72

73

74 75 76 Time step k

77

78

79

80

!

" #

a−99 = s(1), . . . , a0 = s(100), . . . , a100 = s(200) 100th a−1 , a0 , a+1

!! ! "!! # $!! !! ϕ % a0 e(100)

& a+1 e(99)

a−1 e(101)

! ' ( & a−n2 , . . . , a0 , . . . , a+n1 ) F (z) * G−1 (z)% u(k + p) = u(k) + ϕ [a+n1 e(k − n1 ) + . . . + a0 e(k) + . . . + a−n2 e(k + n2 )] + F (z) = an1 z −n1 + . . . + a0 z 0 + . . . + a−n2 z n2 ) * = an1 z 0 + . . . + a0 z n1 + . . . + a−n2 z n1 +n2 /z n1 ,

- ) # ( n1 + n2 & . n1 z −p [1 − ϕG(z)F (z)] * ( # & ϕ & n1 n2 # & ) ϕG(z)F (z) z / # (

ϕ ϕG(z)F (z) z = 1 n1 n2 ϕ

! " #$% &' #

%$&( s(k) ")( * + , , *&& - && - *

. &/ & G(s) 0 *&& - / 1

*

$

2

/

)

'

!

#$% &$ #2% &2 #$% &2 #2% &/ #$% &/ #2% &) #*% &/ #$% &) #2% &' # % &/ #*% &) #$% &' #2% &! #&% &/ # % &) #*% &' #$% &! #2% &# !#% &/ #&% &) # % &' #*% &! #$% &# #2%

& !!% &/ !#% &) #&% &' # % &! #*% &# #

%$& #2%

#/% &/ #/% &) #/% &' #/% &! #/% &# #/%

& #/%

#/%

*

k = 100, 300, . . . p = 200 !! " # !! $ % G(z) &!! ' ! ()! !)* +, & 8 × 10−4- " % . ) /!0 +, 1'- " " ! 2- " . )/!3 p = 10 4" " - - " ϕG(z)F (z) n1 n2 - (

* $ 3 )!/!0 0 3 3 - )'/ % " $- −4

2

x 10

4

0

3 2 Imaginary Axis

Output

−2

−4

−6

1 0 −1 −2

−8 −3

−10 200

220

240

260

280 300 320 Time step k

340

360

380

400

−4 −5

−4

−3

−2

−1

0 1 Real Axis

2

3

4

5

!

1

1

0.8

0.8

0.6

0.6

0.4

0.4

Imaginary Axis

Imaginary Axis

0.2 0 −0.2 −0.4

0.2 0 −0.2 −0.4

−0.6

−0.6

−0.8

−0.8

−1 −0.5

−1 0

0.5

1 Real Axis

1.5

2

2.5

ϕF (z)G(z)

−0.5

0

0.5

1 Real Axis

1.5

2

2.5

ϕF (z)G(z) !

"# $ % & ' ( ) %* +, -... / ) . & * ) + .( 0 ( / %1-/ 2 ! 0 !!! "!# , &(( 3 / 3(* ( +- &( % 4 5 . -... 6 4 $' + % 7 6. .8 2 0 0 9 - :4 *. *. - 5 ;.. ; $.. ;;0 9 0 " # ) , ( < < +) 4 / ) ( -. + .( !9 %;;; /. 5. ( / 0 ! ! "9# ) , $ = ( & ' +)6 4 / ) + .( !9 %;;; /. 5. ( / 0 0 ! " # $ >' & $ $/ ( /* ?? +- ( ) 5. 4 / + 7 5 . ) &

( / 2 0 0 "# +%4 / ( 4 / ; .. + % 7 / ). % %4 / 2 7 ! 9 "# ) 7 = ( +&( $ @ * 1 6 4 / + .( -%--A--) - ( . ). /. & /- - !! "0# ) ) . ( +/ )6 ( ( 1. /( ( 4 / + % 7 -( & . ( / ).. "# ? - , ( ( 7 )6 +@ ) ( ) + .( %;;; /. 5. ( / 0 0

! " # "$% & '

# ( % ) # * + , +

+ " - .

!

" !

#

$

$ !

%

$

G = (V, E) E V

! " qe , e ∈ E " # pv , v ∈ V $ pin

pout !

% & ' ( % % % % %

)

% * p2out = p2in − ( q |q|,

( = ((pout , pin )

! pout = pout (pin , q) pout pin q " * f * f = f (pin , pout , q) + f pin

pout

q

out

in

p1out 2 pipe ! pin pipe2 ! " p1out

1

#$

Λpipe Y pipe

2− pout (pin , q) %

&

P∆ =

λ1 λ2

∈

Ê|Λ |+|Λ | | 1

2

λ1j

j∈Λ1

=

1

λ2j

=

1

p2,j λ2j

=

0

λ2j

≥

0

j∈Λ2

p1,j λ1j −

j∈Λ1

j∈Λ2

λ1j , λ1 , λ2

λ−

#

,

P∆ & p1out,j

!

j ∈ Λ1

!

p2in,i

i ∈ Λ2

' (! )

p1out,1 = 10 p1out,2 2 pin,1 = p2in,2 = p2in,3 =

= 8 p1out,3 = 4 10, . . . *

50.98

60.98

51.01

42.53

51.18

56.97

61.01

51.01

61.01

51.01

61.01

=

70.99

67.87

63.02

61.52 71.01

71.01

71.01

71.01

λ11 =

1 1 1 1 , λ2 = 0, λ13 = 0, λ14 = , λ15 = , λ16 = 0 4 2 4

7 2 13 2 , λ = 0, λ23 = 0, λ24 = 0, λ25 = , λ = 0. 20 2 20 6 λ− λ21 =

λ11 =

1 1 1 1 , λ = 0, λ13 = 0, λ14 = , λ15 = , λ16 = 0 4 2 2 4

7 2 13 , λ2 = 0, λ23 = 0, λ24 = 0, λ25 = 0, λ26 = , 20 20 λ21 λ26 ! in out " # $ %

% & ' ! ( ! λ21 =

1 λ1

1 λ2

10

8

2 λ1 10

4

20

16

1 λ 4

1

λ

1 λ3

5

20

12 1 λ6

2 λ4

2 λ2

2 λ3

10

10

20 2 λ5

2 λ6

20

!" #!$ % &

Ú

Ú

Ú

Ú

Ú

Ú

Ú

Ú

' P∆ ( ' ) *

λ−) % P λ−)

λ− ! " #

" " $ in ∈ N out ∈ N %

" & " ' ( # N i i ∈ {1, 2, . . . , in + out} ( " 1, 2, . . . , in in + 1, in + 2, . . . , in + out ) " Ni ∩ Nj = ∅

(

∀i = j.

N = {N i | i = 1, 2, . . . , in + out}

ÊN i |N i |− " N ÊN # i

Ê

N

=

in+out @

ÊN . i

i=1

( ' λ ∈ ÊN " " ⎛

⎜ ⎜ λ=⎜ ⎝

λ1 λ2

⎞ ⎟ ⎟ ⎟ ⎠

λin+out

" λi ∈ ÊN i ∈ {1, 2, . . . , in + out} ) S i

S = {S 1 , S 2 , . . . , S in+out }

" j ∈

in+out A i=1

Ni

j∈S

∃ i ∈ {1, 2, . . . , in + out}

j ∈ Si.

S ⊆N

⇔ S

∅ = S i ⊆ N i

∀i ∈ {1, 2, . . . , in + out}.

|S| =

in+out

|S i |.

i=1

S

XS ∈

ÊN

XjS =

j∈S

N i , i ∈ {1, 2, . . . , in + out} ni

ni

Ni =

Nki

Nki , k ∈ {1, 2, . . . , ni }

|Nki | ≥ 2.

k=1

n1 = 8 n2 = 9 |Nki | = 3 i, k λ ∈ , λ ≥ 0 i = 1, 2, . . . , in + out ki ∈ {1, 2, . . . , ni }

! "

Ê

N

{j ∈ N i |λij > 0} ⊆ Nki i . !

S

λ−

$

P

P = {λ ∈

A ∈

ÊM×N , b ∈ ÊM

X

S

A∈

ÊM×N

ÊN |Aλ = b, λ ≥ 0},

M A

M

%

Nki

#

A = (aij )

J ⊆ {1, 2, . . . , n} AJ = (aij ) i ∈ M j∈J

m = |M | n = |N | λ ∈ J ⊆ {1, 2, . . . , n} λJ = (λj )j∈J . x ∈

ÊS S ⊆ N

x0 (S) =

xi

ÊN

x0 (S) ∈

ÊN x

i ∈ S, i ∈ N \ S.

N \ S N \ S :⇔ N i \ S i ∀i ∈ {1, 2, . . . , in + out}

S S¯ S = {S 1 , S 2 , . . . , S in+out }

S¯ = {S¯1 , S¯2 , . . . , S¯in+out }

! " S ⊆ S¯ :⇔ S 1 ⊆ S¯1 S 2 ⊆ S¯2 S in+out ⊆ S¯in+out .

# $ % $ % % & in out ' % P P = {λ ∈

ÊN |Aλ = b, λ ≥ 0 , λ

}.

% % $

A b ⎛

(11 )T

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 T ⎜ (p ) ⎜ 1 T ⎜ (p ) ⎜ ⎜ ⎜ ⎜ A = ⎜ (p1 )T ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ (q 1 )T

⎞

(12 )T

(1in )T

(1in+1 )T

−(p

−(p

2 T

(p ) (p2 )T

(1in+out )T

in+1 T

)

−(pin+2 )T

in+1 T

−(pin+out )T

)

−(pin+2 )T

(p2 )T

(1in+2 )T

(pin )T −(pin+1 )T (pin )T −(pin+2 )T

−(pin+out )T

(pin )T −(pin+out )T i T in+1 T in+2 T (q ) . . . (q n) −(q ) −(q ) . . . −(q in+out )T 2 T

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

pi ∈ ÊN+ , i ∈ {1, 2, . . . , in + out} qi ∈ ÊN+ , i ∈ {1, 2, . . . , in + out}

b i

i

b=

1in+out

0in · out+1

ÊN

1m

Êm 0m Êm 1i ∈

i

A ! in λ− out λ−

in + out + 1 in + 2out

in + 2out + 1 (in+ 1)(out+ 1)− 1

−1 A λ b

! " # p q 1 ! # $ %&

$ " %& λ−

' ( $ Aλ = b ) A ' |M | = in + (in + 1)out + 1 = (in + 1)(out + 1) *

Ê

P ⊆ [0, 1]N

λ ≥ 0 in + out A b P

% P∆ + !pipe1 " !pipe2 " # λ− , λ− P∆ pipe1 pipe2 -

λ−

P∆ λ− pipe1 λ− pipe2 1

λ−

λ−

P∆ P rg(A)

A

S ⊆ N

• •

|S| ≤ rg(A) S

! L = ∅ P " # S ⊆ N $ AS λS = b ¯ S λ ¯ S ≥ 0 & ' % λ ¯ λS L % P ( P ) ) * A +!, # , b 1in+out b= 0in+out # in + out ≤ rg(A) ≤ 2(in + out)

P

* ) ' P = {x | Ax = b, x ≥ 0} A ' P ' * P ) %

0

(11 )T

B B B B B B B B B B B B B B B B B B B B B B B B 1 T B (p ) B 1 T B (p ) B B B B B B B B 1 T B (p ) B 1 T B (p ) B B B B B B B B B B @ (q 1 )T

1 C C C C C C C C C C (1in )T C in+1 T C (1 ) C in+2 T C (1 ) C C C C C C C C in+out T C (1 ) C C −(pin+1 )T C in+2 T C −(p ) C C C C C C C C in+out T C −(p ) C 2 T C −(p ) C C (p2 )T −(p3 )T C C C C C C C C in−1 T in T A (p ) −(p ) (q 2 )T ... ... (q in )T −(q in+1 )T −(q in+2 )T . . . . . . −(q in+out )T (12 )T

A

Ì

P S

0|N | AS λS = b

AS λS =

¯ S AS λS = b AS λ ¯S = b λ ¯ ¯ ¯ λS > 0|S| . λ λS

λ¯ P − > 0 λ¯ λ¯ S ∈ S S ⊆ N

Ê

AS λ¯S = b AS (λ¯ S + ) = b AS = 0|N |

¯ = 0|S|

AS = 0|N |

AS = 0|N | ¯ = 0|S| AS (λ¯S + ¯) = b λ¯S + ¯ > 0|S| λ¯S − ¯ > 0|S| !" (λ¯S + ¯)0 (S) λ¯S + ¯ #

¯ (λS + ¯)0 (S) ∈ P S λ− $ "

% AS (λ¯S − ¯) = AS λ¯S − AS ¯ = AS λ¯S − 0|N | = b A(λ¯S − ¯)0 (S) = b (λ¯S − ¯)0 (S) ∈ P &

1 ¯ 1 ¯ ¯S . (λS + ¯) + (λ ) = λ S −¯ 2 2

1 ¯ 1 ¯ ¯ S )0 (S) = λ. ¯ (λS + ¯)0 (S) + (λ )0 (S) = (λ S −¯ 2 2

% λ¯

P &'

# ' ( $ P P ½

P

& $ in + out A #

# ! $ S ⊆ N X S # ) * (X S )T λ ≤ in + out.

& $ P ) P λ− P ) in + out λ¯ = λ0 (S) ∈ P ! λS # # # S #* ¯ = P ∩ {(X S )T λ = in + out}. {λ}

% λ¯ ∈ P (X S )T λ¯ = in + out $ X S ¯ ⊆ P ∩ {(X S )T λ = in + out} $ {λ} ¯ ⊇ P ∩ {(X S )T λ = in + out} % {λ}

¯ ˜ ∈ (P ∩ {(X S )T λ = in + out}) \ {λ} λ

λ˜i = 0 i ∈/ S λ˜ ¯ AS λS = b λ

P

P P ¯ S λ A λ = b A λ = 0 λ¯ ∈ P λ¯ ! λ¯ ∈ P &' P ! " P # λ ! P λ P $ % P $ & ' " n = n = 1 |N | = |N | = 3 A " S

S S

S S

|N |

S

∆

∆

∆

1

1 1

2

1 λ1 p1 1

2 1

2 λ1

1 pipe

1 λ2 p1 2

1 λ3 p1 3

p2 1

2 λ 2 p2 2

2 pipe

2 λ3 p2 3

P∆

⎛

⎞

1 1 1 0 0 0 ⎜ 0 0 0 1 1 1 ⎟ ⎟ A=⎜ ⎝ 15 10 10 −10 −10 −20 ⎠ 0 0 0 0 0 0

⎛

⎞ 15 p1 = ⎝ 10 ⎠ , 10 ⎛ ⎞ 10 p2 = ⎝ 10 ⎠ . 20

⎛ ⎞ 0 q1 = q2 = ⎝ 0 ⎠ . 0

b

⎛ ⎞ 1 ⎜1⎟ ⎟ b=⎜ ⎝0⎠ 0

A rg(A) = 3 S1 = {S 1 , S 2 } S 1 = {1} S 2 = {4, 6} AS 1

⎛

⎞

1 0 0 1 ⎠ AS1 = ⎝ 0 1 15 −10 −20

⎞⎛ 1⎞ ⎛ ⎞ 1 1 0 0 λ1 1 ⎠ ⎝ λ21 ⎠ = ⎝ 1 ⎠ =⎝ 0 1 λ23 0 15 −10 −20 ⎛

AS1 λS1

! " λS1

# $ λS

1

⎛ ⎞ 1 =⎝1⎠ 2 1 2

⎛ ⎞ 1 ⎜0⎟ ⎜ ⎟ ⎜0⎟ ⎜1⎟ ⎜ ⎟ ⎜2⎟ ⎝0⎠ 1 2

P∆ S2 = {S 1 , S 2 } S 1 = {2} S = {4, 5} ⎛ ⎞⎛ 1 ⎞ ⎛ ⎞ 1 0 0 1 λ2 AS2 λS2 = ⎝ 0 1 1 ⎠ ⎝ λ21 ⎠ = ⎝ 1 ⎠ . λ22 10 −10 −10 0

2

rg(AS2 ) = 2 S2 |S2 | > 2 S2 S3 = {S 1 , S 2 } S 1 = {2} 2 S = {4} ⎛ ⎞ ⎛ ⎞ 1 1 0 1 λ 2 AS3 λS3 = ⎝ 0 1 ⎠ = ⎝1⎠, 2 λ1 10 −10 0

1 , 1

!

"#

⎛ ⎞ 0 ⎜1⎟ ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎜1⎟ ⎜ ⎟ ⎝0⎠ 0 $

P∆

%

P∆

$

& & & '( ) $ &

P

$ $ * &

$

& &

P∆ λ− ! "#$ ∆ λ % 8 16 24 32

12 18 24 32

16 49 73 142

18 47 90 10492

25 42 670 50640

& P∆ '%

! ( ) ) P * +

P

l, c P

clin+out , l∗ ∗

l :=

in+out #

nj

-

j=1

nj , j = 1, 2, . . . , in + out λ− + l∗ Nki . - / & / P∆ nj

λ− m ≤ rg(A)

λ−

j ∈ {1, 2, . . . , in j Nmax j Nmax := max{|N1j |, |N2j |, . . . , |Nnj j |} !

j ∈ {1, 2, . . . , in + out}

Pin+out j=1

"

c c

in+out #

c :=

xj ≤m

%in+out j=1

xj c

j Nmax xj

j=1

+ out}

xj ≥ in + out !

# !

λ− λ−

m

S

$ %in+out xj λ− λ− j=1

" cl∗

%

l := max{n1 , n2 , . . . , nin+out }

& ' clin+out ' &

(

c=2 )

c

c

Pin+out j=1

j Nmax

.

*

+ ,-

P∆ ' j

Nmax m ! "

m = 3 P

. . /

c=

l = 40

3 3 3 3 3 3 + + = 27 1 1 1 2 2 1

27 ∗ 41+1 = 432

1 λ1 20

1 λ2

ingoing pipe

2 λ1 20

10

1 λ3

1 λ4 30

40

1 λ5

1 λ6

2 λ5

60

2 λ2

outgoing pipe

20

2 λ3

40

42

2 λ4 40

2 λ6 60

60

S, S¯ λ−

S ⊆ S¯

P

S

P

AS λS = b ! ¯ AS λS = b " S λ− S¯i \ S i i ∈ {1, 2, . . . , in + out}

AS¯λS¯

#

AS¯λS¯ = b

= b

S

S¯

$ % &

S

!

λ− "

S λ− |S i | = 1 ∀i ∈

'

{1, 2, . . . , in + out}

' () ! "

λ−

*

S

|S| = rg(A)

() * ( )

!

" *

rg(A) = rg(AS ) S P∆

P∆ 9 n1 n2 .

P∆ 9 λ− λ− ! ! |S 1 | = |S 2 | = 1 "

λ− ! # " $ % 27 n1 n2 ! 3 3 3 3 3 3 c= + + = 27. 1 1 1 2 2 1

& P v1 , . . . , vk P ' ( ¯ ) * λ

aT x ≤ α ¯−α z ∗ = max aT λ T s.t. a vi ≤ α i = 1, . . . , k

α ∈ {0, 1, −1} ¯−α (¯a, α) ¯ = z ∗ ¯ a ¯T λ ¯ P '( a¯T λ ≤ α

P

v1 , v2 , . . . , vk ' P ( + ∗ %k λ ∈ P β1 , β2 , . . . , βk i=1 βi = 1 ∗

λ =

k i=1

βi vi

a ¯T λ∗ = a ¯T

k i=1

βi vi =

k

βi (¯ aT vi ) ≤

i=1

k

βi α ¯=α ¯

i=1

k

βi = α. ¯

i=1

a¯T λ ≤ α¯ P

z ∗ > 0 z ∗ > 0 a¯T λ ≤ α ¯ α a˜T λ ≤ α˜ λ¯ z ∗ ≥ a˜T λ− ¯ > 0

P ! " source

Compressor Valve sink

control valve

i ∈ Λ yi

λ− y

# # pin,C pout,C qC $ pin,P qP

10 ! " # !

2, y

pin,C pout,C qC pin,P qP 3 3 3 3

3 3 3 3

7 7 7 7

4 4∗ 8 8∗

10 10∗ 20 20∗

29 6 28 7

0 10 0 204

9.39 9.36 9.16 9.15

3.07 0.79 295.9 23.09

$ % &

'

$ () * $ () ' & P∆ * P + ,

- + P P , - . source

Compressor Valve sink

control valve

! "

#$ ! % !

source

Compressor Valve sink

control valve

!"#$% $ & % & " ''( ) * # +% ,& # , - ! "# $ % + ! . " # & *

+ & //0/1/ / ! "# &'( & ' ) % *+ 2 33 4

& 5

1 2

! " # $%$ & '

#( ) * #+, - ./ " / " / . 0 ) - - " "& ( / " " / .) ! & 1 2)3 4 " / / & #+, . . . ( " / . & 2)3 4 . & " " / " & 2 +

!

"#$%&%$ '½ ()*+ ,)--.-/0 1

1 5367631 70 / . 5" " 5" " &

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'()!*+,-( 7

% % ) *+89 ' . /:1 2330 5 % ) )' ' . 2;; &5 - < = $< $ 3◦ >

>- 8(%69T M T M

>- 8(%69T M

8 '%! 8(%69 >- 8(%69 ? ? '%! 8(%69 ? 2 ;5 6

>- 8(%69 >- 8(%69 2 3 4

! " ## " " " $##

% &' ! (# )&&*+,

% - ! (# )&&*+,

% ! " !# $ % & ' # #!! ( )*+#

• • •

,"'' (( (( (#

! "#$%&'( ) & * # +

!!

X(p, t, s)

p

X(p, t) = {X(p, t, s)|s ∈ (1 · · · S)} Mij (p, t) =

t S

X(p, t, i) − X(p, t, j) max(X(p, t, l) − X(p, t, k))

k,l

!

sim(X1, X2)

" #

bsim

"

(p1, t1)

)*#+,

(p2, t2)

(pref , tref )

#

bsim

-...

(p, t) '( sim(X(p, t) X(pref , tref ))

$%&

$%&

!

sim

(p, t) ! " # $ % &''''' (&' ) *' ) + , % - . /01 # 2 " 3 ( 4 $ 5 3 · Cluster(N ) · + · Cluster(N ) . · 4 $ 6 + · Cluster(N ) 7 # 2

2 , tref 2 ) 2 $ Scluster

Ci (tref )

M (C) =

1 M (p, tref ), size(C)

p∈C

size(C) C ! "

" N

A N ◦ P N C P ◦

P A ◦ A * Q A * sim(M (C), X(Q, tref )) ≥ bsim "

Q C " # Q N "

P A * ◦ • N • •

sim(M (C), X(p, tref )) ≥ sim(M (C), X(q, tref )) ∀q ∈ C, P (C) = p ∈ C

and p is the ﬁrst such node in C

N

Scluster T (C) (p, t)

T (C)

C

tref

t sim(M (C), X(p, t)) ≥ bsim C

p T (C)

(p, t)

simcluster

(p, t)

!

simcluster(p, t) =

index of C in the list of clusters, 0,

if(p, t) ∈ T (C) else "

" #$

% &

'' (

$ ) * + " , - . . "' -' / # & 0# , &

& ) 1 )

! "# $ %

& $ '( ) % #* ! "% $ %

% ' * " '* $ ! ") $ %+

! " # $% &

!

" !

#$%& ! ' ' ( ' ( ( !

) *+, - . /0*-1 2 *+,

' *+,

#$& * ( +,3 ( 3 4

-5 + ( + ( 6$7%8599:59;% 7$<<<8 #%& * ( +,3 ( 3 0=* ( .5 / ( + ( 9>7685:>9:<: 7$<<:8

! " # $ %& %!

'())$* + , - . & / &

& % 0 '())(*

1 2 3 0 3

. 4 3

2& '()))* 5 2 0 6

&

0 "

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3!-!:;

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8 3 <= " . & . >3 " '$777* 7 3 <=

. .

..

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. % ; 7$7 %. '$771* $$ "<<

0 / %#0 & &

( @= . A " $(

3 -

6

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& %

6. > ,%

%= '())(* $ 6 % ; ; B : &@ % # '$77+* $+

2 % B& 2 /:!;

# .. &

0 3 :@ 6 '* $5 3 ())) " 3 '()))*

% B

% 6 / ;@ @ =

1,2 1 2

! " # " " $ " %

&' ( ) ! "

α α α ! α

"#$% "#&% α ! ' > 3 × 1015 ( ! ! ! ' ( ) > 1000 −1 ( α * ! + ! , + α - ( - "#.% ! ! "#% "/% 0

1 ! ! ".% "2% 3 ) ( ! ! ) '

! " # $ % & ' " $ 10% (α $ ) * $ & " + , & $ - .

/ $ 0 *

1

' & Ω ⊂ R3 $ I

$ n · ∇x I(x, n) + κ(x)I(x, n) $ " 1 +s(x) I(x, n) − P (n , n)I(x, n ) dt = f (x), 4π S 2

!

x ∈ Ω n dt S 2 κ(x) s(x) P % 1 P (n , n) dt = 1

4π f (x) = κ(x)B(T (x)) + (x)

#!

T (x)

(x) 2 B 3 )4 ! I(x,

n) = g(x, n) 56 7 Γ− = {(x, n) ∈ Γ nΓ · n < 0} nΓ

Γ nΓ ·n 56 7

' g %

A I

AI(x, n) = f (x).

! I " h = 1/1000 1000 1012 # $ % &

'(" ) *#

+,-

W = I ∈ L2 (Ω × S 2 ) n · ∇x I ∈ L2 (Ω × S 2 ) ,

.

L2 / ! g(x, n) = 0

W0 = I ∈ W I = 0 Γ− . , ! # ϕ(x, n) Ω × S 2 0 1 L2

2

Iϕ dt d3 x.

(I, ϕ) = (I, ϕ)Ω×S 2 = Ω S2

# 3 ) I ∈ W0 (AI, ϕ) = (f, ϕ)

∀ ϕ ∈ W0 .

∀ ϕ ∈ W0

4

! s(x) = 0 Ω Ω !

# 5 6 3

(AI, ϕ + δn · ∇x ϕ) = (f, ϕ + δ n · ∇x ϕ)

κ

s

δ

∀ ϕ ∈ W0 .

! "

#$

% %

Wh

W

h % K h|K = hK = K

Ω × S2

%

Ih ∈ Wh

∀ ϕh ∈ Wh .

(AIh , ϕh ) = (f, ϕh )

Th

Wh

&

'(

! ) *

Ω

+

%

%

t

S2

Kx

Ω × S2

x

R

3

Kt

"

!

,

Ω

% -

h

.

Ω

- *

* / %

/

S

T + D + S + χ(x, ν) I(x, n, ν) = f (x, ν).

Ω

T

23 4

D

%

T I(x, n, ν) = n · ∇x I(x, n, ν), ∂ DI(x, n, ν) = w(x, n) ν I(x, n, ν), ∂ν σ(x) ∞ ˆ , νˆ) dˆ SI(x, n, ν) = − R(ˆ n, νˆ; n, ν)I(x, n ω dˆ ν. 4π 0 S2

01

I x n dω S 2 ν χ(x, ν) = κ(x, ν) + σ(x, ν) κ(x, ν) = κ(x)ϕ(ν) σ(x, ν) = σ(x)ϕ(ν) ϕ ∈ L1(R+ ) ! " 1 2 2 ν − ν0 1 ϕ(ν) = √ exp − , ∆νD π ∆νD

##

! ν0 " ! ∆νD " vD vtherm

! vturb ∆νD =

ν0 ν0 vD = c c

2 2 vtherm + vturb .

#$

% ! vturb vtherm " ! & ! ' " % c

f (x, ν) = κ(x, ν)B(T (x), ν) + (x, ν),

#(

! ) f T (x) ! B(T, ν) * + ," - D " . v/c < 0.1 /#01 ) /##1 ! 2 v(x) w(x, n) = −n · ∇x n · #3 c v(x) n 4

w

v S R(ˆ n, νˆ; n, ν)

(ˆ n, νˆ) (n, ν) ) ! ! s(x) SI = − 4π

∞ R(ˆ ν , ν) 0

S2

ˆ , νˆ) dˆ I(x, n ω dˆ ν,

! R(ˆν , ν)

#5

R(ˆ ν , ν) =

1 (4π)2

ˆ , νˆ; n, ν) dˆ R(x, n ω dω.

S2

S2

∞ ∞

R(ˆ ν , ν) dˆ ν dν = 1. 0

0

!

" # $ "

R(ˆ ν , ν) = ϕ(ˆ ν )δ(ν − νˆ)

R(ˆ ν , ν) = ϕ(ˆ ν )ϕ(ν).

%

$

$

S coh I(x, n, ν) = −

σ(x, ν) 4π

$

S2

ˆ , ν) dˆ I(x, n ω

&'

$ "

# $ !

S crd I(x, n, ν) = −

σ(x, ν) 4π

∞

ϕ(ˆ ν) 0

S2

ˆ , νˆ) dˆ I(x, n ω dˆ ν.

&

$ (

$

)

( *α '#

+

$

, "

*α$ Λ := [ν0 , νN +1 ]$ ν0

νN +1

# ! w(x, n) -,# . / w < 0 w > 0 ,

" # ! $

I(x, n, ν) = Icont (x, n, ν)

, "

!

Σ = Ω × S 2 × Λ# + $ " -,# '$

I(x, n, ν) = Iin (x, n, ν) 0 1 2 Γ − × Λ = {(x, n, ν) ∈ Γ | nΓ · n < 0}$ nΓ " Γ Ω # nΓ · n 01 2 #

Icont = 0

Iin = 0 Σ, Γ − × Λ.

I(x, n, ν) = 0 I(x, n, ν) = 0

! " # #$% &'(" ) N

νi ∈ ν1 , ν2 , ..., νN ⊂ Λ" ) * ! + ! * , N -./" # S crd ," " 0 N νi ∈ ν1 , ν2 , ..., νN ⊂ Λ N q1 , q2 , ..., qN * Q(νi ) :=

N

qj ξ(νj )

.

j=1

Λ

ξ(ν )dν " 1 ξ(νj ) =

ϕ(νj ) 4π

S2

ˆ , νj )dˆ I(x, n ω.

2

0 Ii Ij ! σi ϕi qi 4π

N σi ˆ , νj )dˆ Ii dˆ ω+ ϕj qj I(x, n ω. 4π S2 S2

3

j=i

% , " |w|νi $ + Acrd Ii i ∆ν N σi ˜ ˆ , νj )dˆ = fi + ϕj qj I(x, n ω, 4π S2 j=i

4

˜ crd Ii = fˆi . A i Acrd = T + χi + ϕi qi S coh . i

!

Acrd u = f .

" # $ % & ⎛

Acrd

˜ crd A 1

⎜ ⎜ B2 + Q 1 ⎜ ⎜ = ⎜ Q1 ⎜ ⎜ ⎝ Q1

R1 + Q2

Q3

˜ crd A 2

R2 + Q3

...

...

. . . QN

⎞

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

%

˜ crd ... A N

' w(x, n) & Ri Bi ( w(x, n)νi/∆ν # $ & Qj ( ) #

$ *# N + , $

˜ crd ui = ˆfi , A i

ˆfi = fi + Ri ui+1 + Bi ui−1 +

Qj uj .

j=i

$ # # - . $

/# # 0 1 # 2 # 3 % . I = 0

i = 1, .., N ! ! " " [ν1 , νN ] # " ! $ % & ! nout '

! " & ! ( # " ) ! ηK = max(ηK (νi ))|νi & ηK ηK (νi )

K νi *+ , -

# ! !! % " $ & . χ(x) = χ(x, y, z) ! ⎧ r ≤ rc ⎨ χ0 /(1 + αrc2 ) χ(x) = χ0 /(1 + αr2 ) rc < r ≤ rh , ⎩ χ0 /(1 + αrh2 )/103 r > rh r2 = x2 + y 2 + z 2 ' & . rc rh χ0 ! ! rh τ= χ(x)ϕ(ν0 ) n dx / rc

$ rc rh n 0 " $ χ ! $ ! ( rc rh ! ! α τ 1 rc rh α #$ 2 ! " 3 4α ! σ(x) = χ(x) κ(x) = 0 # " ! " 0

rs xi = 0( ϕ(ν) |x − xi | ≤ rs f (x, ν) = . 5 0 |x − xi | > rs

rh

rc

α

rs 3

10

vD −3

10

v0 −3

−10

c

c

r0

R0

ϕ(ν) n

! " " w

# " v0 < 0 # " v0 > 0 vio = v0

" r $l x 0 , r r

$%

" r = |x| v0 r0 w

" r $l 1 |nx| 0 − (l + 1) 3 w(x, n) = v0 $& . r r r "

z '

vrot = v0

R0 R

l

⎛

⎞ y R−1 ⎝ −x ⎠ , 0

$(

" R2 = x2 + y2 ' v0 R0 n = (nx , ny , nz ) w

w = v0

R0 R

l (l + 1)

xy(n2y − n2x ) + nx ny (x2 − y 2 ) R3

.

)*

" + , )

(ν − ν0 )/∆νD = [−4, 6] &* - " 43

" $./ " + τ 0 ' #' Fν τ "

τ = 0.1 τ = 1

" " " " 1 τ = 100

a)

b)

0.1 1 10 100

0 0.5 2

0.034 0.032

0.02

0.02 Fν

Fν

0.030 −0.5

0.00 −4

−2

0

0.0

2

0.5

4

0.00 −6

6

−4

−2

0 (ν−ν0)/νD

2

4

6

−2

0 (ν−ν0)/νD

2

4

6

(ν−ν0)/νD

c)

d)

0.1 1 10 100

Fν

0.02

Fν

0.02

0 0.5 2

0.00 −6

−4

−2

0 (ν−ν0)/νD

2

4

6

0.00 −6

−4

α ! " # !$ # % ! !$ # $ # % ! & # # ' τ & # ! ( & ( & )! # τ = 1 * # τ = 10 *( # # τ = 10& +$ , l ) &

! # $

"

!

l

% &'()

τ =1&

vio

) *

+

"

l

#

τ = 10

,

& )

l = 2

-

!

τ

"

τ ≥ 1

#$

τ

$

' ()*

%&

τ = 100

! % τ

= 10

& l l

= 0 l = 0.5

%

#$ + % $ ,

l = 2

$

% ' & '

&

# '

- ' ' ' #

#

103−4

.α $ τ

≤ 102

' &

' /

•

-

τ ≥1

$

&

0!12

3 '

•

4 ' 4 $ % $ % ,

'

& 5 $

& #

$

! " #$ %

& ' ( ( ) * + , - .-*/ '0 !1# 2( - , 3

$ ,4 , 5 6 78 8!# . #79/ 9$ ': ; :) ; : 4 ;α < : < 6 118 =>8= 8 .9>> / !$ ': ; :) ; : 4 ;α << % : < 6 1=7 #99#!> .9>>9/ 8$ 0& +4 '

* ? @ & - - A B : . ##=/ 1$ - ( ; 6;4 C

' , ? = '

0 : C D .9>>>/ =$ : -* + E F04 % 6 9!= =>#= " . #">/ 7$ + *4 F * F- , A : . ##=/ 4GG G!1#GF

##= "$ & % ( '4 ( << ;α H !#9 "97"!# .9>>9/ #$ & %4 A F- , A : .9>>9/ 4GG G!1#GF

9>>9 >$ -4' ' . #7"/ $ - ) 04 ( :

I A F C D . #"8/ 9$ C -4 , 6 !1> 9 =98 . ##>/ !$ ( ' & % + E C + *4 ( < 0 H !"> 77=7"" .9>> /

!" # $ %"&'( ) " * +

,

α

= 3.6

",

!" - .. /0, 12234

0 (5) 6 $ 7! ( "' ! * 5 ! *)) 5 "' ! ! )5! ' ) " * 5 - .8 .09,.9 12284 3 6!5 5 ! " 6)'5 6 $ 7! ) :(5 " "' ) " " ) $ ; 5 1/<<<4

55(% "5 ) '"( - .02 89<,898

!

! " # ! $ #

% & ' ( &

& % ) * % +

% % *

% , & " %

J(x), x ∈ Oad x Oad ! ! "#$%

& '

M (ζ)xζ = −d(x(ζ)) x(ζ = 0) = x0

"(%

) ! ' • * d = ∇J M = Id • * d = ∇J M = ∇2 J " ! % + " , + % * ' • -(' J ∈ C 1 (Ωad , Ê) • -.' / Jm • -0' ' / ' ∃xm ∈ Ωad , s.t.J(xm ) = Jm • -1' J " J(x) → ∞ |x| → ∞% 2 "(% x0 ∈ Ωad / / Zx J(x(Zx )) = Jm ' 0

0

⎧ ⎨ M (ζ)xζ = −d(x(ζ)) x(0) = x0 ⎩ J(x(Zx0 )) = Jm

".%

345 " / 6% 3 !

Zx0

⎧ ⎨ ηxζζ + M (ζ)xζ = −d(x(ζ)), ˙ = x˙ 0 , x(0) = x0 , x(0) ⎩ J(x(Zx0 )) = Jm

|η| << 1

!"

ζ

x0 = v

#"

x(0) ˙ = v !"" h(v) = J(xv (Zv )) − Jm xv (Zv ) ζ = Zv v A1 (v1 , v2 )

#" !""

• • •

(v1 , v2 ) v ∈ argminw∈O(v2) h(w) h(w) = J(xw (Zw ))−Jm xw (Zw ) # ζ = Zw

w O(v2 ) = → } ∩ Ωad {t− v− 1 v2 , t ∈

Ê

$

A1 % v1 % v2 h2 h2 (v22 ) = minv22 h(v22 ) A1 (v1 , v22 ) A2 (v1 , v22 )

• • •

(v1 , v22 ) v2 ∈ argminw∈O(v22 ) h2 (w) h2 (w) = h(A1 (v1 , w)) O(v22 ) = −−→ {tv1 v22 , t ∈ } ∩ Ωad

v2

Ê

i

hi (v2i ) = minv2i hi−1 (v2i )

Ai−1 (v1 , v2i )

h1 (v) = h(v)

&

Bε (xm )

ε

" h

T

(J(xv (τ )) − Jm )2 dτ,

h(v) = T1

0 < T1 < τ < T

xv (τ ) T1 = T /2 xw (Zw )

[0, Zw ] Jm Jm = −∞

!

"

# $

%

#

$ h(v) = J(xv ) − Jm

v h

! dn = ∇J n !

% 1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

-1

-0.5

0

0.5

1

1.5

2

J(x) = x sin(20x) cos(x) + |x|0.1

Jm = 0 h(v) = J(xv ) v ([−1, 2]) ! " " ! !

h(v) #

J(xv ) − Jm & h(v)

= x sin(20x) cos(x)+|x|0.1 Jm = 0 h(v) [−1, 2]

J(x)

!

"

4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1 4.5-20

-10

0

10

20

30

40

1 4.5 -20

4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 -20

-10

0

10

20

30

40

0 -20

-10

0

10

20

30

40

-10

0

10

20

30

40

! " # ! $ % & ' " ' (

) ) * + & ) ' " '

) ) ! ' " (

) ) * ! " ) #

" # " ! " $ ! "

! "

"

# # $ ! %&' J(x) = 74 + 100(x2 − x21 )2 + (1 − x1 )2 − 400 exp(−10((x1 + 1)2 + (x2 + 1)2 )).

( $ # (1, 1) J(1, 1) = 74& (−0.90, −0.95) J = 34 $ ! )& Jm = 0 J(x)

% I cos(xi − 100) + 10−6 Ii=1 (xi − 100)2 , x ∈ J(x) = 1 − Πi=1 [−600, 600]I I = 5, 10 20 *

4000 3500 3000 2500 2000 1500 1000 500 0 2 1.5 1 -2

0.5 -1.5

-1

-0.5

0 -0.5 0

0.5

1

-1 -1.5

1.5

2 -2

J(x) = 74 + 100(x2 − x21)2 + (1 −

x1 )2 − 400 exp(−10((x1 + 1)2 + (x2 + 1)2 )).

400 RECURSIVE 2 Layers Conjugate Gradient (Recursive 1 layer) 350

300

250

200

150

100

50

0 0

1

2

3

4

5

6

7

! " ! # $ $ ! " %#!

2 1.5 1 0.5 0 600 400 200 -600

I P

-400

0 -200

0

-200 200

400

-400 600 -600

I J(x) = 1 − Πi=1 cos(xi − 100) + 2 I ρ i=1 (xi −100) , x ∈ [−600, 600]

I = 5, 10 20

N=5 N=10 N=20

100000

1

1e-005

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Γi

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V.τ

J(x) : x → q(x) → U (q(x)) → J(x, q(x), U (q(x))).

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∂J ∂J ∂q ∂J ∂U ∂q dJ = + + . dx ∂x ∂q ∂x ∂U ∂q ∂x "

!

#$%

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J

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x

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•

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(

&

) &*

−uxx = 1,

•

n • •

], 1[, u() = 0, u(1) = 0,

u(x) = −x /2 + ( + 1)/2 − /2 2

'

&

(

J

J () = n−1 (nux () + ux()) =

n−1 (−n( + 1) − ). 2

' (

n

˜ ˜ ∼ U (q(x))

U U(x) ! " J x # $ U˜ ˜ /∂x % U

∂ U

& #

˜ (x)( U (q(x)) ). x → q(x) → U ' ˜ U (x)

# U (q(x))/U˜ (x) ˜ U (q(x)) ∂J(U ) ∂J(U ) ∂q ∂J(U ) ∂ U dJ . ≈ + + ˜ (x) dx ∂x ∂q ∂x ∂U ∂x U

(

) ' * U = log(1 + x) J = U 2 dJ/dx = ˜ = x 2U U = 2 log(1 + x)/(1 + x) ∼ 2 log(1 + x)(1 − x + x2 ...) U x = 0 " J ∼ 2U U˜ = 2 log(1 + x) J ∼ 2U U˜ (U/U˜ ) = 2 log(1 + x)(log(1 + x)/x) ∼ 2 log(1 + x)(1 − x/2 + x2 /3...).

)

+ E " J(x) = (

Γi

ds − τ µek |E|

Γo

ds )2 . τ µek |E|

,

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USING THE EXACT GRADIENT USING THE INCOMPLETE GRADIENT

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(x , y ) b

b

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d l

l

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u x

b

SEA

x

! " #

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ktors btors#

∆ϕ#

k b ' #

! ( ! # $ q = (xb , yb , ϕb , ϕl , ϕl )T #

q˙# xb yb # ϕb ϕl # ϕl xl yl d xl = xb + d sin ϕl ) yl = yb − d cos ϕl . % l l0 + u0 ! i * yb li = ⇒ + cos ϕ 1

1

2

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i

i

i

i

i

i

li

sin ϕli y˙ b + yb ϕ˙ l . l˙i = cos ϕli cos2 ϕli i

,

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* & )1 #

k b %

uSEA,i ≥ 0 ∆li =

yb − r + r − uSEA,i − l0 cos ϕli

uSEA,i i

uSEA,i ! i " # utors,1 $ utors,2 % ! ktors $ btors $ & # ! '() ⎛ ⎞⎛ ⎞ x ¨b m 0 0 ml d cos ϕl1 ml d cos ϕl2 ⎜ ⎜ ⎟ 0 m 0 ml d sin ϕl1 ml d sin ϕl2 ⎟ ⎜ ⎟ ⎜ y¨b ⎟ ⎜ ⎜ ⎟ ⎟ 0 0 θb 0 0 ⎜ ⎟ ⎜ ϕ¨b ⎟ = ⎝ ⎝ ml d cos ϕl1 ml d sin ϕl1 0 θl + ml d2 ⎠ 0 ϕ¨l1 ⎠ 2 0 θl + m l d ϕ¨l2 ml d cos ϕl2 ml d sin ϕl2 0 ⎛ ⎞ 2 2 ml d(sin ϕl1 ϕ˙ l1 + sin ϕl2 ϕ˙ l2 ) ⎜ ⎟ −ml d(cos ϕl1 ϕ˙ 2l1 + cos ϕl2 ϕ˙ 2l2 ) − mg ⎜ ⎟ %2 ⎜ ⎟ * ˙ b − ϕ˙ li )) ⎜ ⎟ i=1 (utors,i − ktors (ϕb − ϕli − ∆ϕ) − btors (ϕ ⎝ −utors,1 − ml gd sin ϕl + ktors (ϕb − ϕl − ∆ϕ) + btors (ϕ˙ b − ϕ˙ l ) ⎠ 1 1 1 −utors,2 − ml gd sin ϕl2 + ktors (ϕb − ϕl2 − ∆ϕ) + btors (ϕ˙ b − ϕ˙ l2 ) m m = mb + 2ml utors,1 utors,2 ! + #$ ( &, $ &, -). ('/ x˙ b + (yb + yb tan2 ϕl1 ) ϕ˙ l1 + tan ϕl1 y˙ b = 0.

0

.

! ! (.) 0

m 0 B 0 m B B 0 0 B B ml d cos ϕl ml d sin ϕl 1 1 B @ ml d cos ϕl2 ml d sin ϕl2 1 tan ϕl1

10 1 ml d cos ϕl2 1 0 ml d cos ϕl1 x ¨b C B y¨b C ml d sin ϕl2 tanϕl1 0 ml d sin ϕl1 CB C CB ϕ θb 0 0 0 ¨b C C C B Bϕ 0 θl + m l d 2 0 yb (1 + tan2 ϕl1 ) C ¨l1 C C C B A@ϕ 0 0 θl + m l d 2 0 ¨l2 A 0 yb (1 + tan2 ϕl1 ) 0 0 λ

⎛

⎞ ml d(sin ϕl1 ϕ˙ 2l1 + sin ϕl2 ϕ˙ 2l2 ) + (Fk + Fd ) sin ϕl1 ⎜ ⎟ −ml d(cos ϕl1 ϕ˙ 2l1 + cos ϕl2 ϕ˙ 2l2 ) − mg − (Fk + Fd ) cos ϕl1 ⎜ ⎟ %2 ⎜ ⎟ ˙ b − ϕ˙ li )) ⎟ i=1 (utors,i − ktors (ϕb − ϕli − ∆ϕ) − btors (ϕ =⎜ ⎜ −utors,1 − ml gd sin ϕl + ktors (ϕb − ϕl − ∆ϕ) + btors (ϕ˙ b − ϕ˙ l ) ⎟ 1 1 1 ⎟ ⎜ ⎝ −utors,2 − ml gd sin ϕl + ktors (ϕb − ϕl − ∆ϕ) + btors (ϕ˙ b − ϕ˙ l ) ⎠ 2 2 2 −2 · cos−2 ϕl1 ϕ˙ l1 (y˙ b + yb tan ϕl1 ϕ˙ l1 ) Fk

Fd yb Fk = k ( − l0 − uSEA,1 ) cos ϕl1 tan ϕl1 y˙ b Fd = b ( + yb ϕ˙ l ). cos ϕl1 cos ϕl1 1

slif tof f = l0 + uSEA,1 −

yb =0 cos ϕl1

!

clif tof f = y˙ b > 0.

" ! #

stouchdown = yb − l0 cos ϕl2 = 0. "

$

% & '

ctouchdown = y˙ b + l0 sin ϕl2 ϕ˙ l2 < 0. " ( % (

%

% (

( )

# " % ( ) )

ϕli

ϕl1 ↔ ϕl2

"

) ( %(

&

•

)

% ) &

x˙ contact = x˙ b + l0 cos ϕl,2 ϕ˙ l2 + y˙ b tan ϕl2 + yb ϕ˙ l2 tan2 ϕl2 = 0 •

% &

Htrunk,hip = Θb ϕ˙ b = const. •

*

+

%

Hswingleg,hip = (Θl + ml d2 )ϕ˙ l1 = const.

•

Hrobot,contact = Θb ϕ˙ b − mb (yb − yc )x˙ b + mb (xb − xc )y˙ b +Θl,1 ϕ˙ l,1 − ml (yl,1 − yc )x˙ l,1 + ml (xl,1 − xc )y˙l,1

+Θl ϕ˙ l,2 − ml (yl,2 − yc )x˙ l,2 + ml (xl,2 − xc )y˙l,2 = const. xli yli

xc = xb + l0 sin ϕl2 yc = yb − l0 cos ϕl2 . •

m(x˙ b sin ϕl − y˙ b cos ϕl ) − Fkd = const.

! xb ! "

qred (T ) = qred (0) q(T ˙ ) = q(0) ˙ T # !

$ # ! " % &'( % ! &(! ) # ! " ) *+ # !

, # + ! - ) . C / 0 ! $

% ! &(! ,

+

+ + ! , + 1

Outer optimization loop

Inner optimization loop Stability optimization min φstab modify model parameters (mass, inertia, geometry ...)

Solution of periodic optimal control problem minimize energy for given parameters modify initial values, actuator inputs, cycle time

! " ## $ % ##& # %

min |λmax (C(p))|, p • ! " C " # ! • C $ %" ! & " " ' " " ( ' " ! ) " " $ *+ ! " n + 1 n %

!

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||u||22 dt

min

x,u,T

0

x(t) ˙ = fj (t, x(t), u(t), p) x(τj+ ) = h(x(τj− ))

*+,

gj (t, x(t), u(t), p) ≥ 0

t ∈ [τj−1 , τj ], j = 1, ..., nph , τ0 = 0, τnph = T

#()$ #(-$ #(.$ #(/$

#(0$ #(1$ % 234* #5 6 7 8(9 : 8-9$ ; • • %

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• • •

l0 mb

!

! "#" ! $ !$

!" # $ % &

mb = 2.0 Θb = 0.3465 ml = 0.2622 Θl = 0.182 d = 0.11 l0 = 0.5 ktors = 11.08 ∆ϕl = 0.5

btors = 9.989 k = 606.8 b = 42.48 % '( & T = 0.5476s Tcontact = 0.2533s Tf light = 0.2943s

xT0 = (0.0, 0.4777, −0.1, 0.3, −0.7, 1.240, −2.490, −0.3941, −0.8908, 2.032) xb (0) xb (T ) 0.4637m

)

'

"

uSEA2 xb |λ1 | = 0.6228 |λ2,3 | = 0.8168

|λ6 | = 0.0515

|λ4 | = 0.8168 |λ5 | = 0.5373

|λ7 | = 0.0001 |λ8,9 | = 0.0

! " " # ! $ $ yb ϕl1 % & %'() ) " * x˙ b +'. - '. y˙ b +(. - . ϕ˙ b +'. '. ϕ˙ l1 +/. - '. ϕ˙ l2 +- . . 0 xb ϕb yb ϕl1 ϕl2

+,--. (. −0.046% +0.05% +0.184% −2% +(. - '.

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+

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G(z )

+

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Y (z)

-

YD (z), R(z), U (z), G(z), V (z) Y (z)

u(k) = u(k − p) + ϕe(k − p + γ)

!"

ϕ ! " γ # $ ! " $ % z p U (z) = F (z)[U (z) + z γ Φ(z)E(z)] !$" & !" ϕ Φ(z) F (z) & ' ( # ( R(z) =

F (z)z γ Φ(z) U (z) = p E(z) z − F (z)

!)"

[1 + G(z)R(z)]E(z) = YD (z) − V (z) {z − F (z)[1 − z γ Φ(z)G(z)]}E(z) = [z p − F (z)][YD (z) − V (z)] p

F (z)

! !

z p − F (z)[1 − z γ Φ(z)G(z)] = 0 #

F (z)

"

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!

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F (z)

% &$ '

( !

u(k) = α1 u(k − p) + α2 u(k − 2p) + ϕ[α1 e(k − p + γ) + α2 e(k − 2p + γ)]

!

αi

)

*

+ ! %

, +

R(z) =

-

U (z) F (z)z γ Φ(z)(α1 z p + α2 ) = 2p E(z) z − F (z)(α1 z p + α2 )

.

{z 2p − F (z)[1 − z γ Φ(z)G(z)](α1 z p + α2 )}E(z) = [z 2p − F (z)(α1 z p + α2 )][YD (z) − V (z)]

/

F (z)

z − F (z)[1 − z Φ(z)G(z)](α z + α ) = 0 ! 2p

γ

1

p

2

" # ! $ % & ' ( u(k) = α1 u(k − p) + α2 u(k − 2p) + · · · + αN u(k − N p)

$)

+ ϕ[α1 e(k − p + γ) + α2 e(k − 2p + γ) + · · · + αN e(k − N p + γ)]

R(z) =

F (z)z γ Φ(z)[α1 z (N −1)p + α2 z (N −2)p + · · · + αN ] U (z) = Np E(z) z − F (z)[α1 z (N −1)p + α2 z (N −2)p + · · · + αN ]

$$

{z N p − F (z)[1 − z γ Φ(z)G(z)][α1 z (N −1)p + α2 z (N −2)p + · · · + αN ]}E(z) = {z N p − F (z)[α1 z (N −1)p + α2 z (N −2)p + · · · + αN ]}[YD (z) − V (z)]

$*

z N p − F (z)[1 − z γ Φ(z)G(z)][α1 z (N −1)p + α2 z (N −2)p + · · · + αN ] = 0

%$

+ # , -*). /, & 0 F (z) z E(z) = [1 − z Φ(z)G(z)]E(z) 0 1 2 E (z) p

γ

h

z p Eh (z) = F (z)[1 − z γ Φ(z)G(z)]Eh (z)

M (z) ≡ F (z)[1 − z γ Φ(z)G(z)]

zp

M (eiωT ) = F (eiωT )[1 − (eiωT )γ Φ(eiωT )G(eiωT )] < 1

ω

Ì

Eh (z)

M (eiωT )

! "

Eh (z)

# # $ "

$ ! %&

Æ %&

'

z N p Eh (z) = M (z)[α1 z (N −1)p + α2 z (N −2)p + · · · + αN ]Eh (z)

(

M (z)[α1 z (N −1)p + α2 z (N −2)p + · · · + αN ] $

Æ

) $

Æ

*

*

+

Þ

α1 z (N −1)p + α2 z (N −2)p + · · · + αN

≤ α1 |z|(N −1)p + α2 |z|(N −2)p + · · · + αN

α1 (eiωT )(N −1)p + α2 (eiωT )(N −2)p + · · · + αN

≤ α1 + α2 + · · · + αN = 1

,

- $

Eh (z)

Æ

Æ

R(z)G(z) = −1 Ô

!"# $ %

& '

& F (z)

& ( ) * +,- z p P (z) ≡ 1 − z −p F (z)[1 − z γ Φ(z)G(z)] = 0 +./

0 Þ 1

& 2 & 3 F (z), Φ(z) G(z) P (z)

4 G(z) & Ô 0 & 0 &

2 P (z) = P (−1) + - P (z) Þ

4

P (z) +.- P (z) = 1 − P ∗ (z) +5P ∗ (z) = z −p M (z)

∗ iωT iωT −p

P (e ) = e M (eiωT ) = M (eiωT )

+!" +,- & P (z) Þ '

! z N p P (z) = 1 − M (z)(α1 z −p + α2 z −2p + · · · + αN z −N p ) = 0

"

P (z) ∗

P ∗ (z) = M (z)(α1 z −p + α2 z −2p + · · · + αN z −N p )

""

#

|P ∗ (z)| = |M (z)| α1 z −p + α2 z −2p + · · · + αN z −N p

≤ |M (z)| (α1 |z|−p + α2 |z|−2p + · · · + αN |z|−N p ) = |M (z)|

"!

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dx = w(t, x, β) dt x, w ∈ Rm , β ∈ Rp $

x w = M (t, β)x + f (t).

#

! "

%

& '

(

! )

yi = ϕT x(ti ) + εi , i = 1, 2, · · · , n,

ϕ

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x(t, β, b)

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F (β, b) =

n

ri (ti , β, b)2 ,

0

i=1

ri = yi − ϕT x(ti , β, b).

1

! " B1 , B2 b !

B1 x(0) + B2 x(1) = b, dx = M (t, β)x + f (t). dt B1 , B2

! " #

$ " $ "

t = 0

t=1

% #

B1 = I, B2 = 0.

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min b,β

n

(yi − ϕT x(ti , β, b))2 .

'()

i=1

β, b

#

* % # + " # ! " &

d∆β dt = M ∆β + ∇β M x, B1 ∆β (0) + B2 ∆β (1) = 0, d∆b dt

= M ∆b , B1 ∆b (0) + B2 ∆b (1) = I,

∆β =

∂x ∂x , ∆b = . ∂β ∂b

& $ " , '() - # " # ! ! " " # # . $ " #

min β

n

(yi − ϕT x(ti , β))2 ,

'/)

i=1

0 "

xi+1 − Xi (ti+1 , ti )xi = vi , i = 1, 2, · · · , n − 1, Xi (t, ti ) # " '+) #!

vi (t)

'1)

dXi = M Xi , Xi (ti , ti ) = I, dt ti+1 vi = Xi (ti+1 , u)f (u)du.

ti

n

! " #

" " $ "

m

!

% " ! & %

B1 , B2 %

' # #

! ( )

" ' &

* # + % ! ( )

"" , ""

" & %

- ./0 &

A0i xi+1 + Bi0 xi = c0i

+

" 1

0 A0i−1 0 c0i−1 Bi−1 0 Bi0 A0i c0i

1 Bi/2 0 A1i/2 c1i/2 → . Vi1 −I Wi1 wi1

2

!

" "

xt = Vt x(0) + Wt x(1) + wt ,

3

Gk1 x(0) + Gk2 x(1) = ck1 .

n = 2k

!

B1 , B2

B1 B2 Gk1 Gk2

" #

x(0), x(1)

$

Gk1 , Gk2

% &'( ) * ) *

+

min β

n 2 yt − ϕT (Vt x(0) + Wt x(1) + wt )

t=ti ,i=1

,

Gk1 x(0) + Gk2 x(1) = ck1 .

-

. / $0

m

1

+

•

2 * +

V (0) = I, V (1) = 0, W (0) = 0, W (1) = I, w(0) = 0, w(1) = 0. •

3

Vt , Wt , wt G1 , G2 , c $ )

. Ct , 4 * +

Ct ←

R1 (t) 0 Ct R21 (t) R2 (t)

5

6 7

R2−1 R21 .

7

R1

•

−1 k k −1 k G1 = X(1, 0), Gk2 c1 = v1 . G2

I X(ti+1 , ti ) C= . I −X(t , t ) i+1

i

δ = ti+1 − ti !

! " # δ $ % & ' d2 dt2

V X −1 = 0, ⇒ V = X(t, 0)(1 − t), W = X(t, 1)t. W

( " )* V W

R2−1 R21 = S1 = δS + O(δ 2 ).

C ← RC

d2 V dV dM 2 + M + 2 (S − M ) − 2SM − V = 0. dt2 dt dt

)

# +# # " ( C T RT RC = I

" # δ S=

M + MT 2

))

)

d + MT dt

d −M dt

V =0 W

) # Y V, W

d Y Y M I =N , N= . Z −(2S − M ) dt Z

),

d w w f =N + . z 0 dt z x 2m × 2m V W w

dx − Mx − f, dt dV dW dw = − M V x(t1 ) + − M W x(tn ) + − Mw − f, dt dt dt = ZV (t)x(t1 ) + ZW (t)x(tn ) + z(t).

0=

! " t

ZV , ZW , z t = t2 t = t1 Z(t2 , t1 ) Z dZ = − (2S − M ) Z. dt

# $ %

&' ( & $ ) * +, -./ 0 ( -!/ %

&' 1

m

1 • 1 yt ∈ R , t ∈ T • 1 f (yt , η(x, t)) • 1 η(x, t) : Rp × T → Rq x 0 $ xT = arg min KT (x), !2 KT (x) = − Lt , !. t∈T

Lt = log f (yt , η(x, t)).

!3

• • n = |T | m = dim x • J = ∇2 K(x), h = −J

−1

T

(x)∇K(x) ,

x → x + h.

!

" # xˆ , ∇K(ˆx) = 0 J (ˆ x) # " $ % & ∇2 K(x) # ' () * ∇2 K(x) +

1 2 E ∇ K(x) , n 1 h = −I(x)−1 ∇K(x)T , n x → x + h. I=

"

# ,- xˆn x∗ • • E{∇2 K(x)} = E{∇K(x)T ∇K(x)}

.

I

+

& # ,- limn→∞ In

/ 0 In +

I 1 1 E{∇2 Kn } = E{∇LTi ∇Li } n n i 1 1 ∇LTi ∇Li − E{∇LTi ∇Li } + = − ∇LTi ∇Li n i n i 1 → ∇LTi ∇Li , n → ∞. n i

In ∇Kn h < (=)0 Kn In Kn ∇Kh 1 ! ∇Kh < − I " # K $ n

% & xi+1 = F(xi );

1 F(x) = x − In (x)−1 ∇Kn (x)T n

' xˆ n $(∇F(ˆ xn )) < 1

∇Kn (ˆxn ) = 0 1 ∇F(ˆ xn ) = I − In (ˆ xn )−1 ∇2 Kn (ˆ xn ) n 1 2 −1 (In (ˆ = (In (ˆ xn )) xn ) − ∇ Kn (ˆ xn )) n = ∇F(x∗ ) + O(ˆ xn − x∗ ), , n → ∞

( ∇F(x∗ ) = o(1), n → ∞ x∗ ⇒ $(∇F(ˆ xn )) → 0, n → ∞

)* + &

min Kn ; Cx = d, C : Rp → Rm , x

(C) = m.

∇Kn = λT C

λ ! " n → ∞ 1 1 {∇Kn − E{∇Kn }} + E{∇Kn } = (λ/n)T C n n

1

−

E{∇L(y, x, t)}dω(t), 0

ω # $ " −

1

E{∇L(y, x, t)}dω(t) = λ∗T C

0

Cx = d

%

λ∗ = limn→∞ λ/n" x = x∗ , λ∗ = 0.

&' & !$ # " ( )

*) * # " + , , , " $ " -

# ) " ! min K(x); c(x) = 0. x

. ! l(x, λ) = K(x) + λT c(x).

! Bk ' ∇2x l(xk , λk )

1 min∇K(xk )d + dT Bk d, S = {d; c(xk ) + A(xk )d = 0} d∈S 2

xk+1 := xk + γdk . ! "

λ

T λk+1 = −A+ k (∇Kk + Bk dk ))

#$ %

Bk

"

& )

%n

∂ 2 ri i=1 ri ∂xj ∂xk ' (

%n−1 i=1

2

ci λTi ∂x∂j ∂x k

$ % #$ % * + "

λi → 0'

"

, " " " ' - , &

dλ = −∇x w(t, x, β)T λdt + σϕdω. * " (

. " " " (% ( " " % /! #$ " , 0 1! " $ %

#$ . "

⎤ 1 − β1 cos(β2 t) 0 1 + β1 sin(β2 t) ⎦ 0 β1 0 M (t, β) = ⎣ 1 + β1 sin(β2 t) 0 1 + β1 cos(β2 t) ⎡ ⎤ −1 + 19(cos(2t) − sin(2t)) ⎦ −18 f (t) = et ⎣ 1 − 19(cos(2t) + sin(2t)) ⎡

x(t) = et e

x(t) + σrnd rnd σ = 5., 1., .01 234 [19, 2]

.

" ' " " "

σ

5

2 + 1 27 + 1 210 + 1

!

"

#

$ %

η

τ

&

n (yi − η(ti ))2 + τ

min

η(t)

1 0

i=1

dk η dtk

2 dt.

'(

" ) *(+&

η(t) = E {y(t)|y1 , y2 , · · · , yn , λ} , √ dω dk η = σ λ . dtk dt

∞

λ = 1/τ % λ

,-

η(t) → E{y(t)}, n →

.

#

min η

"

τ

n

(yi − η(ti )) + τ 2

1

2

(Mk η) dt.

,/

0

i=1

η Mk %

Mk

" ) " 0 */-+ *//+ 1 %

Mk

dx = Mk x dt ,-

b = ek

η(t) = E {x1 (t)|y1 , y2 , · · · , yn , λ} , √ dx = Mk xdt + σ λbdω.

ϕT x(ti )

yi = M (t, β) b

X(t, ξ)

!

"!

#

√ xi+1 = X(ti+1 , ti )xi + σ λui , ti+1 ui = X(ti+1 , s)bdω(s),

ti

∼ N (0, σ 2 R(ti+1 , ti )), ti+1 R(ti+1 , ti ) = λ X(ti+1 , s)bbT X(ti+1 , s)T ds.

$ %

ti

& ! #! # '

x(t|n)

! ( ) # )

& !

xi|i = E {x(ti )|y1 , y2 , · · · , yi , λ}

σ 2 Si|i

'

# # #

*

ti ≤ t ≤ ti+1

x( t|n) = X(t, ti )xi|i + A(t, ti ) xi+1|n − xi+1|i , −1 A(t, ti ) = X(t, ti )Si|i Xi + Γ (t, ti ) Si+1|i ,

Γ (t, ti ) = R(t, ti )X(ti+1 , t)T . &

+ , -

x1|0 # . x1|0 = 0, S1|0 ↑ ∞ 0 #

! /! #

#! # ) ! # # ! # 1&

# 2 ! & 3 "! * 4! # 5

& # /! #

min rT1 V −1 r1 + rT2 R−1 r2 , x ⎤ ⎡ T ϕ1 ⎥ ⎢ ϕT2 ⎥ ⎢ ⎥ ⎢ · · · · · · ⎥

⎢ T ⎥ ⎢ y r1 ϕ n ⎥ ⎢ x− , =⎢ ⎥ 0 r2 −X I 1 ⎥ ⎢ ⎥ ⎢ −X2 I ⎥ ⎢ ⎦ ⎣ ··· ··· Xn−1 I

$6

$

V = σ2 I R = σ2 {R1, R2 , · · · , Rn−1 } Ri = R(ti+1 , ti ) R ϕ, b

x(t|n)

dx(t|n) = M x(t|n) + bbT X(ti+1 , t)T v, dt −1 v = Si+1|i (xi+1|n − xi+1|i ).

ti ! bbT X(ti+1 , t)T !

X(ti, t)" dj X(ti , t) = X(ti , t)Pj (M ), dtj dPj−1 − M Pj−1 , j = 1, 2, · · · . P0 = I, Pj = dt

$

#

ti %&#'

dj x(t|n) dtj

( )

Pj−1 (M )T ϕ j < k * +

ϕT Pj−1 (M )b = 0, j = 1, 2, · · · .

ϕ = e1 , b = ek ,

⎡

0

⎢ ⎢ M =⎢ ⎢ ⎣

1 0

−mk −mk−1

⎤

⎥ 1 ⎥ ⎥ ··· ··· ⎥ 0 1 ⎦ · · · · · · −m1

!

R(ti+1 , ti) , R(ti+1 , ti ) + δ ! +

R(t + δ, t) = t

t+δ

(s − (t + δ))i+j i,j

i!j!

Pi (M )bbT Pj (M )T ds

- δ → 0 & ) R(ti+1 , ti)" πk = λδbT b + O(δ 2 ).

! → b

R(ti+1 , ti ) bT Pj−1 (M )T ϕ = 0, j = 1, 2, · · · , k − 1,

→ ϕ π1 =

(bT Pk−1 (M )T ϕ)2 δ 2k−1 λ + O(δ 2k ). ((k − 1)!)2 ϕT ϕ 2k − 1

!"# $%&'( ) *

+,# $%-'($%%' .*

/ * 0

ζi = yi −ϕT xi|i−1 0 (

σ 2 Vi Vi = (1+ϕT Si|i−1 ϕ) 1 ζ2 log σ 2 + log Vi + 2i . σ Vi i + 1 σ 2 N ≤ n # σ ˆ2 =

1 ζi2 . N i Vi

* *

ζ 2 i GM L = log Vi + N log . Vi i i 2 2 %n T /n i=1 yi − ϕ xi|n GCV = , 2 {{I − T }/n} T 3 ) yi ϕT xi|n 4 λ 5 O(n2 ) ( +, O(n)

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s.t.

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xR (P )

x = xR z ∗ (P ) z ∗ z ∗ z ∗ (P ) P ∈ P z ∗ (P ) ! z ∗ " x(k) x

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* + , % ! - , * , '

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−

+

+

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2

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∀ i ≥ 2, p, ω , 3

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Nifp ,ω=1 = Nifp ,ω=2 = . . . = Nifp ,ω=Ω

∀ i, fp , p | i ≤ 3 ,

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F : Ên → Êm m ≥ n

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t

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(l = 1, . . . , L)

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l=1

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.

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2

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G := G(x) := F (x)T F (x), S := S(x) := F (x) ◦ F (x) :=

m

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2

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.

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G

/

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2

Z = [ Z1 | Z2 ]

Λ1 = diag(λ1 , . . . , λn−q ) ∈ Rp×p Λ2 = diag(λn−q+1 , . . . , λn ) ∈ Rq×q

G

1

Sˆ11 Sˆ := Z T SZ = [ Z1 | Z2 ]T S [ Z1 | Z2 ] = Sˆ21

Sˆ12 Sˆ22

2

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B := Z

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#

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∀u, v ∈ im Z2 .

) B S im Z2 G

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q(q − 1)/2

2 Dij := F (x)[zp+i , zp+j ] =

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i = j i < j

, , δ > 0 x = xk - xopt f .

H(xopt ) / 0 / ARN

g(xopt ) = 0 G(xopt ) 1 ARN (xopt ) = G(xopt ) + B(xopt )

2 .30 .40 h {zi } G h = Zu u = Z T h = [ uu12 ] ˆ = uT1 Λ1 u1 + uT2 Λ2 u2 , hT Gh = uT Gu ˆ = uT1 Sˆ11 u1 + 2uT1 Sˆ12 u2 + uT2 Sˆ22 u2 , hT Sh = uT Su ˆ = uT Sˆ22 u2 . hT Bh = uT Bu 2

.10

|uT1 Sˆ11 u1 + 2uT1 Sˆ12 u2 + uT2 Sˆ22 u2 | = |hT Sh| ≤ %GN (hT Gh) = %GN (uT1 Λ1 u1 + uT2 Λ2 u2 )

u1 = 0

∀ u1 , u2 .

|hT Bh| = |uT2 Sˆ22 u2 | ≤ %GN (uT2 Λ2 u2 ).

hT ARN h = hT (G + B)h = uT1 Λ1 u1 + uT2 Λ2 u2 + uT2 Sˆ22 u2 ≥ uT1 Λ1 u1 + uT2 Λ2 u2 − %GN (uT2 Λ2 u2 ) = uT1 Λ1 u1 + (1 − %GN ) uT2 Λ2 u2 ≥ (1 − %GN )(hT Gh)

0 ≤ %GN < 1

λi (ARN ) ≥ (1 − %GN )λmin (G) > 0 & '

sRN = −A−1 RN g xopt !

"

#

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M := T (xopt ) = −A(xopt )−1 (H(xopt ) − A(xopt ))

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BA SA MKL

160

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140 120 100 80 60 40 20 0

Task graphs (ranges from 1,000 to 50,000 vertices)

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! " " #" $ $ % " & a) Comparision of edge−cut of BA, MKL and SA

Edge−cut Ratio

BA/MKL BA/SA 1

0 Task graph (range from 1,000 to 50,0000 vertices)

b) Comparision of load imbalance of BA, MKL and SA

Load−Imbalance Ratio

1 BA/MKL BA/SA

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Load Imbalance (%)

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j=k

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{Mk } {Mk } Mk+1 Mk ) limk→+∞ Mk = 0, ∗ ∩∞ k=1 Mk = {x }.

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Ê

γ(M ) := inf{F (x, y)| G(x, y) ≤ 0, x ∈ M ∩ C, y ∈ D}.

#

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0

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∗

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inf{F (x∗ , y)| G(x∗ , y) ≤ 0, y ∈ D} > β ∗ .

(

) ∗

∗

sup {F (x , y) + !λ, G(x , y) } =

λ∈

Ê

m +

F (x∗, y) if G(x∗ , y) ≤ 0 +∞ otherwise

inf{F (x∗ , y)| G(x∗ , y) ≤ 0, y ∈ D} = inf sup {F (x∗ , y) + !λ, G(x∗ , y) }. y∈D λ∈

Ê

m +

*

D, inf sup {F (x∗ , y) + !λ, G(x∗ , y) } = sup inf {F (x∗ , y) + !λ, G(x∗ , y) }.

y∈D λ∈

Ê

m +

λ∈

Ê

m +

y∈D

sup inf {F (x∗ , y) + !λ, G(x∗ , y) } > β ∗ .

y∈D λ∈Rm +

λ˜ ˜ G(x∗ , y) > β ∗ . min{F (x∗ , y) + !λ, y∈D

˜ (x, y) → {F (x, y)+ !λ, G(x, y) } ! ! y ∈ D, " Uy Ên x∗ " Vy Êp y ˜ G(x , y ) > β ∗ F (x , y ) + !λ,

∀x ∈ Uy ∩ C, ∀y ∈ Vy .

# " Vy , y ∈ D D ! S ⊂ D " Vy , y ∈ S, D. $

U = ∩y∈S Uy y ∈ D y ∈ Vy y ∈ S, x ∈ U ⊂ Uy

˜ G(x, y) > β ∗ F (x, y) + !λ,

∀x ∈ U ∩ C, ∀y ∈ D.

Mk ⊂ U % k, " ∩k Mk = {x∗ }. & " ' sup inf{F (x, y) + !λ, G(x, y)| x ∈ Mk ∩ C, y ∈ D} > β ∗ .

λ∈

Ê

m +

β(Mk ) > β ∗ , ( ) *( " α < +∞ inf{F (x∗ , y)| G(x∗ , y) ≤ 0, y ∈ D} ≤ max{F (x, y)| x ∈ C, y ∈ D} < +∞.

$ D " " " ( " Ì +

D , D {y ∈ D| (∃x ∈ C) G(x, y) ≤ 0} ⊂ D ⊂ D.

D D β(M ) = sup inf{F (x, y) + !λ, G(x, y) | x ∈ M ∩ C, y ∈ D}. λ∈

Ê

m +

min{F (x, y)| Gi (x, y) ≤ 0 (i = 1, . . . , m), x ∈ C, y ∈ D}

(P)

& '

sup{F (x, y)| x ∈ C, y ∈ D} ≤ α.

∗ ! λ∗ ∈ Êm + inf x∈C {F (x, y)+!λ , G(x, y) } → +∞ y ∈ D, y → +∞. ! D D∗ := {y ∈ D| ϕ(y) ≤ α}, ϕ(y) := inf {F (x, y) + !λ∗ , G(x, y) } x∈C

" # D∗ $ #

{yν } ⊂ D∗

y ν → +∞ # ϕ(y) → +∞ # ! % # y ∈ D G(x, y) ≤ 0 x ∈ C inf x∈C {F (x, y) + !λ∗ , G(x, y) } ≤ inf x∈C {F (x, y)| G(x, y) ≤ 0, y ∈ D} ≤ α, ϕ(y) ≤ α, '

$ y ∈ D∗ . $ D & &

C Rn , D Êp F (x, y), Gi (x, y), i = 1, . . . , m x y x. ! "

# x∗ ∈ C y ∗ ∈ D Gi (x∗ , y ∗ ) < 0, i = 1, . . . , m.

# '

( x∗ ∩k Mk = {x∗ }.

! {F (x∗ , y) + !λ∗ , G(x∗ , y) ≥ inf x∈C {F (x, y) + !λ , G(x, y) } → +∞ y ∈ D, y → +∞. ) $ F (x∗ , y ∗ ) + !λ, G(x∗ , y ∗ ) → −∞ λ → +∞. $ & (y, λ) → F (x∗ , y) + !λ, G(x∗ , y)

&$ ∗

inf y∈D supλ∈Êm {F (x∗ , y) + !λ, G(x∗ , y) } + = supλ∈Êm inf y∈D {F (x∗ , y) + !λ, G(x∗ , y) }. + ˜∈ $ ( & λ

Êm+

˜ G(x∗ , y) } > β ∗ . inf {F (x∗ , y) + !λ,

y∈D

(D) ∗

˜ G(x , y) !λ,

D.

y → F (x∗ , y) +

˜ G(x∗ , y) } > β ∗ . min {F (x∗ , y) + !λ,

y∈vert(D)

y ∈ vert(D), ˜ G(x∗ , y) x∗ U (y) n y → F (x∗ , y)+!λ, ∗ x ˜ G(x, y) > β ∗ ∀x ∈ U (y). F (x, y) + !λ,

Ê

U = ∩y∈vert(D) U (y)

˜ G(x, y) > β ∗ F (x, y) + !λ, !

k

∀x ∈ U, ∀y ∈ D.

Mk ⊂ U :

˜ G(x, y) > β ∗ F (x, y) + !λ,

∀x ∈ Mk ∩ C, ∀y ∈ D,

˜ G(x, y) | x ∈ Mk ∩ C, y ∈ D} > β ∗ , β(Mk ) = sup inf{F (x, y) + !λ, λ∈

"

Ê

m +

β(Mk ) 0 β ∗ . & '

# $ %&&

α

'

( )

min{f (x)| gi (x) ≤ 0 (i = 1, . . . , m), x ∈ C}

(SP )

C Rn , f, gi : Rn → R, i = 1, . . . , m. D = {y ∗ } ⊂ Rm F (x, y) ≡ f (x), Gi (x, y) ≡ gi (x) ∀y ∈ Rm * * D. $ %&& α + M1 ⊃ C, M ⊂ M1

β(M ) = sup inf{f (x) + λ∈

Ê

m +

m

λi gi (x)| x ∈ M ∩ C}.

i=1

$ ! ,

f (x), gi(x), i = 1, . . . , m, Mk ⊂ Mk x∗ Mk ν

ν

x∗ x∗ ∈ C lim β(Mk ) = inf{f (x)| gi (x) ≤ 0 (i = 1, . . . , m), x ∈ C}. lim β(Mk ) < +∞, x∗ ∗ x ∈ C x∗ ∗ gi (x ) ≤ 0, i = 1, . . . , m. gi0 (x∗ ) > 0 i0 ∈ {1, . . . , m}, gi0 (x), 1 ∗ ∗ W x gi0 (x) > ρ := 2 gi0 (x ) ∀x ∈ W. k Mk ⊂ W β(Mk ) = supλ∈Êm inf x∈Mk ∩C {f (x) + + %m i=1 λi gi (x)} ≥ supλi0 ≥0 inf x∈Mk ∩C {f (x) + λi0 ρ} = +∞, ! β(Mk ) < +∞. & '

" # $%

& '()

F (x, y), Gi (x, y)

'

∗ f[r,s] = min{!c(x), y | A(x)y − b(x) ≤ 0, r ≤ x ≤ s, y ≥ 0}

Ê

Ê

x ∈ Rn , y ∈ Rp , c : Rn → Rp , A : n → m×p , b : {x| r ≤ x ≤ s} ⊂ Rn . " '()

F (x, y) = !c(x), y ,

G(x, y) = A(x)y − b(x),

y

$#%)*

(PL)

Êm → Rp, [r, s] :=

p C = [r, s], D = R+ .

M ⊂ [r, s]

β(M ) = sup inf inf {!c(x), y + !u, A(x)y − b(x) } u≥0 x∈M y≥0

'#+)

= sup{−!b(x), u + inf inf {!c(x) + (A(x))T u, y }} x∈M y≥0

u≥0

= sup{−!b(x), u + h(u)} u≥0

h(u) =

0 if (A(x))T u + c(x) ≥ 0 ∀x ∈ M −∞ otherwise

,

β(M ) = sup{−!b(x), u | (A(x))T u + c(x) ≥ 0 ∀x ∈ M }. u≥0

'#-)

A(x) = [aij (x)] ∈ Êm×p, aij (x), cj (x), bi (x) α := sup{!c(x), y | x ∈ [r, s], y ≥ 0} < +∞

Ê Ê

∗ T ∗ ∗ (∀x∗ ∈ [r, s]) (∃u∗ ∈ m + ) (A(x )) u + c(x ) > 0; p ∗ ∗ ∗ ∗ ∗ (∀x ∈ [r, s]) (∃y ∈ + ) A(x )y − b(x ) < 0; α {Mkν }

x∗ ∈ C

inf{!c, y | A(x∗ )y ≤ b(x∗ ), y ≥ 0}

y ∗ ∈ Êp+ (x∗ , y ∗ ) ! !

T

∗

minx∈U∩C (A(x)) u + c(x) > 0.

U

x∗

inf x∈U∩C [!c(x), y + !u∗ , A(x)y − b(x) ] = inf x∈U∩C [!c(x) + (A(x))T u∗ , y − !u∗ , b(x) ] → +∞ y ≥ 0, y → +∞,

' &

aij (x), c(x) ≡ c, b(x) ≡ b,

!" # $ % $ & $ !'" ( )

* # *

a1 (x), . . . , am (x)

U x∗

A(x), 0 ∈ intconv{a1 (x), . . . , am (x), c(x)} ∀x ∈ U $ u∗ # (A(x∗ ))T u∗ + c(x∗ ) > 0 u∗ > 0, (A(x))T u∗ + c(x) > 0

x + x∗ ).

W, r > 0, , W ⊂ conv{a1 (x), . . . , am (x), c(x)} ∀x ∈ U. - x {a1 (x), . . . , am (x), c(x)} {y| A(x)y ≤ e, !c(x), y ≤ 1} e = (1, . . . , 1) ∈ m ),

x ∈ U

1/r ,

x ∈ U {y| A(x) ≤ b(x), !c(x), y ≤ α}

. {y| A(x) ≤ b(x), !c(x), y ≤ α, x ∈ U }

Ê

$ #

$ !1"2

/0

min!c, x + !d, y m G0 + yj Gj 1 0 L0 +

j=1 n

xi Li0 +

i=1

m

yj L0j +

j=1 n

n m

xj yj Lij ≺ 0

i=1 j=1

x ∈ X = [p, q] ⊂ R , y ∈ Rm +

x, y

G0 , Gj , L0 , L0i , Lj0 , Lij G 1 0, L ≺ 0

G

⎡

A00 (x) = ⎣ L0 +

%Gn0 i=1

!x, c

A B

L

d

A 0 0 B

⎡

⎤

xi Li0 ⎦ , Aj0 (x) = ⎣ L0j + d

Q00

Gj % n i=1

⎤ xi Lij ⎦ ,

dj

d

⎡ ⎤ 0 = ⎣0⎦ . 1 d

! "#$

min{t| A0 (x, p, q) +

m

yj Aj (x, p, q) 1 tQ, y ≥ 0, x ∈ X}

j=1

Aj0 (x) Q00 , Q= , Q01 = 0 Aj1 (x, p, q) d Q01 d ⎤ ⎡ (x1 − p1 )Gj ⎢ (q1 − x1 )Gj ⎥ ⎥ ⎢ ⎥ , j = 0, 1, . . . , n. ··· Aj1 (x, p, q) = ⎢ ⎥ ⎢ ⎣ (xn − pn )Gj ⎦ (qn − xn )Gj d

Aj (x, p, q) =

%

&'%

! ( ) ( * ! * "#$ +* , -&&

α

* * * ! * ) ( * *

(∀x ∈ X)(∃Z1 2 0)

Tr(Z1 Q00 ) = 1, Tr(Z1 Aj0 (x) > 0, j = 1, . . . , m

⎧ ⎤⎫ ⎡ m ⎨ ⎬ max min yj Aj (x, p, q) − tQ)⎦ t + Tr ⎣Z(A0 (x, p, q) + Z0 t∈R,y≥0,x∈M ⎩ ⎭ j=1

!" max {t| Tr(ZA0 (x, p, q)) ≥ t, Tr(ZAj (x, p, q)) ≥ 0 ∀x ∈ vertX, j = 1, . . . , m, Tr(ZQ) = 1, Z 2 0}

X # X. $ #

$ % &

' ( % # # ) * & + ) # # " # , #& # # # - . / ∗ f[r,s] = min{!c(x), y + c0 (x)| A(x)y + B(x) ≤ b, r ≤ x ≤ s, y ≥ 0}

. (GPL) x ∈ Rn , y ∈ Rp , c : Rn → Rp , c0 : Rn → R, A := Rn → Rm×p , B : n R → Rm×n , b ∈ Rm , [r, s] ⊂ Rn+ . A(x) = [aij (x)], i& B(x) Bi (x), # min s.t.

p j=1 p j=1

yj cj (x) + c0 (x) yj aij (x) + Bi (x) ≤ bi (i = 1, . . . , m)

y ≥ 0,

r ≤ x ≤ s.

- # )

(GPL)

F (x, y) = !c(x), y +c0 (x), G(x, y) = A(x)y+B(x)−b, C = [r, s], D = {y ≥ 0}.

(P L)

B(x) c0 (x) = !c0 , x , c0 ∈ Rn , B(x) = Bx B = [bik ] ∈ Rm×n . j, cj (x), aij (x), i = 1, . . . , m, [r, s], [r, s]. j, cj (x), aij (x), i = 1, . . . , m x c0 (x)

(GP L)

ϕ∗[r,s] = sup inf{!y, c(x) +!c0 , x +!λ, A(x)y+Bx−b | x ∈ [r, s], y ≥ 0}.

!

λ≥0

"

λ≥0

inf{!y, c(x) + !c0 , x + !λ, A(x)y + Bx − b | x ∈ [r, s], y ≥ 0} = −!b, λ + inf inf {!Bx, λ + !c0 , x + !c(x) + (A(x))T λ, y } x∈[r,s] y≥0

= −!b, λ + h(λ)

h(λ) =

inf x∈[r,s] [!Bx, λ + !c0 , x ] −∞ .

q ∈

c(x) + (A(x))T λ ≥ 0 ∀x ∈ [r, s], #

Ên :

inf !q, x = !q, r + max{!r − s, t | t ≥ 0, t ≥ −q}.

r≤x≤s

$ min{!q, % x | r ≤ x ≤ s} = min{!q, r + !q, x − r | 0 ≤ x − r ≤ s − r} = !q, r + qi <0 qi (si − ri ) = !q, r + max{!r − s, t | t ≥ 0, t ≥ −q}. ' & %

inf r≤x≤s {!Bx, λ + !c0 , x } = !B T λ + c0 , r + max{!r − s, t | t ≥ 0, t ≥ −B T λ − c0 }. &

∀x ∈ [r, s]

j ' A Aj , !A(x), λ + c(x) ≥ 0

!Aj (x), λ + cj (x) ≥ 0

∀x ∈ [r, s], ∀j = 1, . . . , p.

( J+

j

cj (x), aij (x), i = 1, . . . , m J−

j

cj (x), aij (x), i = 1, . . . , m

J + ∪ J − = {1, . . . , p}. j = 1, . . . , p, !Aj (x), λ + cj (x) ≥ 0 ∀x ∈ [r, s] m ⇔ λi aij (x) + cj (x) ≥ 0 ∀x ∈ [r, s] i=1

%m λi aij (r) + cj (r) ≥ 0 if j ∈ J+ ⇔ %i=1 m i=1 λi aij (s) + cj (s) ≥ 0 if j ∈ J−

ϕ∗[r,s] = sup{−!b, λ + inf {!Bx, λ + !c0 , x | x∈[r,s]

!Aj (r), λ + cj (r) ≥ 0 (j ∈ J+ ); !Aj (s), λ + cj (s) ≥ 0 (j ∈ J− ), λ ≥ 0}

i B Bi . ! " " " " !c0 , r + max{!r − s, t +

m

λi [!Bi , r − bi ]}

i=1

%m

− λi Bi − t ≤ c0

%i=1

− m λi aij (r) ≤ cj (r) (j ∈ J+ )

%i=1 m −

− i=1 λi aij (s) ≤ cj (s) (j ∈ J )

λ ≥ 0, t ≥ 0

" !c0 , r + min{

cj (r)yj +

j∈J+

aij (r)yj +

j∈J+

cj (s)yj + !c0 , z }|

j∈J−

aij (s)yj + !Bi , r + z ≤ bi (i = 1, . . . , m),

j∈J−

y ≥ 0, 0 ≤ z ≤ s − r.

x = r + z, " min {

cj (r)yj +

j∈J +

j∈J+

aij (r)yj +

cj (s)yj + !c0 , x }

#

aij (s)yj + !Bi , x ≤ bi (i = 1, . . . , m),

$

j∈J −

j∈J−

y ≥ 0, r ≤ x ≤ s.

" (GP L)

%

aij (x) ← aij (r),

cj (x) ← cj (r)

j ∈ J+ )

aij (x) ← aij (s),

cj (x) ← cj (s) j ∈ J− ). % % j (r)yj + j (s)yj j∈J+ c% j∈J− c% !y, c(x) j∈J+ aij (r)yj + j∈J− aij (s)yj

!A(x), y ,

Ì

(GP L) (GP L).

!" #

$%& ' ' " (

)) ' *

!

'

'" +' $%&

"" $,&"

-

" !λ, B(x) + c0 (x)

x,

.

x.

j = 1, . . . , n

!Aj (x), λ

+ cj (x)

"

( *

'

[r, s]

inf {!B(x), λ + c0 (x)} = min{!λ, B(wi ) + c0 (wi )| i = 1, . . . , 2n }

x∈[r,s]

n

w1 , . . . , w2

'

[r, s],

,

max{−!b, λ + t

−!B(wi ), λ + t ≤ c0 (wi ), i = 1, . . . , 2n

%m

− %i=1 λi aij (r) ≤ cj (r) (j ∈ J+ ) ""

− m λi aij (s) ≤ cj (s) (j ∈ J − ) i=1

λ≥0 !/

.

!Aj (x), λ + cj (x) ≥ 0 ⇔ !Aj (wk ), λ + cj (wk ) ≥ 0

j = 1, . . . , n :

k = 1, . . . , 2n ,

!c0 , r + max{!r − s, t + !Br − b, λ } s.t. !Aj (wk ), λ + cj (wk ) ≥ 0 λ ≥ 0, t ≥ 0

j = 1, . . . , n, k = 1, . . . , 2n

c(x) ≡ c ∈ Êp , c0 (x) ≡ 0, B(x) ≡ 0

! " # $ % # " & ' # (

) * # #

+ xil # i

l, ylj # l j, zij # i j, plk , # k , # , l, Cik , # k i. # , ci , dj # i j, , xil , ylj , zij , plk & ) −

i

ν

ci xiν +

ν

j

dj yνj +

i

(dj − ci )zij .

j

% % x + z ≤ Ai %ν il % j ij x − y il i j lj = 0 % % %i xiν ≤ Sν (p − P )y %ν νk %jk νj + i (Cij − Pjk )zij ≤ 0 ν yνj + i zij ≤ Dj xiν , yνj , zij , pνk ≥ 0

Ai , Sν , Pjk , Dj , # ,

& , Cik , # k i. - " & , , plk . ! , # " # -

% ) , ' .α

/ ," min f1 (x, y) + f2 (x, y) + u(x) s.t. gi (x, y) + hi (x, y) + vi (x) ≤ 0 (i = 1, . . . , m) x ∈ C, y ∈ D.

(PC)

C ⊂ [a, b] ⊂ Rn+ , D Rp , u(x), vi (x) f1 (x, y), f2 (x, y), gi (x, y), hi (x, y) (i = 1, . . . , m) y x, x y. f1 (x, y), gi (x, y) x f2 (x, y), hi (x, y) M = [r, s] ⊂ [a, b] (M )) x ∈ C x ∈ C ∩ [r, s]. Ì (M )) β(M ) := min f1 (s, y) + f2 (r, y) + u(x) s.t. gi (s, y) + hi (r, y) + vi (x) ≤ 0 (i = 1, . . . , m) r ≤ x ≤ s, y ∈ D.

(RC(M ))

α f1 (s, y), f2(r, y), gi (s, y), hi(r, y) f1 (x, y), f2 (x, y), gi (x, y), hi (x, y), !(M ) (M ) " β(M ) # " " $ % {Mk = [rk , sk ]} x∗ lim β(Mk ) = β ∗ ,

k→∞

where

β ∗ = min{f1 (x∗ , y) + f2 (x∗ , y) + u(x∗ )| gi (x∗ , y) + hi (x∗ , y) + vi (x∗ ) ≤ 0 (i = 1, . . . , m), y ∈ D}.

#&

k (xk , yk ) !(Mk )), f1 (sk , y k ) + f2 (rk , y k ) + u(xk ) = β(Mk ), gi (s , y k ) + hi (rk , y k ) + vi (xk ) ≤ 0 (i = 1, . . . , m), k

rk ≤ xk ≤ sk , y k ∈ D. ∗ k ∗ k ∗ ' ∩+∞ k=1 Mk = {x }, r → x , s → x k → +∞. ( D % yk → y∗ ∈ D. "

f1 (x∗ , y ∗ ) + f2 (x∗ , y ∗ ) + u(x∗ ) ≤ lim β(Mk ), k→+∞

gi (x∗ , y ∗ ) + hi (x∗ , y ∗ ) + vi (x∗ ) ≤ 0 (i = 1, . . . , m).

* !(Mk )) (Mk )), β(Mk ) ≤ min{f1 (x, y) + f2 (x, y) + u(x)| gi (x, y) + hi (x, y) + vi (x) ≤ 0 (i = 1, . . . , m), rk ≤ x ≤ sk , y ∈ D},

#)

x∗ ∈ ∩+∞ k=1 Mk

k : β(Mk ) ≤ min{f1 (x∗ , y) + f2 (x∗ , y) + u(x∗ )| . gi (x∗ , y) + hi (x∗ , y) + vi (x∗ ) ≤ 0 (i = 1, . . . , m), y ∈ D} = β ∗ .

k : β(Mk ) ≤ β ∗ ≤ f1 (x∗ , y ∗ ) + f2 (x∗ , y ∗ ) + u(x∗ ).

lim β(Mk ) = f1 (x∗ , y ∗ ) + f2 (x∗ , y ∗ ) + u(x∗ ) = β ∗ ,

k→+∞

& ' (M )) ! ! " (M )) # !

$ # % # &'( ! !) # * # H, + qi Ji i = 1, . . . , s, , f (q, H, J) := b1 Li (KLi /Ji )β/α (qi /c)βλ/α + b2 |a1 |e1 H e2 + b3 |a1 |H i

% % q − i∈out(k) qi = ak k = 1, . . . , n + # %i∈in(k) i ±Ji = 0 p = 1, . . . , m # %i∈loop p min ±J p = 1, . . . , m . i ≤ H + h1 − hk i∈r(k) qimax ≥ qi ≥ qimin ≥ 0 i = 1, . . . , s H max ≥ H ≥ H min ≥ 0 KLi (qi /c)λ /dα max ≤ Ji Ji ≤ KLi (qi /c)λ /dα min

#

i = 1, . . . , s ,)/

.

Li i, K c ,)/

0 −a1 b1 , b2 , b3 , e1 , e2 , β 0 < e1 , e2 < 1 1 < β < 2.5, λ = 1.85 α = 4.87). * #1 f (q, H, J) # ! J ! (q, H), KLi (qi /c)λ , q !) M = M {(q, H)| q M ≤ q ≤ q M , H M ≤ H ≤ H } 23

f (q M , H M , J) ≤ f (q, H, J) KLi (q M /c)λ i

≤ KLi (qi /c) ≤ λ

λ KLi (q M i /c) ,

i = 1, . . . , s

24

(q, H) ∈ M.

(q, H, J) (q, H) ∈ M

KLi (q M /c)λ i !!"α# (q, H)

KLi (qi /c)λ M KLi (q i /c)λ

"$%#"$&#

! '

(' )*+

, )&&+

-

. )&/+ 0 )&/+ 1 "2# ! )&+

3 " # )&+ " "21## "

c(x), aij (x),

#

)&4+ 5

)&$+ )&$+ )6%+

min{F (x, y)| Gi (x, y) ≤ 0 (i = 1, . . . , m), x ∈ C, y ∈ D}

D→

C, D

Ê

Ên, Êp,

(GP ) F, Gi : C ×

x ∈ C y, !"#$ F (x, y), Gi (x, y)

x

m

∗

inf sup{F (x , y) +

y∈D λ≥0

i=1

∗

∗

λi Gi (x , y)} = sup inf {F (x , y) + λ≥0 y∈D

m

λi Gi (x∗ , y)},

i=1

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M %..!α# !(# .. β(M ) = sup inf{F (x, y) + !λ, G(x, y) | x ∈ M, y ∈ D}. λ≥0

0 .. !# !# - .. 1 Q(M ) := M ∩ C

M M M ∩ C = ∅.

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" # $ i q x y

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1−bit N T (n) 1−bit N kT (n) k k = 3log2 |Q|4 N

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" t = 0 #

1

Q

" t = 0 & ! M & {tn | n = 1, 2, 3...} k M

!

1

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5

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7

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9

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11

12

13

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18

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" # $ $ % &"1−bit '( ) $ # ! $ * "'+ % ,' F (⊆ Q) t = ktn k M k = 1

1−bit

!" 1−bit Ì [19] 1−bit {n2| n = 1, 2, 3, ...} [19] 1−bit

[20] 1−bit 1−bit 1−bit #

1−bit

${2n| n = 1, 2, 3, ...} ! 1−bit {2n | n = 1, 2, 3, ...} 1 a i %i ≥ 2& ' q

Current state

Input from right link

Input from left link

(next state, left output, right output)

1

2

a

R =0

R= 1

q

R =0

R= 1

L =0

(a,0,0)

(a,1,0)

L =0

(q,0,0)

(q,1,1)

L =1

--

--

L =1

(q,1,1)

(q,0,0)

{2n | n = 1, 2, 3, ...}

1−bit

" ( ' )

{2n| n = 1, 2, 3, ...} 1−bit 1−bit

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*

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27

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28

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31

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32

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33

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1−bit

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

{2n | n = 1, 2, 3, ...}

! " # " $ " % & "' " " " () * + +" & m × n ) (i, j) (i,j , - . /" + & - " . + & + ) + & + & & 0 & m× n (1,1 + t = 0 + + - " & . & + & & m n /

1

2

3

4

n

1

C11

C12

C13

C14

C1n

2

C21

C22

C23

C24

C2n

m

Cm1

Cm2

Cm3

Cm4

Cmn

m n !"# !$# %&!'

( )*1−bit ! )*1−bit + , !-#

)*1−bit 2n − 2 n ! . +, /

Ì [13] 1−bit n

2n − 2 1−bit . $ . "

[22] 1−bit

n k 1 ≤ k ≤ n k

1−bit

n+(k, n − k + 1)

0 (2n − 1) n × n %

!"# %&!'

( n 1 ! i 1 2n − 2i + 1 (1 ≤ i ≤ n) n ! % i 1 Ci,i t = 2i − 1

step 0

step 1

1

2

3

4

5

6

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1

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step 3

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step 13

1

step 11

step 10

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7

8

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step 15

step 14

1

QLA

p0

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p0

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p1

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p0

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PWLT

1

2

3

4

5

6

7

8

1

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2

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3

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(2n−1) ! " # # $ # % & ' ()* *+ t = 2i − 1 + 2(n − i + 1) − 2 = 2n − 1 ! " 2 × 2 1000 × 1000 # $%1−bit " &' ()* + ,

&'- " ! " 8 × 8 .

Ì 1−bit n × n 2n − 1

step 0

step 1

1

2

3

4

5

6

7

8

1

xJ

Q

Q

Q

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2

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2

3

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step 4

step 3

step 2

1

1

2

3

4

5

6

7

8

1

JD1

HS

xH

Q

Q

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step 10

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7

8

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1

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step 11

1

2

3

4

5

6

7

8

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1

2

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6

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step 13

step 12

1

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1

2

3

4

5

6

7

8

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7

Q

HQR2

6

Q

1

6

Q

2

5

xH

JD1

5

xH

HQR1

4

HS

1

4

HS

1

3

HQRS

step 9

3

HQRS

JD2

2

HQR2

step 8

2

JD2

1

1

JD1

1

1

1

step 7

step 6

step 5

VKX

VKX

xVQX1 xVQX1 xVQX1

2

3

4

5

6

7

8

JQRo1

HAr1

HQRd

HFGW

HB>

HBL1

HQLE0

HKX

1

VARA VsARD VsARA

VKX

VKX

VKX

VKXs

xVQX1 xVQX1 xVQX1

HKX

JQX

step 15

step 14

1

1

2

3

4

5

6

7

8

1

2

JQRo2

HAr2

HQRa

HG

HFB>

HfBL1

HALA

HKX

1

JQRo1

HAr3

VQRe2 JQRe2

4

5

HG

HQLa

HQLb

HFBL1 HfALA

HKX

HBr2

3

HQRc

HG

HFB>

6

HfAL3

7

HKX

8

HKX

VQRe1

JAr2

HFW

HW

HQR1

HG0

HAL1

HKX

2

VRe

JAr3

HFW

HGW

HQLA

HAL2

HKX

2

HBr1

HQRb

HFGW

HB>

HKX

2

3

VBRa

VQRo2

JFXB

HW

HQR1

HQR2

HQRS

HKXs

3

VBRb

VQRo1

JBr2

HFW

HW

HQR1

HGX

HKX

3

VBRc

VRo

JBr3

HQRd

HFW

HGW

HAL1

HKX

3

VBRe

HAr1

HQRd

HFGW

HfAL1

4

VfARA VSARD VSARA

JX

HQRS

HS

xH

xHQX1

4

VARa

VARd

VARa

JFXA

HW

HQRS

HS

xCQX

4

VARb

VARe

VARb

JAr2

HFW

HW

HQRS

HKXs

4

VARa

VARf

VARc

JAr3

HQRb

HFW

HGXX

HKX

VKX

xH

JQX

5

VKX

VKX

VKX

VKX

HPX

xH

xVQX1

JQX

5

VKX

VKX

VKX

VKX

HfPX

HS

xH

JQX

5

VKX

VKX

VKX

VKX

HFPX

HtSX

HS

xJQX

2

5

VKX

VKX

VKX

xVQX1 xVQX1

step 16

1

HQRb

2

3

4

5

6

7

8

JRo

HK1d

HKA

HQLb

HQLc

HBl1

HFALA

HKX

1

HQLb

HFAL3

HG

VQRe0 JQRe1

HAL3

1

2

3

4

5

6

7

8

JG

HK1

HK1

HI

HQLd

HBl2

HALa

HKX

1

HKA

HQLb

VQRo0 JQRo1

step 19

step 18

step 17

1

1

2

3

4

5

6

7

8

JQLa

HK1

HK1

HQRa

HI

HBl3

HALb

HKX

1

1

2

3

4

5

6

7

8

JAl1

HK1

HK1

HAr1

HKA

HK0d

HALc

HKX

HAr1

HG

HQLa

HKX

2

VRe

JRe

HK0d

HAl3

HKX

2

HK0

HK0

HI

HBl1

HQLe0

HKX

2

HK0s

HALa

HKX

3

VBRd

VBRa

JQRo2

HAr2

HQRa

HFAL1

HKX

3

VBRe

VBRb

JQRo1

HAr3

HG

HQLa

HAl1

HKX

3

VBRf

VBRc

JRo

HK1d

HKA

HQLb

HAl2

HKX

3

VK0d

VK0s

JG

HK1

HK1

HI

HAl3

HKX

4

VARb

VQRe0

VARe

JQRe0

HBr1

HQRb HFGOX

HKX

4

VARa

VARa

VARd

JARa

HBr2

HQRc

HGOx

HKX

4

VARb

VARb

VARe

JARb

HBr3

HG

HQLa

HKX

4

VARc

VARa

VARf

JARc

HK0d

HKA

HAl1

HKX

5

VKX

VKX

VKX

VKX

HKX

HTSX

HKXs

5

VKX

VKX

VKX

VKX

HKX

HAr1

HTSX

HKX

5

VKX

VKX

VKX

VKX

HKX

HAr2

Hsubr

HKX

5

VKX

VKX

VKX

VKX

HKX

HAr3

HGOx

HKX

2

VQRe1 JQRe1

HBr3

HtSX

HQLc

VG

JG

VKA

JAl1

HK0

HK0

step 21

step 20 1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

JK0

HK1

HK1

HK0

HK0

HK0

HK0

HKX

1

T

T

T

T

T

T

T

T

2

VK0

JK0

HK0

HK0

HK0

HK0

HK0

HKX

2

T

T

T

T

T

T

T

T

3

VK0

VK0

JKA

HK1

HK1

HKA

HK1

HKX

3

T

T

T

T

T

T

T

T

4

VK0

VK0

VK1

JK0

HK0

HK0

HK0

HKX

4

T

T

T

T

T

T

T

T

5

VKX

VKX

VKX

VKX

HKX

HK1

HKA

HKX

5

T

T

T

T

T

T

T

T

! " # $ %& '

(m, n) ! " # $ % &! 5 × 8

1−bit

Ì 1−bit m × n

m + n+ (m, n)

! " r, s 1 ≤ r ≤ m 1 ≤ s ≤ n

t = 0 Cr,s # $ % &'($ $ $ )* + $, m×n - mn m+n−1 gk 1 ≤ k ≤ m + n − 1 . / gk = {Ci,j |(i − 1) + (j − 1) = k − 1}.

g1 = {C1,1 }, g2 = {C1,2 , C2,1 }, g3 = {C1,3 , C2,2 , C3,1 }, . . . , gm+n−1 = {Cm,n }.

" M $ 1−bit . T (, k) k k 1 ≤ k ≤ - M m+n−1 - $ $ i gi i Ci M gi ↔ Ci 1 ≤ i ≤ m + n − 1 - $, 1−bit N gi i Ci $ N . m × n r,s t = T (m + n − 1, r + s − 1) M . $, m + n − 1 r+s−1 t = T (m + n − 1, r + s − 1) # $, ) * $, . T (m, n, r, s) 1−bit 011 000 ! 2 011$ 5 × 8 3,4

1−bit

m × n T (m, n, r, s) (r, s) T (m, n, r, s) T (m, n, r, s) = m + n − 2 + max(r + s, m + n − r − s + 2) ± O(1)

step 0

step 1

step 3

step 2

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

QX

QXT

QXT

QXT

QXT

QXT

QXT

QXX

1

QX

QXT

QXT

QXT

QXT

QXT

QXT

QXX

1

QX

QXT

QXT

ctrl

QXT

QXT

QXT

QXX

1

QX

QXT

L

QLS

D2

QXT

QXT

QXX

2

QXL

Q

Q

Q

Q

Q

Q

QXR

2

QXL

Q

Q

ctrl

Q

Q

Q

QXR

2

QXL

Q

L

QLS

D2

Q

Q

QXR

2

QXL

L

QLS

QL2

D1

D2

Q

QXR

3

QXL

Q

Q

P

Q

Q

Q

QXR

3

QXL

Q

ctrl

D1

ctrl

Q

Q

QXR

3

QXL

ctrl

QLS

D2

QRS

ctrl

Q

QXR

3

ctrl

QLS

QL2

D1

QR2

QRS

ctrl

QXR

4

QXL

Q

Q

Q

Q

Q

Q

QXR

4

QXL

Q

Q

ctrl

Q

Q

Q

QXR

4

QXL

Q

D2

QRS

S

Q

Q

QXR

4

QXL

D2

D1

QR2

QRS

S

Q

QXR

QX

5

QXX

QXB

D2

QRS

S

QXB

QXB

QX

QXX

5

QXB

QXB

QXB

QXB

QXB

QXB

QX

step 4

QXB

QXB

QXB

QX

QXX

5

4

5

6

7

8

1

KXs

QLS

QL2

QL1

QL2

D1

D2

QXX

1

QXR

2

QLS

QL2

QL1

QL2

D1

QR2

QR1

D2

ctrl

3

QL2

QL1

QL2

D1

QR2

QR1

QR2

QRS

S

QXR

4

QL1

QL2

D1

QR2

QR1

QR2

QRS

QXB

QX

5

D2

D1

QR2

QR1

QR2

QRS

S

5

6

7

8

QX

L

QLS

QL2

QL1

D2

QXT

QXX

2

L

QLS

QL2

QL1

D2

QR1

D2

3

QLS

QL2

QL1

D2

QR1

QR2

QRS

4

D2

QL1

D2

QR1

QR2

QRS

5

QXX

D2

QR1

QR2

QRS

S

step 8 2

3

4

5

6

7

8

KX

AR2

QRA

I0

QL1

D2

QR1

QR2

2

AR2

QRA

I0

QL1

D2

QR1

QR2

QR1

3

QRA

I0

QL1

D2

QR1

QR2

QR1

G0

4

I0

QL1

D2

QR1

QR2

QR1

G0

AL1

D2

QR1

QR2

QR1

G0

AL1

1

3

QXB

QXB

step 7 3

4

5

6

7

8

1

2

3

4

5

6

7

8

KX

IX

QL1

QL2

QL1

D2

QR1

D2

1

KX

AR1

I0

QL1

QL2

D1

QR2

QR1

2

IX

QL1

QL2

QL1

D2

QR1

QR2

QR1

2

AR1

I0

QL1

QL2

D1

QR2

QR1

QR2

3

QL1

QL2

QL1

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QR1

QR2

QR1

QR2

3

I0

QL1

QL2

D1

QR2

QR1

QR2

QR1

S

4

QL2

QL1

D2

QR1

QR2

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4

QL1

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GX

QX

5

QL1

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QR1

QR2

QR1

QR2

QRS

KXs

5

QL2

D1

QR2

QR1

QR2

QR1

GX

KX

step 10 2

3

4

5

6

7

8

KX

AR3

QRB

QRA

I0

D1

QR2

QR1

2

QRA

1

1

step 11

2

KX

3

QRE0 BR1

QRE0 BR1

4

5

6

7

8

QRB

X

QR1

G0

QRB

1

2

3

4

5

6

7

8

1

KX

ARA

BR2

FXA

GW

QLA

BR2

AR3

QRB

I0

D1

QR2

QR1

G0

2

X

QR1

G0

QLA

2

ARA

FXA

GW

QLA

3

QRB

QRA

I0

D1

QR2

QR1

G0

QLA

3

BR1

QRB

X

QR1

G0

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QLB

3

BR2

FXA

GW

QLA

QLB

BL1

4

QRA

I0

D1

QR2

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G0

QLA

AL2

4

QRB

X

QR1

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AL3

4

FXA

GW

QLA

QLB

BL1

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5

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QR2

QR1

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AL2

KX

5

X

QR1

G0

QLA

QLB

AL3

KX

5

FXA

GW

QLA

QLB

BL1

QLE0

KX

3

4

step 13

2

1

step 14

2

QLB

7

8

1

2

3

4

5

6

7

8

G

FB>

1

KX

fARA

BRd

QRe1

Ro

K1d

KA

QLb

Ar3

G

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2

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Ro

K1d

KA

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3

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KA

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4

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5

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KA

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KX

5

6

7

8

QRo2

Ar2

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B>

1

QRo2

Ar2

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B>

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2

QRo2

Ar2

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BL2

3

SBRD QRe2 QRo1

KX

ARA SBRD QRe2 QRo1

2

ARB sBRD

3

sBRD

4

QRo2

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B>

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BL2

ALA

4

QRe2 QRo1

5

QRo2

Ar2

FGW

B>

QLC

BL2

ALA

KX

5

QRo1

step 16

5

ARA SBRD QRe2 QRo1

Ar3

step 17

step 18

1

2

3

4

5

6

7

8

1

KX

ARc

K0d

KA

Al1

K1

K1

Ar1

I

2

ARc

K0d

KA

Al1

K1

K1

Ar1

Bl3

3

K0d

KA

Al1

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KA

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4

KA

Al1

K1

K1

Ar1

KA

K0d

KX

5

Al1

K1

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KA

K0d

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1

2

3

4

5

6

7

8

1

KX

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2

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4

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QLa

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5

QLa

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step 15

6

Ar3

4

ARB sBRD

QXB

KX

step 12 KX

ctrl

2

1

1

QXB

1

step 9

1

1

QXB

step 6 3

4

1

QXB

2

3

QL1

QXB

1

2

5

QXB

step 5

1

1

QXX

5

1

2

3

4

5

6

7

1

KX

ARa

BRe

Re

G

K1

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2

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G

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G

K1

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5

G

K1

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step 19

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

KX

K0

K0

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K0

K1

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1

T

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3

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5

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5 × 8 3,4 ! " # $ % & m × n m + n+(m, n)− (r, m − r + 1)−(s, n − s + 1) − 1 ! " # $ "$ # % & %

1−bit ' ! # # ()1−bit ( () ! % ! ! ! &! # ! % * # " ! + %!, %

!"# $ %& '# ( ) * +# ,! - !. /01−bit 1,!

/01−bit 2 n 3 4 !.

! " ! #$1−bit % & ' ! " ( $) *+

Ì 1−bit 0 5 $ 5 2 ,! 3 6 /01−bit 3 5 ! /01−bit - 2 /01−bit 7 ,! - 2 "8 0 !. /01−bit

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!!

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$ $ $ # ! " a(u, q)(ϕ) = f (ϕ)

∀ϕ ∈ V.

!0"

1 q ∈ Q = Rnp u ∈ g(q) + V ˆ 2 g : Q → V ˆ V V 1 V ⊂ Vˆ $ a(·, ·)(·) 1 Vˆ × Q × V 3 $

a(·, ·)(·) au (·, ·)(·, ·) aq (·, ·)(·, ·) C : Vˆ → Z u Z = Rnm

nm ≥ np # !·, · Z Z ·Z 3 Q $ C¯ ∈ Z 4

? 1? ?C(u) − C¯ ?2 Z 2

Rm (u) := C¯ − C(u).

Th Vh ⊂ V Vˆh ⊂ Vˆ Vh ⊂ Vˆh ! uh ∈ gh (qh ) + Vh qh ∈ Q

? 1? ?C(uh ) − C¯ ?2 Z 2

"

a(uh , qh )(ϕh ) = f (ϕh )

∀ϕh ∈ Vh ,

#

gh : Q → Vˆh $ g $ gh = ih ◦ g ih : Vˆ → Vˆh % &'( ) E : Vˆ × Q → R ) E(u, q) * )

E(u, q) − E(uh , qh ) = ηh + R,

+

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" # Thk → Thk+1 ηhk $ k

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& *

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au (v, p)(w, w) ≥ γ wV 2

∀(v, p) ∈ B(u, q),

∀w ∈ V.

$

0 1 0 2 3 4 * .

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j

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)

* + & , + -)+ .+ /0

* , + & E(u, q) 1 + E(u, q) − E(uh , qh ) = ηh + R, /. ηh R + + 2 * E(u, q) − E(uh, qh ) = E(u, q) − E(S(qh ), qh ) + E(S(qh ), qh ) − E(uh , qh ). / 1 u! = S(qh ) ∈ g(qh ) + V , qh + a(! u, qh )(ϕ) = f (ϕ) ∀ϕ ∈ V. // (+ + E (1) : Q → R E (2) : Vˆ → R E (1) (r) = E(S(r), r) /3

E (2) (v) = E(v, qh ),

E(u, q) − E(uh , qh ) = E (1) (q) − E (1) (qh ) + E (2) (! u) − E (2) (uh ).

E

(2)

E (1)

! " # $ %&

Ì E (1) (q) − E (1) (qh ) =

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'

y ∈ V au (u, q)(ϕ, y) = −!J(J ∗ J)−1 ∇E (1) (q), C (u)(ϕ)

ρ(·)(·)

ρ∗ (·)(·)

∀ϕ ∈ V

(

ρ(uh )(ϕ) := f (ϕ) − a(uh , qh )(ϕ) ρ∗ (uh , yh )(ϕ) := −!Jh (Jh∗ Jh )−1 ∇E (1) (qh ), C (uh )(ϕ) − au (uh , qh )(ϕ, yh ). )

R1

P ! eu + eq + δh v + δh z¯ Rm (u) , |P | ≤ C V V V Z Q

*

eu := u − uh eq := q − qh δh ϕ := ϕ − ihϕ Vˆ

Rm (u) +

v ∈ Vˆ np ∗ −1 (J J) ∇E (1) (q) j wj v=−

+,

j=1

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Rm (u) , C (u)(ϕ) Rm (u)

Z

∀ϕ ∈ V,

+-

Rm (u) z¯ = 0 C!

h C¯

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1 1 ! ρ(uh )(! y − ih y!) + ρ!∗ (uh , yh )(! u − ih u !) + R, 2 2

y! ∈ V

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(

∀ϕ ∈ V,

ρ!∗ (·)(·)

ρ(uh )(ϕ) := f (ϕ) − a(uh , qh )(ϕ) ρ!∗ (uh , y!h )(ϕ) := Eu (uh , qh )(ϕ) − au (uh , qh )(ϕ, y!h ).

R! ) y − ih y *

u! − ihu! + , ! + * ! , "# $ % &' - * u − ih u + uh ) u − ih u* y! − ih y!

.

δu := u − ih u ≈ u ! − ih u !.

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∀ϕ ∈ V.

E(u, q) − E(uh , qh ) ≈ ηh =

1 1 ρ(uh )(y − ih y) + ρ∗ (uh , y h )(δu), 2 2

0 1

ρ(·)(·) ρ∗ (·)(·) ρ(uh )(ϕ) := f (ϕ) − a(uh , qh )(ϕ) ρ (uh , yh )(ϕ) := −!Jh (Jh∗ Jh )−1 ∇E (1) (qh ), C (uh )(ϕ) + Eu (uh , qh )(ϕ) − au (uh , qh )(ϕ, y h ). ∗

! " #! $ !! %& ' ( )*+!

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ξ2 = (0.5, 0.25), ξ4 = (0.5, 0.75),

ξ5 = (0.5, 0.5).

C

Ci (v) = v(ξi ),

(u, q) ∈ V × Q

V = H01 (Ω)

Q = R2

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p1s

bR1S

QR0S

PWLT

6

QLB

QLA

AL

bR11

7

bL00

bL00

QLA

AL0

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

PWLT

BR0v0

QR11

BR00

QRA

AL

QLA

PWRB

1

PWLT

BR0v1

QR10

BR0S

AL

QLA

BL01

PWRB

2

bR0v0

PWLT

BR0u0

BR1S

QRD

QRC

AL0

PWRB

2

bR0v1

PWLT

BR0uS

QR10

BR01

AL

BL01

PWRB

3

QR11

bR0u0

PWLT

QR0S

BR11

QRB

subH

PWRB

3

QR10

bR0uS

PWLT

BR0u1

BR10

QRC

AL1

PWRB

4

bR00

bR1S

QR0S

PWLT

BR00

subH

AR’

xPWRB

4

bR0S

QR10

bR0u1

PWLT

BR0S

odd

subH

PWRB

5

QRA

QRD

bR11

bR00

PWLT

AR’

xPWLT

QW

5

AL

bR01

bR10

bR0S

PWLT

BR01

AR’

xPWRB

6

AL

QRC

QRB

subV

aR’

xPWLT

Q

QW

6

QLA

AL

QRC

odd

bR01

7

QLA

AL0

subV

aR’

xPWLT

Q

Q

QW

7

bL01

bL01

AL1

subV

aR’

QW

QW

QW

QW

8

8

step 13

1

PWRB PWRB PWRB xPWRB

2

3

4

5

6

7

8

1

PWLT

BR0vS

AL

P1

P1

AR

BL0S

PWRB

1

PWRB

2

bR0vS

PWLT

BR0v1

AL

P1

AR

BL0S

PWRB

PWRB

3

AL

bR0v1

PWLT

BR0uS

P0d

PA

BL01

PWRB

AL0

PWRB

4

p1

AL

bR0uS

PWLT

BR0u1

P0s

BL01

BR00

subH

PWRB

5

p1

p1

p0d

bR0u1

PWLT

BR0S

bR00

PWLT

AR’

xPWRB

6

AR

AR

pA

p0s

bR0S

subV

aR’

xPWLT

QW

7

bL0S

bL0S

bL01

bL01

AL1

QW

QW

8

PWRB PWRB PWRB PWRB PWRB

1

2

3

4

5

6

7

8

PWLT

P1

PA

P1

P1

PA

P1

PWRB

2

p1

PWLT

P1

PA

P1

PA

P1

PWRB

3

pA

p1

PWLT

P0

P0

P0

P0

PWRB

PWRB

4

p1

pA

p0

PWLT

P0

P0

P0

PWRB

AL1

PWRB

5

p1

p1

p0

p0

PWLT

P1

P1

PWRB

PWLT

BR01

PWRB

6

pA

pA

p0

p0

p1

PWLT

P0

PWRB

bR01

PWLT xPWRB

7

p1

p1

p0

p0

p1

p0

PWLT

PWRB

PWRB xPWRB

QW

8

PWRB PWRB PWRB PWRB xPWRB

PWLT xPWLT

QW

xPWLT

Q

QW

QW

QW

QW

step 15

step 14

1

PWRB PWRB PWRB PWRB PWRB xPWRB

step 11

step 10

1

8

step 12

8

6

8

PWLT

8

5

7

bR00

QW

4

6

PWLT

8

3

5

1

Q

2

4

2

7

1

3

step 4

step 3

step 2 2

1

2

3

4

5

6

7

8

1

T

T

T

T

T

T

T

T

2

T

T

T

T

T

T

T

T

3

T

T

T

T

T

T

T

T

4

T

T

T

T

T

T

T

T

5

T

T

T

T

T

T

T

T

6

T

T

T

T

T

T

T

T

7

T

T

T

T

T

T

T

T

8

T

T

T

T

T

T

T

T

PWRB PWRB PWRB PWRB PWRB PWRB PWRB xPWRB

(2n − 1)

! " "

step 0

step 1

1

2

3

4

5

6

7

8

1

xJ

Q

Q

Q

Q

Q

Q

CQX

2

Q

Q

Q

Q

Q

Q

Q

3

Q

Q

Q

Q

Q

Q

Q

4

Q

Q

Q

Q

Q

Q

5

CQX

VQX

VQX

VQX

VQX

VQX

2

3

4

5

6

7

8

1

JP

xH

Q

Q

Q

Q

Q

CQX

HQX

2

xV

Q

Q

Q

Q

Q

Q

HQX

3

Q

Q

Q

Q

Q

Q

Q

Q

HQX

4

Q

Q

Q

Q

Q

Q

VQX

JQX

5

CQX

VQX

VQX

VQX

VQX

VQX

step 4

step 3

step 2

1

1

2

3

4

5

6

7

8

1

JD1

HS

xH

Q

Q

Q

Q

CQX

HQX

2

VL

xJ2

Q

Q

Q

Q

Q

HQX

HQX

3

xV

Q

Q

Q

Q

Q

Q

HQX

Q

HQX

4

Q

Q

Q

Q

Q

Q

Q

HQX

VQX

JQX

5

CQX

VQX

VQX

VQX

VQX

VQX

VQX

JQX

HQX

Q

HQX

Q

HQX

4

VAR1

VQLS

xV

xJ2

xJ2

Q

Q

HQX

VQX

JQX

5

VKX

VKXs

VQX

VQX

VQX

JQX

HQX

2

VQL1

JD1

HS

xH

xJ2

Q

Q

HQX

2

VQL2

JD2

HQRS

HS

xH

xJ2

Q

Q

HQX

3

VQL2

VL

xJ2

xJ2

Q

Q

Q

HQX

3

VQL1

VQLS

xJ

xJ2

xJ2

Q

Q

4

VL

xJ2

Q

Q

Q

Q

Q

HQX

4

VQLS

xV

xJ2

Q

Q

Q

Q

HQX

4

VIX

VL

xJ2

xJ2

Q

Q

5

xCQX

VQX

VQX

VQX

VQX

VQX

VQX

JQX

5

VKXs

xVQX1

VQX

VQX

VQX

VQX

VQX

JQX

5

VKX

xCQX

xVQX1

VQX

VQX

VQX

step 10

HQR1

HGX

HKX

1

JQX

xJ2

Q

Q

HQR2

VQX

xH

Q

Q

HQR1

VQX

xJ2

Q

Q

HQR2

VQX

HS

xJ2

xJ2

HQR1

VQX

xJ2

xH

xV

JX

VQX

xH

JP

VQLS

1

VQX

HQRS

VQL2

3

HKXs

CQX

JP

2

HQRS

HQX

5

HQR2

CQX

HQR2

Q

JD1

xH

HQR1

Q

VQL2

8

HS

HQR2

Q

VI0

7

HQRS

HQR1

Q

VQL1

6

HQR2

HQR2

Q

2

5

HQR1

JD1

Q

3

4

HQR2

1

xV

HQX

3

JD1

8

HQX

4

HQX

2

1

7

HQX

Q

8

1

CQX

6

Q

Q

xCQX

8

Q

5

Q

Q

7

7

xH

4

Q

Q

HS

6

HS

3

Q

Q

6

5

HQRS

2

xJ2

xJ2

HQRS

4

HQR2

1

xJ

VL

5

3

HQR1

step 9

VQLS

3

HQR2

2

JD2

8

2

4

1

1

7

CQX

HQR1

8

CQX

6

8

Q

3

7

Q

5

7

Q

HQR2

6

Q

4

6

Q

2

5

xH

3

5

xH

HQR1

4

HS

2

4

HS

1

3

HQRS

1

3

HQRS

JD2

2

HQR2

step 8

2

JD2

1

1

JD1

1

1

1

step 7

step 6

step 5

xVQX1 xVQX1

step 11

1

2

3

4

5

6

7

8

JFXB

HW

HQR1

HQR2

HQR1

HG0

HAL1

HKX

1

1

2

3

4

5

6

7

8

JBr2

HFW

HW

HQR1

HG0

HQLA

HAL2

HKX

2

VI0

JD2

HQR1

HQR2

HQRS

HS

xH

xHQX1

2

V
JD1

HQR2

HQR1

HQR2

HQRS

HS

xCQX

2

V
JX

HQR1

HQR2

HQR1

HQR2

HQRS

HKXs

2

VQRe2

JFXA

HW

HQR1

HQR2

HQR1

HGX

HKX

3

VQRA

VQL1

JD1

HS

xH

xJ2

xJ2

HQX

3

VQRB

VI0

JD2

HQRS

HS

xH

xJ2

xHQX1

3

VsBRA

V
JD1

HQR2

HQRS

HS

xH

xHQX1

3

VSBRA

V
JX

HQR1

HQR2

HQRS

HS

xCQX

4

VAR2

VIX

VL

xJ2

xJ2

xJ2

Q

HQX

4

VAR3

VAR1

VQLS

xJ

xJ2

xJ2

xJ2

HQX

4

VQRE0 VAR2

xH

xJ2

xJ2

xHQX1

4

HS

xH

xJ2

xHQX1

5

VKX

VKX

xCQX

xVQX1 xVQX1

VQX

VQX

JQX

5

VKX

VKX

VKXs

VQX

JQX

5

JQX

5

step 13

step 12 1

2

3

4

5

6

7

8

1

JBr3

HQRd

HFW

HGW

HQLA

HQLB

HAL3

HKX

1

2

VQRe1

JAr2

HFW

HW

HQR1

HG0

HAL1

HKX

2

3

VBRa

VQRo2

JFXB

HW

HQR1

HQR2

HQRS

HKXs

3

4

VfARA VSARD VSARA

JX

HQRS

HS

xH

xHQX1

4

VKX

xH

JQX

5

VKX

5

VKX

VKX

xVQX1 xVQX1

step 16

1

xVQX1 xVQX1 xVQX1

2

3

4

5

6

7

8

JRo

HK1d

HKA

HQLb

HQLc

HBl1

HFALA

HKX

1

VKX

VIX

JP

VKX

xCQX

xVQX1 xVQX1 xVQX1

VARA VsARD VsARA

VKX

VKX

VKX

JD1

VKXs

xVQX1 xVQX1 xVQX1

1

2

3

4

5

6

7

8

1

2

3

4

5

1

JQRo2

HAr2

HQRa

HG

HFB>

HfBL1

HALA

HKX

1

JQRo1

HAr3

HG

HQLa

HQLb

HKX

2

VQRe0 JQRe1

HBr1

HQRb

HFGW

HB>

HAL3

HKX

2

VQRe2 JQRe2

HBr2

HQRc

HKX

3

VBRc

VRo

JBr3

HQRd

HFW

HGW

HAL1

HKX

3

VBRe

HAr1

HS

xCQX

4

VARb

VARe

VARb

JAr2

HFW

HW

HQRS

HKXs

4

VARa

VARf

VARc

JAr3

xVQX1

JQX

5

VKX

VKX

VKX

VKX

HfPX

HS

xH

JQX

5

VKX

VKX

VKX

VKX

2

3

4

5

6

7

8

JQRo1

HAr1

HQRd

HFGW

HB>

HBL1

HQLE0

HKX

VRe

JAr3

HQRb

HFW

HGW

HQLA

HAL2

VBRb

VQRo1

JBr2

HFW

HW

HQR1

HGX

VARa

VARd

VARa

JFXA

HW

HQRS

VKX

VKX

VKX

VKX

HPX

xH

2

3

4

5

6

7

8

JG

HK1

HK1

HI

HQLd

HBl2

HALa

HKX

1

VQRo0 JQRo1

6

7

8

HFBL1 HfALA

HKX

HG

HFB>

HfAL3

HKX

HQRd

HFGW

HfAL1

HKX

HQRb

HFW

HGXX

HKX

HFPX

HtSX

HS

xJQX

step 19

step 18

1

JQX

step 15

step 14

1

step 17

1

VKX

1

2

3

4

5

6

7

8

JQLa

HK1

HK1

HQRa

HI

HBl3

HALb

HKX

1

1

2

3

4

5

6

7

8

JAl1

HK1

HK1

HAr1

HKA

HK0d

HALc

HKX

HBr3

HG

HQLa

HQLb

HFAL3

HKX

2

VRe

JRe

HK0d

HKA

HQLb

HQLc

HAl3

HKX

2

VG

JG

HK0

HK0

HI

HBl1

HQLe0

HKX

2

VKA

JAl1

HK0

HK0

HAr1

HK0s

HALa

HKX

3

VBRd

VBRa

JQRo2

HAr2

HQRa

HG

HFAL1

HKX

3

VBRe

VBRb

JQRo1

HAr3

HG

HQLa

HAl1

HKX

3

VBRf

VBRc

JRo

HK1d

HKA

HQLb

HAl2

HKX

3

VK0d

VK0s

JG

HK1

HK1

HI

HAl3

HKX

4

VARb

VQRe0

VARe

JQRe0

HBr1

HQRb HFGOX

HKX

4

VARa

VARa

VARd

JARa

HBr2

HQRc

HGOx

HKX

4

VARb

VARb

VARe

JARb

HBr3

HG

HQLa

HKX

4

VARc

VARa

VARf

JARc

HK0d

HKA

HAl1

HKX

5

VKX

VKX

VKX

VKX

HKX

HTSX

HKXs

5

VKX

VKX

VKX

VKX

HKX

HAr1

HTSX

HKX

5

VKX

VKX

VKX

VKX

HKX

HAr2

Hsubr

HKX

5

VKX

VKX

VKX

VKX

HKX

HAr3

HGOx

HKX

2

VQRe1 JQRe1

HtSX

step 21

step 20 1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

JK0

HK1

HK1

HK0

HK0

HK0

HK0

HKX

1

T

T

T

T

T

T

T

T

2

VK0

JK0

HK0

HK0

HK0

HK0

HK0

HKX

2

T

T

T

T

T

T

T

T

3

VK0

VK0

JKA

HK1

HK1

HKA

HK1

HKX

3

T

T

T

T

T

T

T

T

4

VK0

VK0

VK1

JK0

HK0

HK0

HK0

HKX

4

T

T

T

T

T

T

T

T

5

VKX

VKX

VKX

VKX

HKX

HK1

HKA

HKX

5

T

T

T

T

T

T

T

T

!

step 0

step 1

step 3

step 2

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

QX

QXT

QXT

QXT

QXT

QXT

QXT

QXX

1

QX

QXT

QXT

QXT

QXT

QXT

QXT

QXX

1

QX

QXT

QXT

ctrl

QXT

QXT

QXT

QXX

1

QX

QXT

L

QLS

D2

QXT

QXT

QXX

2

QXL

Q

Q

Q

Q

Q

Q

QXR

2

QXL

Q

Q

ctrl

Q

Q

Q

QXR

2

QXL

Q

L

QLS

D2

Q

Q

QXR

2

QXL

L

QLS

QL2

D1

D2

Q

QXR

3

QXL

Q

Q

P

Q

Q

Q

QXR

3

QXL

Q

ctrl

D1

ctrl

Q

Q

QXR

3

QXL

ctrl

QLS

D2

QRS

ctrl

Q

QXR

3

ctrl

QLS

QL2

D1

QR2

QRS

ctrl

QXR

4

QXL

Q

Q

Q

Q

Q

Q

QXR

4

QXL

Q

Q

ctrl

Q

Q

Q

QXR

4

QXL

Q

D2

QRS

S

Q

Q

QXR

4

QXL

D2

D1

QR2

QRS

S

Q

QXR

QX

5

QXX

QXB

D2

QRS

S

QXB

QXB

QX

QXX

5

QXB

QXB

QXB

QXB

QXB

QXB

QX

step 4

QXB

QXB

QXB

QX

QXX

5

4

5

6

7

8

1

KXs

QLS

QL2

QL1

QL2

D1

D2

QXX

1

QXR

2

QLS

QL2

QL1

QL2

D1

QR2

QR1

D2

ctrl

3

QL2

QL1

QL2

D1

QR2

QR1

QR2

QRS

S

QXR

4

QL1

QL2

D1

QR2

QR1

QR2

QRS

QXB

QX

5

D2

D1

QR2

QR1

QR2

QRS

S

5

6

7

8

QX

L

QLS

QL2

QL1

D2

QXT

QXX

2

L

QLS

QL2

QL1

D2

QR1

D2

3

QLS

QL2

QL1

D2

QR1

QR2

QRS

4

D2

QL1

D2

QR1

QR2

QRS

5

QXX

D2

QR1

QR2

QRS

S

step 8 2

3

4

5

6

7

8

KX

AR2

QRA

I0

QL1

D2

QR1

QR2

2

AR2

QRA

I0

QL1

D2

QR1

QR2

QR1

3

QRA

I0

QL1

D2

QR1

QR2

QR1

G0

4

I0

QL1

D2

QR1

QR2

QR1

G0

AL1

D2

QR1

QR2

QR1

G0

AL1

1

3

QXB

QXB

step 7 3

4

5

6

7

8

1

2

3

4

5

6

7

8

KX

IX

QL1

QL2

QL1

D2

QR1

D2

1

KX

AR1

I0

QL1

QL2

D1

QR2

QR1

2

IX

QL1

QL2

QL1

D2

QR1

QR2

QR1

2

AR1

I0

QL1

QL2

D1

QR2

QR1

QR2

3

QL1

QL2

QL1

D2

QR1

QR2

QR1

QR2

3

I0

QL1

QL2

D1

QR2

QR1

QR2

QR1

S

4

QL2

QL1

D2

QR1

QR2

QR1

QR2

QRS

4

QL1

QL2

D1

QR2

QR1

QR2

QR1

GX

QX

5

QL1

D2

QR1

QR2

QR1

QR2

QRS

KXs

5

QL2

D1

QR2

QR1

QR2

QR1

GX

KX

step 10 2

3

4

5

6

7

8

KX

AR3

QRB

QRA

I0

D1

QR2

QR1

2

QRA

1

1

step 11

2

KX

3

QRE0 BR1

QRE0 BR1

4

5

6

7

8

1

2

3

4

5

6

7

8

QRB

X

QR1

G0

1

KX

ARA

BR2

FXA

GW

QLA

BR2

AR3

QRB

I0

D1

QR2

QR1

G0

2

X

QR1

G0

QLA

2

ARA

FXA

GW

QLA

3

QRB

QRA

I0

D1

QR2

QR1

G0

QLA

3

BR1

QRB

X

QR1

G0

QLA

QLB

3

BR2

FXA

GW

QLA

QLB

BL1

4

QRA

I0

D1

QR2

QR1

G0

QLA

AL2

4

QRB

X

QR1

G0

QLA

QLB

AL3

4

FXA

GW

QLA

QLB

BL1

QLE0

5

I0

D1

QR2

QR1

G0

QLA

AL2

KX

5

X

QR1

G0

QLA

QLB

AL3

KX

5

FXA

GW

QLA

QLB

BL1

QLE0

KX

3

4

step 13

2

1

QRB

step 14

2

QLB

7

8

1

2

3

4

5

6

7

8

G

FB>

1

KX

fARA

BRd

QRe1

Ro

K1d

KA

QLb

Ar3

G

FB>

D>

2

fARA

BRd

QRe1

Ro

K1d

KA

QLb

FD>

Ar3

G

FB>

D>

BL3

3

BRd

QRe1

Ro

K1d

KA

QLb

FD>

fBL3

Ar3

G

FB>

D>

BL3

ALB

4

QRe1

Ro

K1d

KA

QLb

FD>

fBL3

ALC

G

FB>

D>

BL3

ALB

KX

5

Ro

K1d

KA

QLb

FD>

fBL3

ALC

KX

5

6

7

8

QRo2

Ar2

FGW

B>

1

QRo2

Ar2

FGW

B>

QLC

2

QRo2

Ar2

FGW

B>

QLC

BL2

3

SBRD QRe2 QRo1

KX

ARA SBRD QRe2 QRo1

2

ARB sBRD

3

sBRD

4

QRo2

Ar2

FGW

B>

QLC

BL2

ALA

4

QRe2 QRo1

5

QRo2

Ar2

FGW

B>

QLC

BL2

ALA

KX

5

QRo1

step 16

5

ARA SBRD QRe2 QRo1

Ar3

step 17

step 18

1

2

3

4

5

6

7

8

1

KX

ARc

K0d

KA

Al1

K1

K1

Ar1

I

2

ARc

K0d

KA

Al1

K1

K1

Ar1

Bl3

3

K0d

KA

Al1

K1

K1

Ar1

KA

FALC

4

KA

Al1

K1

K1

Ar1

KA

K0d

KX

5

Al1

K1

K1

Ar1

KA

K0d

ALc

1

2

3

4

5

6

7

8

1

KX

ARb

BRf

G

QLa

K1

K1

QRa

2

ARb

BRf

G

QLa

K1

K1

QRa

3

BRf

G

QLa

K1

K1

QRa

I

4

G

QLa

K1

K1

QRa

I

Bl3

5

QLa

K1

K1

QRa

I

Bl3

FALC

step 15

6

Ar3

4

ARB sBRD

QXB

KX

step 12 KX

ctrl

2

1

1

QXB

1

step 9

1

1

QXB

step 6 3

4

1

QXB

2

3

QL1

QXB

1

2

5

QXB

step 5

1

1

QXX

5

1

2

3

4

5

6

7

1

KX

ARa

BRe

Re

G

K1

K1

I

2

ARa

BRe

Re

G

K1

K1

I

QLd

8

3

BRe

Re

G

K1

K1

I

QLd

FBL3

4

Re

G

K1

K1

I

QLd

FBL3

fALC

5

G

K1

K1

I

QLd

FBL3

fALC

KX

step 19

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

1

KX

K0

K0

K0

K0

K1

K1

K0

1

T

T

T

T

T

T

T

T

KA

2

K0

K0

K0

K0

K1

K1

K0

K0

2

T

T

T

T

T

T

T

T

K0d

3

K0

K0

K0

K1

K1

K0

K0

K0

3

T

T

T

T

T

T

T

T

ALc

4

K0

K0

K1

K1

K0

K0

K0

K0

4

T

T

T

T

T

T

T

T

KX

5

K0

K1

K1

K0

K0

K0

K0

KX

5

T

T

T

T

T

T

T

T

5 × 8 ! 3,4

"#$ % % 1−bit

Printing: Mercedes-Druck, Berlin Binding: Stein + Lehmann, Berlin