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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! ! ! 0 l2 ! ,-℄ 6 072! l1 ' $
l1 tk ≡ 1
! "
l1 # l1 $
J(x) =
J1 (x) J2 (x)
l1
" %&' (
)
⎞ D10 D11 . . . . . . D1m ⎟ ⎜ Gl0 Gr0 ⎟ ⎜ ⎟ ⎜ l ⎜ ⎟ G1 0 ⎟ J(x) = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ 0 Glm−1 Grm−1 ⎛
* " ( "
" +
F1 (x) , - ./℄ .1℄ l1 l1
2 3 4 5+* 3 $
6
M1
min f (Y ) =
i=1
|ATi Y + ci |,
ATi Y + ci = 0, i = M1 + 1, ..., M1 + M2 M1
m1
l1 M1 2 × M1
min fLP (Y ) =
M1
ξi ,
i=1
ξi − ATi Y ≥ ci , i = 1, ..., M1 , ξi + ATi Y ≥ −ci , i = 1, ..., M1 ,
ATi Y + ci = 0, i = M1 + 1, ..., M1 + M2 , ξ ≥ 0,
!
"
#
$ %
ϕ(λ) = cT λ,
min
λ∈ RM1 +M2
&
Aλ = 0, |λi | ≤ 1, i = 1, ..., M1 .
' " $ %
(℄ '
#
A
'
*
min f (Y ) =
M1 i=1
|ATi Y + ci |,
+,
ATi Y + ci = 0, i = M1 + 1, ..., M1 + M2 , L ≤ Y ≤ U,
L
U
Y
-
+, . $ % (/℄ "
# 0 .
!"# !$ %# $! & '(℄ *+, & l1 # # - *+, &- . *+, &- , !" # *+, &- / l2 l1
1 2 3 4 5 6 7 8 9 10 !"
-
/0
-/
-
-1
(
2
2
3
2
-
-
4
-/
4
*+, &-
l2 l1 p1 [km] l2 l1
p2 [km]
p3 [km] p4 [km/s] p5 [km/s] p6 [km/s]
0 -1(- 4 /0/0 /31- /
42(/13
/ 124 /23
0 -1(( 4 /0/2 /31 (
42(//3
/ 2 / 1
1
l1 l2
℄ ! " # $ % &
'
()
* + ,- ./++/0 /℄
1 23 ( . 0 #
4 % ' ) )4 ) % 5 4 .,60 7℄
1 ! # ) .,680 9℄
1 ( : ; 5 ) > 7 ./++/0 6℄ # 1 (
)
)4 4 * A 1 ./+++0 ,℄ * & # # 1 /+ * 4 +℄ ) B4 & / ./++/0
!"# $ %
& ' (
) *
'
+ , &* - * + - '. , ( ( ) * ) *( ,
) * /
, &
)
, - * , '"
!" # ! $%& ' %(℄ # * ! $+ , %% %'℄ # - . - / - 0 " -
℄
!
"
# # $ %# & ' (
& ) $ *# !
+ S ⊂ Êd d = 2, 3
L D := Êd\S S
C VC Ω F
S t
S(t) R(t) $ F
L ρ ∂∂t + ρ( · ∇) = ρg + ∇ · T ( , ) (x, t) ∈ [Êd \S(t)] × {t}, ∇· =0 t>0 ρ L# ,
L# T -
ρg
! +
. (& -
T ( , ) := −1 + µ(∇ + (∇ )T ),
µ
(x, 0) = 0,
lim
|x|→∞
(x, t) = 0
x ∈ Êd \S(t)
(x, t) = VC (t) + Ω(t) × (x − xC (t)) x ∈ ∂S(t).
! ˙ mS VC = mS g − T ( , ) · N dσ, ∂S(t) ⎪ d(J ·Ω) ⎪ ⎩ S(t) (x − xC ) × [T ( , ) · N ] dσ, =− dt ⎧ ⎪ ⎪ ⎨
"
∂S(t)
mS
# N ∂S(t) JS
C $
VC (0) = 0# Ω(0) = 0 %&"
L & R y P R x F # x = Q(t) · y + xC (t), Q(0) = 1, xC (0) = 0, ' ( ) ' % ρ{ ∂v ∂t + ((v − V ) · ∇)v + ω × v} = ∇ · T (v, p) + ρG(t)
(y, t) ∈ [Êd \S(0)] × (0, ∞)#
∇·v =0
*
v(y, t) := QT · (Q · y + xC , t), p(y, t) := (Q · y + xC , t), G := QT · g + V (y, t) := QT (VC +Ω×(Q·y)), T (v, p) := QT ·T (Q·v, p)·Q, ω := QT ·Ω. ,
ω × v (*)1 ' )
"
⎧ ⎪ ˙ ⎪ m V + m (ω × V ) = m G(t) − T (v, p) · n dσ, ⎪ S C S C S ⎪ ⎪ ∂S ⎨ IS · ω˙ + ω × (IS · ω) = − y × [T (v, p) · n] dσ, ⎪ ⎪ ⎪ ∂S ⎪ ⎪ ⎩ dG dt = G × ω,
%-
VC := QT · VC , n := QT · N, IS := QT · JS · Q, ∂S := ∂S(0).
d = 2 ω := (0, 0, ω) y × [T · n] = (0, 0, −y2 (T · n)1 + y1 (T · n)2 ) d = 2 ()2
R G t
! "# " $%℄
S
VC ω L R(t) '
( $%℄ v p VC ω G )
* ( ρ{((v − V ) · ∇)v + ω × v} = ∇ · T (v, p) + ρG ∇·v =0
Êd \S],
y∈[
+
lim v(y) = 0
|y|→∞
v(y) = V (y) := VC + ω × y y ∈ ∂S T (v, p) · n dσ, mS (ω × VC ) = ms G − ∂S y × [T (v, p) · n] dσ, ω × (IS · ω) = −
, -
% G × ω = 0. ! ! .
' $/℄ . ω = 0 0 ! '
d = 3 G
ω
∂S
d = 3 ρ T = T (v, p) |G| = |g| IS mS G = |g||ω|−1 ω ω = 0
v p VC ω G
Ω ⊂ Êd L2 (Ω) Ω (f, g)Ω :=
1 ||f ||Ω := ( |f |2 dx) 2 .
f g dx,
Ω
Ω
! L2 (∂Ω) ! ∂Ω " L2 #
L2(Ω) $ H 1 1 1 H0 = v ∈ H (Ω), v|∂Ω = 0
" % & ω = 0 " %! % ! ' ! ! ! % ( " ! ) * ! + 1 H1 (D) := (v, V, ω) : v ∈ [Hloc (D)]d , V ∈ Êd , ω ∈ Êd , v = V + ω × y ∂S , D := Êd\S " p L20 (D) :=
q ∈ L2 (D) :
q=0 .
D′
-
D′ ⊂ D u := {(v, VC , ω), p} ∈ H1 (D) × L20 (D) ϕ := {(ϕ, ϕ1 , ϕ2 ), q} ∈ H1 (D) × L20 (D)
A1 (u; ϕ) := ρ(((v − (VC + ω × y)) · ∇)v, ϕ)D + (ω × v, ϕ)D −(p, ∇ · ϕ)D + 2µ D(v) : D(ϕ) − (ρ|g||ω|−1 ω, ϕ)D D −1
−ϕ1 · [mS (|g||ω|
ω − ω × VC )] + ϕ2 · [ω × (IS · ω)] −(∇ · v, q)D ,
.
ϕ ∈ H1 (D) ×
()1 D(v) D(v) := 12 (∇v + (∇v)T ) u := {(v, VC , ω), p} ∈ H1 (D) × L20 (D) A1 (u; ϕ) = 0 ∀ϕ ∈ H1 (D) × L20 (D). !" #
!"
{(0, ϕ1 , 0), 0} {(0, 0, ϕ2), 0} # !"
$ %
&'( !℄ L20 (D)
( D := Êd\S * L Ω ⊂ Êd\S + * ∂Ω\∂S
, ( v(y) = 0 y ∈ ∂Ω\∂S. - * . ( Ω
&!( /℄ # . * W1h ⊂ H1 (Ω) × 2 L0 (Ω) * Th = {K} $ K Ω 0 {Th }h h → 0 W1h ⊂ H1 (Ω) × L20 (Ω) 1 # * &"℄ W1h := ((v, V, ω), p) ∈ [C(Ω)]d ×
Êd × Êd × C(Ω),
v|K ∈ [Q2 ]d , p|K ∈ Q1 , v|∂S = V + ω × y ,
Qr
r
&2℄ #
$
*
&℄ 3
( .
( * ( * .
′ uh := W1h
A1 (uh ; ϕh ) = 0 ∀ϕh ∈ W1h .
S
! " #$℄ ! !
& ' u := {(v, VC , ω), p} ∈ H1 (D) × L20 (D) Jψ (u) :=
∂S
[T (v, p) · n] · ψ dσ,
ψ := ψ1 + ψ2 × y ∈ Ê3 ψ1 , ψ2 ∈ Ê3 ψ = ψ1 ψ = ψ2 × y Jψ (u)
Jψ1 (u) = ψ1 · Jψ2 ×y (u) = ψ2 ·
∂S
∂S
[T (v, p) · n] dσ
y × [T (v, p) · n] dσ.
(
) & A(u; ϕ) := ρ(((v − (VC + ω × y)) · ∇)v, ϕ)D + (ω × v, ϕ)D D(v) : D(ϕ) −(p, ∇ · ϕ)D + 2µ D
*
−(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 } .
#,℄-
+
u Jψ (u) = A(u; w) ∀w ∈ H1ψ (D) × L20 (D).
H1ψ (D) × L20 (D) W1ψ,h := W1h ∩ {((v, V, ω), p) : V = ψ1 , ω = ψ2 } .
uh ∈ W1h
′ J˜ψ (uh ) := A(uh ; w) ∀w ∈ W1ψ,h ,
A(uh ; w) ψ w !
" J˜ψ (uh ) = Jψ (uh ).
# $℄& J˜ψ (uh )& Jψ (uh ) ' Jψ (u) ( &
Jψ (uh ) − J˜ψ (uh ) !
Jψ (uh ) − J˜ψ (uh) & " ) ( z := (z v , z V , z ω ), z p ∈ H1ψ (D) × L20(D) L(u, uh; z, ϕ) = 0 ∀ϕ ∈ H1ψ=0 (D) × L20 (D). * + & L(u, uh; z, ϕ) z ϕ " L(u, uh ; z, u − uh ) = A(u; z) − A(uh , z) ∀z ∈ H1 (D) × L20 (D), ,- ′ u uh
. A(·; ·)& L(u, uh; ·, ·) '
$*℄ / "
Jψ (uh ) − J˜ψ (uh ) z 0 Π : H1ψ (D) × L20 (D) → W1ψ,h Jψ (uh ) − J˜ψ (uh ) = A(uh , z − Πz). , 1 $*℄ !
,
" " ψ1 ψ2 ψ = ψ1 + ψ2 × y & z ψ ' "
.
z v |∂S = ψ C
l = 6.10 m L = 1.10 µ = 0.1 ρ = 1
∂Ω\∂S Ω !℄ #
$
% & −2
−2
ν = 0.1 D = 800 ! " # ! $! %&!
℄
℄
! " #$
( ) *$ % 0 ! 1 % 2 ,
! %
& '
+
,' - ,./
,℄ 3 4
2 ' % 2'5$ 6 //7
7℄ - 88 - %& ( 2 ' '+ &
!! 9:,; $℄ , / 7 " - ) 1-
"- > #$!&%
- 2 8 , 6
"> # % " - , , 6
' 8 2. A 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 $ " # (I − Pjk ) A−1 Rj = I − 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
i=j+1 xl ∈N ei ,xl ∈E k i−1
i
i
j
j=1 xk ∈N ej ,xl ∈E k j
i
j
"
$3/2 " ≤C hli /hkj hli /hkj
$1/2
≤C
xkj ∈ N!j ,
xli ∈ N!i.
# $ Πj : XJ → Xj J
j=1 z∈N ej
|(Πj v − Πj−1 v)(z)|2 ≤ CA(v, v)
∀v ∈ XJ .
!
℄ %℄ %℄ &
' ()* + 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 = j 1/2 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∈Z
!
n
An1 ei(αn +α)x1 +iβ1 x3 , x ∈ Ω1 .
" Ω2 u=
n∈Z
#!
n
An2 ei(αn +α)x1 −iβ2 x3 , x ∈ Ω2 . %
$% 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
( % ∆ˆ u+k 2 (x)ˆ u = 0 D = {(x1 , x3 ) : 0 < x1 < L, b2 −δ2 < 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)
ˆα (b1 )ei(αn +α)x1 Γ1 uˆ(x1 , b1 ) = uI (x1 , b1 ) + n∈Z u n n
A1 , B1
ˆ = uI Γ1PML
u
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 n ζ1 (x3 )
=e
−iβ1n
Rb
1 +δ1 x3
s(τ )dτ
u ˆ=
−e
iβ1n
Rb
1 +δ1 x3
s(τ )dτ
ζ n (x3 ) 2 u ˆ(n) (b2 )ei(αn +α)x1 ζ2n (b2 ) α
Ω2PML ,
n∈Z
ζ2n (x3 ) = e
−iβ2n
R x3
s(τ )dτ
−e
iβ2n
R x3
s(τ )dτ
% (n) i(αn +α)x1 e n∈Z f PML & # $ % ! Tj !
b2 −δ2
f
b2 −δ2
"!
f =
PML iβjn coth(−iβjn σj )f (n) ei(αn +α)x1 , f (x1 ) = Tj
'
n∈Z
coth(τ ) =
eτ +e−τ eτ −e−τ
σ1 =
b1 +δ1
s(τ )dτ,
σ2 =
b2
s(τ )dτ.
(
b2 −δ2
b1
)
∂(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
!
x3 ≥ b1
!
x3 ≤ b2
,
m ≥ 1.
" #
σjR
ℜσjm = 1+ δj , m+1
σjI =
ℑσjm δj . m+1
$%
& # ' δj ! ' ℜσjm ℑσjm
" ! ' '# !
( # !
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 aD (ˆ u, ψ) =
¯ g ψdx
D
∀ ψ ∈ X0 (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 ) =
g ψ¯h dx
D
A(x) =
A11 0 0 A22
=
∀ ψh ∈ Vh0 (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 |T2 ) · νe , uh |T1 − A∇ˆ
( ( 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 ′) , (ˆ u | ) − e · (ˆ u | Je = A11 ∂x h T h T ∂x 1 ' 1 ( ∂ ∂ ′) . Je′ = A11 eiαL · ∂x (ˆ u | ) − (ˆ u | h T h T ∂x1 1
T ∈ Mh ηT $1/2 ( ' "1 , ηT = max ρ(x3 ) · hT RT L2 (T ) + he Je 2L2 (e) 2 x∈T˜ e⊂T
T˜ T |s(x3 )|e−Rj (x3 ) x ∈ ΩjPML , ρ(x3 ) = 1 x ∈ Ω.
Rj (x3 ) (j = 1, 2) x3 x3 + − s1 (τ )dτ , s2 (τ )dτ, ∆1 R1 (x3 ) = min ∆1 b1 b1 R2 (x3 ) = min ∆− 2
b2
x3
s2 (τ )dτ, ∆+ 2
x3 ≥ b1 ,
b2
s1 (τ )dτ
x3 ≤ b2 .
,
x3
C > 0 Mh ˆ 1 ˆ ˆ 2 ˆ |u − uˆh |Ω ≤ CM uh − uI L2 (Γ1 ) + CM uh L2 (Γ2 ) 1/2 2 ˆ +CM3 Ih uI − uI L2 (Γ1PML ) + C ηT , T ∈Mh
Cˆ M3 = max
Mj (j = 1, 2) −
I
−
I
−∆1 σ1 2∆− 1e
1 − e−2∆1 σ1
,
+
R
+
R
−∆1 σ1 2∆+ 1e
1 − e−2∆1 σ1
.
℄ ! "#$ 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
EFEM
" - . ! *
δj σjm Mj L1/2 ≤ 10−8 ! # ! * + . / " * ! * " "
$%,' 0 " T ∈ Mh ! *
!) EPML
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 x
1.6
2
1
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
" " # $
θ = π/6% ω = π
"
L = 2
!
ε1 = 1, ε2 = (0.22 + 6.71i)2 %
! & " ' !(
) & " !( ! ) " " # * # + ! " "
" , " ( ! )
" *- " " ! ,+,
(
*+.
! " "
" %
"
/℄ % % "% 1 2 ( 3 45 6 4 4 7 % % -8*+ 9-:,; /$℄ ?
"'6&"69 ("%%$*
""℄ + 3 21 ! A ,
, -
")9&"9" ("%%0*
/ / ,
/
/ -
/ ?5 != 3
"$℄ ! ! A
,
*
@
5
!= ! !
'66&'00 ($###*
452B
/ <
= C / , ? ($###*
"℄ ! 42 : ! ?
, , -
,
'0&'% ("%%#*
"'℄ > 1 : C , / , 8- / < < ,
-
")℄ D / 4 3 , > ! = / ! <
"'&$# ("%%)*
"6℄ D :7 / 4
B ,
E
, 1 !
)9&%9 ($##$*
!"#$
% &
' ' '
(
)
) * ' + *
,
' ' -
* '
*
. *
/ ,
0 .
& 1 .
&
2 *
3 1 .
! " ! ! ! ! # "
$
%
& '
("$ ) !
! * ! ' ! "$
+,-. /+0.℄
!"℄
!$%&'℄ (
)
* +
, - . - . -
.
/ / * - . /0 !&%1 &&%&1℄ 2
3
4 /
-5*). , &678 !&%8℄ 5*) 5*)
- &. -& 9.
+ :
-1.
; &
9 9
& 9 ; &
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 )
Φi = (p − pi )S, i = 1, . . . , n! #! 4
! 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
67kg
" 48kg
"
!
30kg
) & *+ ,℄ ! " " ./0 $ ( 1 ! " )2 *3℄ ! " & & & 4
$
4 ! "
( !
"
" " 5 " "
( " " 6
" " 5 " ! " *3℄ 7 8 "
9
℄ !
"
# $ ! " ω % # ! α
#
"& ' " (
"
℄ v1 = Lω L ! " # !
$ α → π/2 "
% v2 = 2(21/2 + 1)π−1 Lω = 1.54Lω α = π/4
" #& # # ' # &
" ( ) *℄ " # # $ # & + ! " # # , ! # -
" # # # " # # - . & # " # # #
# /
# # # #
℄
! !
" # $
! %&℄
'
O1 O2
() *+ !
( + #
# #
O1
O2
F
O1
O2 " M1
P
O1 O2
"
M2
"
N
F = f1 M 1 + f2 M 2 + f0 ,
fi
ni , i = 0, 1, 2
N = n1 M 1 + n2 M 2 + n0 ,
℄!
" M1 M2
|M1 | ≤ M10 , |M2 | ≤ M20 F ∗ # $ % |F | ≤ f N & F = max F = F ! ' ( & )
&
O1
±M10
±M20
M2 =
F = F∗
!!
! ) ℄!
M10 , M20
M1 =
F∗
x
P
ℓ1 = O1 O2
&
ℓ2 = O2 P
*
"! +! ,
x
/
! '
#
x0
x0 − s/2
O1
M10 , M20
s
.
F∗
.
0 ! 1
#&
s
ℓ1
h
!
2&0
min
x∈[x0 −s/2,x0 ]
! -
x0 ℓ2
F∗
+
#& !
2&0 +
x0
ℓ2 !
)
& ! 4
℄! 3
M10 = 68N m, M20 = 27N m, f =
1, s = 0.28m, h = 0.24m, ℓ1 = 0.15m! ) ( ℓ2 = 0.15m, x = s/2 = 0.14m F ∗ = 642N ! ) ( . ∗ 5 0 x0 x0 = 0.105m, F = 786N ! ' ( 6 0 x0 ℓ2 x0 = 0.02m, ℓ2 = 0.21m, F ∗ = 1047N ! ) 0 .
! 7
! )
f
ℓ2
M10
&
M20
F∗
h
℄!
ℓ2
F ∗ x M10 = 45N m, M20 = 8.6N m, f = 0.2, h = 0.24m
F ∗ x ℓ2 = ℓ1 + 0.01(i − 1)m i = 1, . . . , 8
i
! " # " # $ %℄ '
v
(
T ) T /2 ) * ) ) + ) , " # - ) )
T /2
℄ ∆t ! " # $ % v #$ x O1 & ' s % ℓ2 ( % (# ) ℓ1 = 0.15m $
s x ›
s x
0.14 0.11 0.08 0.05 0.02 −0.01 −0.04 −0.07 −0.1 −0.13
0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.36 0.41 0.46
0.003 0.004 0.004 0.020 0.026 0.030 0.048 0.053 0.069 0.070 0.081 0.078 0.085 0.080 0.085 0.078 0.082 0.076 0.080 0.077
0.005 0.032 0.055 0.070 0.075 0.074 0.073
0.005 0.034 0.057 0.069 0.071 0.070
0.006 0.036 0.059 0.068 0.067
0.006 0.005 0.005 0.004 0.034 0.032 0.026 0.057 0.048 0.065
l2 s x l2 s x v
0.13 0.24 0.06 0.076
0.15 0.30 0.12 0.087
0.17 0.30 0.15 0.103
0.19 0.30 0.18 0.129
0.21 0.24 0.18 0.156
0.23 0.24 0.18 0.196
0.25 0.12 0.15 0.245
* + , - . ℓ2 (# ) ℓ2 = ℓ1 = 0.15m / # 0
v = 0.085m/s s = 0.26m x = 0.11m ℓ2, s x ℓ2 = 0.25m
! "#℄ %
&
'
( )
"#℄
)
"*℄ !
' "℄ "+,*℄
(
) ' -
& !
. ! !
"+ / #℄ .
"0℄
(
! 1 !
"℄ ) ! O1 C1 C2 O2 ( ! Oxy . .
O1 C1 , C1 C2 C2 O2
- C1 C2
' m1 O1 O2
' m0 m = 2(m0 + m1 ) C1 C2 2a
' ℓ 2 x, y
θ x! αi Oi Ci , i = 1, 2 .
O , C , i = 1, 2 Oxy m m f f M M C C
α α ! α (t) = ±α (t) ! " ω ε #
$ ω = max |α˙ (t)|, ε = max |¨ α (t)|, i = 1, 2. %&'( ) # i = 1, 2 # *+℄$ , + ≤ m gf . m ℓ [ω + (ε + gf ℓ ) ] + (ε + gf ℓ )ℓa %&&( ω ε %&&( m f (a + ℓ) < m f a !
τ T $ τ ≪ T M M $ |M | ≫ m gf ℓ , i = 1, 2, m = max(m , m ), %&-( f = max(f , f ), a = max(a, ℓ). i
i
0
1
0
1
2
1
1
2
1
1
2
2
0
0
0
4 0
0
i
0
0
0
−1 2 1/2
i
0
0
−1
−1
1
0
0 0
1
1 1
2
∗
1
∗
∗ ∗
∗
0
1
∗
0
1
1
0
x S F αi α0i → α1i , i = 1, 2
! S, α1 : 0 → γ, α2 (t) ≡ 0 γ ∈ (−π, π) " #! $%& 1. F, α1 : 2. S, α1 : 3. F, α1 : 4. S, α1 :
γ → 0, α2 0 → −γ, α2 −γ → 0, α2 0 → γ, α2
: 0 → γ; : γ → 0; : 0 → −γ; : −γ → 0.
' ( " " $ $)(
" α1 = α2 = 0 S : α1 : γ → 0, α2 (t) ≡ 0
x
* +,℄ x $)( ∆x = 8m0 m−1 ℓ sin2 (γ/2). m = 2(m0 + m1 ). #$.& y $)( ∆y = 0, ∆θ = 0
v1 = ∆x(2T )−1 .
℄ ! "! ! #! ! $ % ! ! %
!
!
&' ( '% )℄ ! ! *
ω(t) = |α˙ i (t)| = ε0 t, ω(t) = ε0 (T − t),
t ∈ [0, T /2], t ∈ [T /2, T ],
ω0 = ε0 T /2 = 2γT −1,
+
ε = 4γT −2 .
, ! ! + #!
m0 ℓ{[(2γT −1)4 + P 2 ]1/2 + P ℓa−1 } ≤ m1 gf1 , - !
ℓ
m1 . 2a γ $ m0 ! T
&
P = 4γT −2 + gf0 ℓ−1 .
/
f0
f1
&' v1 0 $ !. f − ≤ f0 ≤ f + , f − ≤ f1 ≤ f + 1 ! / *
f − , f1 = f +
f0 =
℄ V1 = v1 (gaf + )−1/2
γ χ = f − /f + λ = ℓ/a
λ = ℓ/a ! " γ χ
λ
# # $
%&'$
( |Mi| ∼ 10m1gf + ℓ ) $ * a = 0.2m, m1 = 1kg + ,
γ $ # # 40f + N m f − /f + γ,deg l, m m0 , kg m = 2(m0 + m1 ), kg T1 , s v1 , m/s
0.2/0.2 0.1/0.5 0.2/1 0.5/1 0/1 60 0.32 0.17 2.34 0.77 0.030
60 0.57 0.40 2.80 1.10 0.075
90 60 90 0.64 0.4 ∞ 0.39 0.26 1 2.78 2.52 4 1.08 0.49 ∞ 0.167 0.084 0.28
- . / 0 ) " 1 1 "
!
" # $ " $ % $
%&' '
( '
# & ) ( ' * * + , , , + - " - %&'
.
. & / 010200102
32℄ + " 5 6 " " 7 2708291 2:;: 31℄ , + , . ' ?# .344A/
5℄ (!" ' 0 = 1 B # ! $ -!
%& # !
%CCC % ( # * $
0
' 6 $ 6 #
* ? ! @ 0;//12 =℄ # ! 4 ) A B6 $ )B # ## %
' 5
'#
#
8
< 9/ -;.:1 0;//2 ,℄ # ! 4 # ) ) 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℄ #℄ ℄ ℄ $℄ ℄ ℄ %℄
℄ +℄ 1℄ #℄
!
" # #$ %$ ! " $% ! & '() *& *+ ' (! , $$
! " # $
, - .$$/ #0 $$1
! % & ' &
( ) ) - - 2 3 ( 4! ! "5 ! - 6 , 7 ! 8 $$ + 0#$ "
# * + ' , 9 2 :!!! 3 !; $$ $ " - , ! $$ % & ' ( &%) " ) &! 4! 4 * . / +#01$1 * . $ ) +% ? 6 & ! * ! $$ + $ &! ! 3 ! 3 ! 1 . / 1 01 , , 8 1 - %/ '$ 9 ! !6 '?-7 *
- & ' ( ! ' *
! 3 : 6! -' ; ; !
- 3 ( ' @ 6! % +%01 + , ' " " " ! 0 ) , ! "! !! * 1 , , . / ! ! ' '
/ / 1 .$$$/ +0 1+
$ $ ) , . 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
n
j ∈ 1 − n, . . . , l − m
1
∧
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
c21
−1
1 c 10
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 =
cik+1 ik
!"#
k=1
(j,i)⊂P
$% &'℄ am−i,j ≡
¯ k−1 #
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
¯ $ x ∈ n y˙ x ( 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 = i≻j = 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
4
c˜31 = c32 · c21 · c1−1
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′ j
j
i
h′
+ i
+ h k
k j
+ j +
i
k′
h i h
k
k′ d iv 1 j
i
v di
′
k
k
k j
∗
i
′
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"
#$℄ & #'℄
( ) * + (j, i) ∈ E (k, i) ,
* ) m = n
F ′ (x)x˙ = y˙
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" #.℄ &
"
"
/ " 0 1 2 3 × 3 3 4 6 + 5
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k0
i0
k0
j
j
k1
i1
k2
i2
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G G˜ ! 2 " # (−1, 1) # G˜ $
% −2
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10 8 (−2, 1) (2, 4) 1 Ê3×3 8 ! " ! ! −2
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# 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
˜ vn˜ −1,j,i j=1...˜n,i=1...˜n # m = 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
.
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#
7
8
9
4
5
6
1
2
3
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Tc
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i
i,σ
i
" t # U $ c†iσ σ % i niσ = c†iσ ciσ & '
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! ! -
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& !
! .
% % i ni↑ +ni↓ z
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u = 1, . . . , Del ; v = 1, . . . , Dph },
/
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ph ,
mi,v ∈ [0, ∞]
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j
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ij
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∆c = ∆s > 0; P = +1
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$
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 ∂T2 (r, z) ∂T1 (r, z) X ∈ Γ1,2 : λ1 − λ2 = U · ρ2 · L · ∂r ∂r ∂z
∂T (r, z) 1 ∂ ∂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∈
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k = I, II, . . . , n = x
n = y
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k Co ρo
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∂T ∂T +w ∂t ∂z
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z +
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0
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2 2 T0f +1 − T0f Tef − T0f Φ Ψ + qof Φe (Ψe − 1) = e e f ∆t R oe e=1 e=1
∈ Ω0
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P σT σT = 1, 2 1, 5
!" #$ %&℄(
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( Rliq ( 2.03.10×−2 m. $ + # ( Lcrit ( * Lcrit = Rcast − Rliq = 75 − 20, 3 = 54, 7mm.
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⎧ 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)
% %
%
$ %
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# $ %
z
&
' & &
K
A
'
& '
b(y)
yt + Ay + b(y) = K(y). ( & ) *
κ
K z
&
|y(t) − z(t)|H 1 ≤ ce−κt
c
+ ,,℄ &
- & & , . / 0 ,1 ,! ,2 ,+℄ &
+ ,,℄ -
& & 2℄ & &
& & &
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C
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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)}.
# b(u, v, w) :=
(u · ∇)v w dx.
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 ) −
h 1 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
Ω
|wk+1 | +
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
L2 *
""
1 J(y, u) = 2 %
t
γ |y(x, t)) − z(x, t)| dx + 2 Ωo
2
Ωo
Ωc
|u(x, t)|2 dx,
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)|:
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
* (
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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′
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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) γ r(b , a ) r(b , a ) = V (b ) := E / 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
zt+1
π
ρ(zt+1 |bt , at ) V π (bt+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
$ %
& ' &
()℄ ∗
Qπ (bt , at ) := r(bt , at ) + γ
zt+1
∗
ρ(zt+1 |bt , at ) V π (bt+1 )
+
∗
∗
V π (bt ) = arg max Qπ (bt , at ).
,
at
- & .
($℄ (//℄ - 0 1 1 (/2℄ 1 1 (3℄ 1 1 (+℄ % 1
1
4
/,,# (/℄
4 '
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8
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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 1 − co 0 co 0 z=2 := := , , O 0 co 0 1 − co −100 cr 10 −1 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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% # " "! !& !$ ' cr % 0.2# " ! # $
V π (s) ρ(s|π)
π !V π !Rπ := % t tγ
!V π =
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t
t0
ρ(s|π) !" # $
%
!R co & '
π1 (z) π2 (z) = 3 & ( & ) *
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! "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
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TSa=1
a=1 TH
T a=2
T a=3
⎛
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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
! " #$% & & '( ) * + ! " # $" "% &'"
, ⎧ ⎪ ⎨3 z π(z ) = 1 z ⎪ ⎩ 1 z t
t t t
=1 =2 =3
-& , ⎧ ⎪ ⎨3 z π(z ) = 2 z ⎪ ⎩ 2 z t
t t t
=1 =2 =3
( , " . #& a = 1 a = 3 & , ( & , " & ,
( & , a = 1
% t |at − at−1 | !"#$!
d ! R
" #$# π(b) P t |at − at−1 | π(z)
% !"#$! 0.2 !"#$!
& ' ( %
!"#$!
) !"#$!
!"#$! !"#$! *
+ & , - . /
!
" " #$℄ & & '& ( ) * +
,-./$ $001 #2℄ 34 $.15 $0,1 #6℄ 4 + $007 #-℄ & & $526.$527 $00- #,℄ 89 +
! $077 #/℄ : : + !
2552 #1℄ 34 ; "#$% $ 72*6,.-, $0/5 #7℄ & & 9 * 2?*272.65- $017 #$2℄ 3 # ## # 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 (1in+2 )T
−(p
2 T
−(p
(p ) (p2 )T
(p2 )T
(1in+out )T
in+1 T
)
−(pin+2 )T
in+1 T
)
−(pin+2 )T
(pin )T −(pin+1 )T (pin )T −(pin+2 )T
−(pin+out )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
(12 )T
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 ) C (1 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
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 ǫ− ǫ > 0 λ¯ λ¯ S ǫ ∈ S S ⊆ N
Ê
AS λ¯S = b AS (λ¯ S + ǫ) = b AS ǫ = 0|N |
ǫ¯ = 0|S|
AS ǫ = 0|N |
AS ǫ = 0|N | ¯ǫ = 0|S| AS (λ¯S + ǫ¯) = b λ¯S + ǫ¯ > 0|S| λ¯S − ǫ¯ > 0|S| !" (λ¯S + ǫ¯)0 (S) λ¯S + ǫ¯ #
¯ (λS + ǫ¯)0 (S) ∈ P S λ− $ "
% AS (λ¯S − ¯ǫ) = AS λ¯S − AS ǫ¯ = AS λ¯S − 0|N | = b A(λ¯S − ǫ¯)0 (S) = b (λ¯S − ǫ¯)0 (S) ∈ P &
1 ¯ 1 ¯ ¯S . (λS + ǫ¯) + (λ ǫ) = λ S −¯ 2 2
1 ¯ 1 ¯ ¯ S )0 (S) = λ. ¯ (λS + ǫ¯)0 (S) + (λ ǫ)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
pipe1
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
AS1 λS1
⎛
⎞⎛ 1⎞ ⎛ ⎞ 1 0 0 λ1 1 1 ⎠ ⎝ λ21 ⎠ = ⎝ 1 ⎠ =⎝ 0 1 λ23 15 −10 −20 0
! " 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 λ2 1 AS2 λS2 = ⎝ 0 1 1 ⎠ ⎝ λ21 ⎠ = ⎝ 1 ⎠ . 10 −10 −10 λ22 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
!
"#
$
P∆
⎛ ⎞ 0 ⎜1⎟ ⎜ ⎟ ⎜0⎟ ⎜ ⎟ ⎜1⎟ ⎜ ⎟ ⎝0⎠ 0 %
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 i λ− |S | = 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 ¯ = z ∗ ¯ a ¯T λ (¯a, α)
¯ 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 i=1
βi (¯ aT vi ) ≤
k
βi α ¯=α ¯
k
βi = α. ¯
i=1
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
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x n dω S 2 ν χ(x, ν) = κ(x, ν) + σ(x, ν) κ(x, ν) = κ(x)ϕ(ν)
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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
2
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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k b %
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uSEA,i i
uSEA,i ! i " # utors,1 $ utors,2 % ! ktors $ btors $ & # !
'() ⎛ ⎞⎛ ⎞ m 0 0 ml d cos ϕl1 ml d cos ϕl2 x ¨b ⎜ ⎜ ⎟ 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 ml d cos ϕl2 ml d sin ϕl2 0 ϕ¨l2 ⎛ ⎞ 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 + yb ϕ˙ l ). Fd = b ( 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
"
) ( %(
&
•
)
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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 ! "
˙ ) = q(0) ˙ T qred (T ) = qred (0) q(T # !
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, # +
! - ) . 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 %
!
" ! , ! ' ! ' ' " " " -! , ' " ! " '
! !
"
# u$ %
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' min
x,u,T
T 0
||u||22 dt
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(℄ : 8-℄$ ; • • %
' • < 3=7 # : 8-℄$ • "
#5 8>℄$ req (x(0), .., x(T ), p) = 0 rineq (x(0), .., x(T ), p) ≥ 0.
• • •
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 |λ4 | = 0.8168 |λ5 | = 0.5373
|λ6 | = 0.0515
|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% +(. - '.
!"! # !"$ ! " !
% " " !& "!
ϕb
! "
Θb ml Θl d ktors
# $#
∆ϕ btors k b
% $ % % $ # %
$#
% $ %
%
$%
& $ # #
$&
& $ '
6 5
x_b
4 3 2 base solution perturbed solution
1 0 0
1
2
3
4
5
t
0.3
0.65
0.2 0.1
0.55
phi_b
y_b
0.6
base solution perturbed solution
0.5
0 -0.1 -0.2 base solution perturbed solution
-0.3
0.45
-0.4 0.4
-0.5 0
1
2
3
4
5
0
1
2
t
3
4
5
t
0.4 0.2 0.2 0 phi_l2
phi_l1
0 -0.2 -0.4
-0.2 base solution perturbed solution
-0.4 base solution perturbed solution
-0.6 0
1
2
3 t
4
-0.6 5
0
1
2
3
4
5
t
ϕb
#$%$ & %' $%#(# !)*
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6 '(
V (z) Y D(z )
E(z )
+
R (z )
U (z)
G(z )
+
+
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)
!)"
p
[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)]
F (z)
! !
z p − F (z)[1 − z γ Φ(z)G(z)] = 0
#
F (z)
"
!
! ! # !
!
$
"
!
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) =
U (z) F (z)z γ Φ(z)[α1 z (N −1)p + α2 z (N −2)p + · · · + αN ] = 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) = M (z)(α1 z −p + α2 z −2p + · · · + αN z −N p )
""
P (z) ∗
#
|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)|
"!
$ % $ P ∗ (eiωT ) & & $ P (z)
' $ $ % $ ( )" ℄ % $ $ ( $ + % $ ,$ --- .$ --- / M (z) % & ∆ω
∆ωT - $ % 0 &
& P ∗ (z) 1
∆ωT p 2 !3
& # $ P ∗ (z)
$ z −N p ∆ωT N p$ ' $ % #
$ %
4 & 2 5
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% - 1 % / 0 ' F
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!" #! ! ! $%
Pm 2 F (x)2 = 21 i=1 [Fi (x)] ! ' ( H = G + S ! f # G = (F ′ )T F ′ S = F ◦ F ′′ T T !" ) ARN = G + B # B = Z2 Z2 SZ2 Z2 ! ! ! ! S ! ) im Z2 ) % ! ! #! " G #
) !% ! q % ! G' *! !
! &
f (x) =
1 2
! !" ! %! ! !
! ! ! ! '
! ! ! & !"
#! ! #! ! ! !
f (x) :=
m 1 1 2 2 [Fi (x)] −→ Minn F (x) = x∈Ê 2 2 i=1
F : Ên → Êm m ≥ n
m ≫ n %m . 2 y = i=1 yi ! " y = r(t, x), r : Êdim × Ên → Êdim # t ∈ Êdim y ∈ Êdim x ∈ Ên $ % x t
t
y
y
(tl , yl ) ≈ (tl , yl∗ ),
(l = 1, . . . , L)
(tl , yl∗ ) yl∗ = r(tl , x∗ )
x∗
x∗ f (x) :=
L 1 2 yl − r(tl , x) −→ Minn . x∈Ê 2
l=1
m := L × dimy !
F ⎛ ⎞ y1 − r(t1 , x) ⎟ ⎜ F (x) := ⎝ ⎠. yL − r(tL , x)
" # $ % %& ' x ( ) y˙ = Φ(t, y, x)
t ∈ [ t0 , te ],
y(t0 ) = y0 (x)
*
y = y(t, x) t + x
Φ(t, y, x) ( y0 (x)
)
r(t, x) := y(t, x) , y = y(t, x) * t + x dimt = 1 t0 ≤ t1 < · · · < ti < ti+1 < · · · < tL ≤ te y : [ t0 , te ] × n → dimy , y0 :
Ê
n
→
Ê
Φ : [ t0 , te ] ×
Ê
Ê
dimy
Ê
,
dimy
×
Ên → Êdim . y
- Φ y0 . y . x / 0/1 + x+ = x + s,
s f (x+ ) ≤ f (x)
.
x = xk s = sk x+ = xk+1 s & ϕ(s) := f + g T s + 21 sT As ≈ f (x + s)
2
f x = xk f := f (x) g := g(x) := ∇f (x) = F ′ (x)T F (x) A = AT H := H(x) := ∇2 f (x)
H = H(x) = ∇2 f (x) = F ′ (x)T F ′ (x) + F (x) ◦ F ′′ (x) =: G + S
G := G(x) := F ′ (x)T F ′ (x), S := S(x) := F (x) ◦ F ′′ (x) := ′
F (x)
′′
F (x)
m i=1
Fi (x) · ∇2 Fi (x)
F
x
! "
s
! # $
%$&
AGN := G
&
AN := G + S = H
' &
B ≈ S
G
G
ARN = G + B
q
!
"
q = qk
( )
λmax (G)
qk qk = 0
$
(
qk > 0
%$&
%$&
%$& $ # !
G
)
s
! "*
sfull
! (
Min{ ϕ(s) : s ∈
Ên}
+
A α > 0 s s := α · sfull f s Min{ ϕ(s) : s ∈
Ên, s ≤ ∆}
s ∆ > 0 A ! "#$℄
"&℄ '
( A H ) (
( AGN := G = (F ′ )T F ′ F *
ϕGN (s) := f + g T s + 12 sT (F ′ )T F ′ s =
1 2
F + F ′ s
2
+
F := F (x) F ′ := F ′ (x) , - 2
Min{ 12 F + F ′ s : s ∈
Ên}.
.
sfull = sGN = −(F ′ )† F / 0 sGN , J † 1
2 ( J * F ′ 3 n G = (F ′ )T F ′ sGN = −G−1 g . 4 sGN + ker F ′ sGN F ′ * 3 / 0 F ′ = F ′ (xk ) 3 n ′ cond(F ) = (cond(G)) F = F (xk ) 5 6 7 / 0 x+ = x + sGN
8 ̺GN xopt f rank(F ′ (xopt )) = n ̺GN := ̺(G(xopt )−1 S(xopt )) = max h=0
|hT S(xopt )h| 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 !
"
#
̺RN = ̺(MRN ) < ̺(MGN ) = ̺GN
B = 0
MRN := −(G + B)−1 (S − B),
MGN := −G−1 S
M := T ′ (xopt ) = −A(xopt )−1 (H(xopt ) − A(xopt ))
T
x+ = T (x) := x − A(x)−1 g(x) $
̺(M ) %
&℄
)
q
G(xopt )
B(x) ( q
℄ λmax (G(xk )) λ = λ(G(xk )) Λ2 λ ≤ tol × λmax (G(xk )) tol ℄ !
"# $ $ %&'(% ℄ n ! ) * +," ,- .℄ ℄
/ 0 qk qk = 0 B ! "# & 0 B(xk ) # G(xk ) # / F (xopt ) # S ' B
1 ( H 2 B 3 • , / 3 # G
#
! 4 5 6℄ • 7 $ 8 ! / G 8 ! / )9 * Lp p ! # $ im Z2 ! Z2 = [er(p+1) , . . . , er(n) ] ej :# r(p + 1), . . . , r(n) q = n − p 8 ! / p# q = qk
℄
; 0 # !'&8 *7# B
D 5 ( 8 ' B ! - ..)0 ℄ 8# 8 E 1. +67) - . $ ? @$6 8, 8 ? $ "6 $@$6%$ ! ,=
"6 "@℄ ?(7 2 ?(7 ' "@ 2 > " "% ?(7 A 3* "" "B℄ C & 8 ( >' 25
5 ' 7 D
, + ) - . ,
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E (
"$4#"::
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5
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!
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4 5 6 "
7
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+ 2 A0
- "4
! ! "
# ! $ #% & #!
! ! $ ! " ! ! !#
% ' !
! % # !(
$%
) #! "
# ! ! ! !%
! ! "
# f : Ên+ → Ê f (x′ ) ≥ f (x) x′i ≥ xi (i = 1, . . . , n), f (x′ ) ≤ f (x) x′i ≥ xi (i = 1, . . . , n); $ % &'(℄ &'*℄ &(℄ &++℄ &,℄ #
- "
min{F (x, y)| Gi (x, y) ≤ 0 (i = 1, . . . , m), x ∈ C, y ∈ D}
(P)
C X ⊂ Rn +, D p Y ⊂ R+ , F (x, y) : X × Y → R, Gi (x, y) : X × Y → R y x x y. n ! P (x), x ∈ Rn + , R+ ,
P+ (x) + P− (x), P+ (x) (P− (x), "
P (x) # " $
#%"
min{!c, y : A(x)y ≤ b, y ≥ 0, x ∈ X}
& '(℄ * #%" + F (x, y), Gi (x, y), i = 1, . . . , m, y x #%"
& ,- . / 0 F (x, y) Gi (x, y), i = 1, . . . , m.
x ∈ Rn + x # " 1
#
" M ⊂ Rn + β(M )
γ(M ) := inf{F (x, y)| Gi (x, y) ≤ 0 (i = 1, . . . , m, x ∈ M, y ∈ D}. (P(M))
{Mk } {Mk } Mk+1 Mk ) limk→+∞ Mk = 0, ∗ ∩∞ k=1 Mk = {x }.
!! " #
$ % & ' !!
& ( & ) !!
* +
,
%
& & - $
. $ / % $ 0 & 1
!! * 2 % & 3 +
.
!!
x, y
x "#$℄ "#&℄ ' x n (x, y) n + p) "#(℄ ) X * + G(x, y) = (G1 (x, y), . . . , Gm (x, y)), G(x, y) ≤ 0 Gi (x, y) ≤ 0 i = 1, . . . , m.
α
, '
α ≥ sup{F (x, y)| x ∈ C, y ∈ D}. - α ∈ α = +∞. - .!,/(¯ x0 , y¯0 ) , M1 = X, P1 = S1 = {M1 }, k = 1. # M ∈ Pk β(M )
Ê
γ(M ) := inf{F (x, y)| G(x, y) ≤ 0, x ∈ M ∩ C, y ∈ D}.
#
xk , y¯k ) 2 0 1
.!, (¯ 3 * M ∈ Sk β(M ) ≥ min{α, F (¯ xk , y¯k )} k k k k x , y¯ ) = +∞ (¯ x , y¯ ) + 4 Rk
F (¯ Sk . 5 - Rk = ∅ 6 .!, = (¯ xk , y¯k ) .!,= ∅). & . Mk ∈ argmin{β(M )|M ∈ Rk }. , Mk
4 Pk+1 Mk . $ 4 Sk+1 = (Rk \ {Mk }) ∪ Pk+1 . , k ← k + 1 , #
' ! β(M ) 6 (a) M ′ ⊂ M ⇒ β(M ′ ) ≥ β(M );
(b)
β(M ) < +∞ ⇒ M ∩ C = ∅. 0
1 β(M ) # '
F (x, y) Gi (x, y), i = 1, . . . ,
C, D ! γ(M ); " β(M ) = sup inf{F (x, y) + !λ, G(x, y) | x ∈ M ∩ C, y ∈ D}. λ∈
Ê
m +
"
# ! ! " ! $ % & ! {(x, y)| Gi (x, y) ≤ 0 (i = 1, . . . , m), x ∈ M ∩ C, y ∈ D} β(M ) < +∞. ' !
β(M ) < +∞ (℄$
Ì
α
{Mkν } ⊂ {Mk }
∗
∗
x∗ ∈ C.
lim β(Mk ) = inf{F (x , y)| G(x , y) ≤ 0, y ∈ D},
k→∞
*
β ∗ := limk→∞ β(Mk ) y * (x∗ , y ∗ ) α < +∞ *
∗
$ + ! {Mkν } ⊂ {Mk } ! ! ! $$ ( ,℄$ -
x∗ . M β(M ) ≥ α β(Mk ) < +∞ ! Mk ∩ C = ∅ ∀k. + Mk ∩ C ! & !
+∞ ∗ . ∩+∞ k=1 (Mk ∩ C) = (∩k=1 (Mk ) ∩ C = ∅. +! x ∈ C. ∗ ' ! & β(Mk ) β = limk→+∞ β(Mk ) β ∗ ≤ α. /
γ := inf{F (x, y)| G(x, y) ≤ 0, x ∈ C, y ∈ D}.
+ β ≤ γ. - ∗
∗
0
! *
∗
β = inf{F (x , y)| G(x∗ , y) ≤ 0, y ∈ D} ≥ ≥ inf{F (x, y)| G(x, y) ≤ 0, x ∈ C, y ∈ D} = γ,
β = γ. '! α < +∞, β ∗ ≤ α < +∞ ! $ ⊓ ⊔ . * $ +
! * β ∗ = +∞$ 1 ! α < +∞. ∗
α
β(M )
! α
Ì C Ên , D
Êp , F (x, y), Gi(x, y), i = 1, . . . , m,
y x. α
Mk , ν = 1, 2, . . . , x∗ ∈ C ν
min{F (x∗ , y)| G(x∗ , y) ≤ 0, y ∈ D}
y (x∗ , y ∗ )
∗
∈D
"
"
# $ lim β(Mk ) = inf{F (x∗ , y)| G(x∗ , y) ≤ 0, y ∈ D}
k→∞
sup{F (x, y)| x ∈ C, y ∈ D} ≤ α < +∞.
%
& '
β(Mk ) ≤ inf{F (x, y)| G(x, y) ≤ 0, x ∈ Mk ∩ C, y ∈ D} ≤ inf{F (x∗ , y)| G(x∗ , y) ≤ 0, y ∈ D}
β(Mk ) ր β ∗ ≤ inf{F (x∗ , y)| G(x∗ , y) ≤ 0, y ∈ D} ≤ α < +∞. %
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 ) ր 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 i=1
λi gi (x)| x ∈ M ∩ C}.
$ ! ,
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 yj Gj 1 0 G0 + L0 +
j=1 n
xi Li0 +
m
yj L0j +
i=1 j=1
j=1 n
i=1
x ∈ X = [p, q] ⊂ R , y ∈ Rm +
x, y
n m
xj yj Lij ≺ 0
G0 , Gj , L0 , L0i , Lj0 , Lij G 1 0, L ≺ 0
G
L
A B
d
A 0 0 B
⎡
A00 (x) = ⎣ L0 +
%Gn0
i=1
!x, c
⎡
⎤
xi Li0 ⎦ , Aj0 (x) = ⎣ L0j + d
Q00
⎡ ⎤ 0 = ⎣0⎦ . 1 d
Gj % n
i=1
dj
⎤
xi Lij ⎦ , d
! "#℄
min{t| A0 (x, p, q) +
m j=1
yj Aj (x, p, q) 1 tQ, y ≥ 0, x ∈ X}
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 ⎬ ⎨ yj Aj (x, p, q) − tQ)⎦ t + Tr ⎣Z(A0 (x, p, q) + max min ⎭ 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 ;:5 A $ .%$$= $ )
>%$ " @ " 6 *+ 6-77
℄ #:
A $ %$$= $ ;5 #$ %% )
67℄ #:
9# -( % > # 19 .%$$= $ )
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" # $ i q x′ y′
" % i i−1 i+1 & q ∈ Q δ(q, 0, 0) = (q, 0, 0) ' F (⊆ Q) ( # 1 & # 1−bit ) N ( δ : Q3 → Q # N #
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" ( 1−bit N ⌈log2 |Q|⌉ #
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T (n) 1−bit N kT (n) k k = ⌈log2 |Q|⌉ N
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WZ D1
S
S
C
R
A1
Q
Q
Q
Q
Q
34
N0
H0
Z
WZ
Z
C
Z
Z
WZ D2
S
C
Z
B
R
A0
Q
Q
Q
Q
35
N1
H1
C
WZ
Z
Z
C
Z
WZ
Z
U0
Z
Z
P
B
A1
Q
Q
Q
Q
36
N0
Z
Z
WC
Z
Z
Z
C
WZ
Z
U1
C
Z
P
S
WV A0
Q
Q
Q
37
So
Z
C
WZ
Z
Z
Z
Z
WC
Z
U2
Z
C
P
C
WY
Q
Q
Q
Q
38
N0
C
Z
WZ
Z
Z
Z
C
WZ
Z
U3
Z
Z
C
Z
WY
Q
Q
Q
Q
39
N1
Z
Z
WZ
Z
Z
C
Z
WZ
Z
C
Z
Z
Z
C
WY
Q
Q
Q
Q
40
N0
C
Z
WZ
Z
C
Z
Z
WZ
C
Z
Z
Z
Z
Z
WT
Q
Q
Q
Q
U0 WY
" # $ $ % &"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
) * )
*
+ , *
1
2
3
4
5
6
7
0
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
1
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
2
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
3
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
4
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
5
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
6
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
7
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
8
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
9
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
10
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
11
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
12
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
13
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
14
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
15
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
16
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
17
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
18
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
19
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
20
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
21
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
22
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
23
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
24
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
25
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
26
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
27
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
28
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
29
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
30
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
31
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
32
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
33
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
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
7
8
1
2
3
4
5
6
7
8
PWLT
AR’
xPWLT
Q
Q
Q
Q
QW
1
PWLT
BR01
AR’
xPWLT
Q
Q
Q
QW
2
aR’
xPWLT
Q
Q
Q
Q
Q
QW
2
bR01
PWLT xPWLT
Q
Q
Q
Q
QW
3
xPWLT
Q
Q
Q
Q
Q
Q
QW
3
aR’
xPWLT
Q
Q
Q
Q
Q
QW
QW
4
Q
Q
Q
Q
Q
Q
Q
QW
4
xPWLT
Q
Q
Q
Q
Q
Q
QW
QW
5
Q
Q
Q
Q
Q
Q
Q
QW
5
Q
Q
Q
Q
Q
Q
Q
QW
Q
Q
Q
Q
Q
Q
Q
QW
5
6
7
8
PWLT xPWLT
Q
Q
Q
Q
Q
QW
1
2
xPWLT
Q
Q
Q
Q
Q
Q
QW
3
Q
Q
Q
Q
Q
Q
Q
QW
QW
4
Q
Q
Q
Q
Q
Q
Q
QW
5
Q
Q
Q
Q
Q
Q
Q
3
4
5
6
7
8
1
PWLT
Q
Q
Q
Q
Q
Q
QW
1
2
Q
Q
Q
Q
Q
Q
Q
QW
3
Q
Q
Q
Q
Q
Q
Q
QW
4
Q
Q
Q
Q
Q
Q
Q
5
Q
Q
Q
Q
Q
Q
Q
2
1
4
2
step 3
step 2 3
1
6
Q
Q
Q
Q
Q
Q
Q
QW
6
Q
Q
Q
Q
Q
Q
Q
QW
6
Q
Q
Q
Q
Q
Q
Q
QW
6
7
Q
Q
Q
Q
Q
Q
Q
QW
7
Q
Q
Q
Q
Q
Q
Q
QW
7
Q
Q
Q
Q
Q
Q
Q
QW
7
Q
Q
Q
Q
Q
Q
Q
QW
8
QW
QW
QW
QW
QW
QW
QW
QW
8
QW
QW
QW
QW
QW
QW
QW
QW
8
QW
QW
QW
QW
QW
QW
QW
QW
8
QW
QW
QW
QW
QW
QW
QW
QW
step 5
step 4 1
2
3
4
5
6
7
8
1
PWLT
BR00
subH
AR’
xPWLT
Q
Q
QW
2
bR00
PWLT
AR’
xPWLT
Q
Q
Q
QW
3
subV
aR’
xPWLT
Q
Q
Q
Q
QW
4
aR’
xPWLT
Q
Q
Q
Q
Q
QW
xPWLT
5
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
QW
7
Q
Q
Q
Q
Q
Q
Q
QW
8
QW
QW
QW
QW
QW
QW
QW
QW
2
3
4
5
6
7
8
PWLT
BR0u0
BR1S
QRD
QRC
QRB
subH
PWRB
bR0u0
PWLT
QR0S
BR11
QRB
subH
AR’
xPWRB
3
bR1S
QR0S
PWLT
BR00
subH
AR’
xPWLT
QW
QRD
bR11
bR00
PWLT
AR’
xPWLT
Q
QW
5
QRC
QRB
subV
aR’
xPWLT
Q
Q
QW
7
8
QRB
subV
aR’
Q
QW
subV
aR’
xPWLT
Q
Q
Q
Q
QW
QW
QW
QW
QW
QW
QW
PWRB xPWRB
xPWLT
Q
3
4
5
6
7
8
BR0u1
BR10
QRC
odd
subH
AR’
xPWRB
QW
2
bR0u1
PWLT
BR0S
odd
subH
AR’
xPWLT
QW
QW
3
bR10
bR0S
PWLT
BR01
AR’
xPWLT
Q
QW
Q
QW
4
QRC
odd
bR01
PWLT xPWLT
Q
Q
QW
Q
QW
5
odd
subV
aR’
xPWLT
Q
Q
Q
QW
Q
Q
QW
6
subV
aR’
xPWLT
Q
Q
Q
Q
QW
Q
Q
Q
QW
7
aR’
xPWLT
Q
Q
Q
Q
Q
QW
QW
QW
QW
QW
QW
8
xPWRB
QW
QW
QW
QW
QW
QW
QW
3
4
5
6
7
8
1
2
3
4
5
6
7
8
BR0S
odd
subH
AR’
xPWLT
Q
QW
1
PWLT
QR0S
BR11
QRB
subH
AR’
xPWLT
QW
2
bR0S
PWLT
BR01
AR’
xPWLT
Q
Q
QW
2
QR0S
PWLT
BR00
subH
AR’
xPWLT
Q
3
odd
bR01
PWLT xPWLT
Q
Q
Q
QW
3
bR11
bR00
PWLT
AR’
xPWLT
Q
Q
4
subV
aR’
xPWLT
Q
Q
Q
Q
QW
4
QRB
subV
aR’
xPWLT
Q
Q
5
aR’
xPWLT
Q
Q
Q
Q
Q
QW
5
subV
aR’
xPWLT
Q
Q
Q
6
xPWLT
Q
Q
Q
Q
Q
Q
QW
6
aR’
xPWLT
Q
Q
Q
7
Q
Q
Q
Q
Q
Q
Q
QW
7
xPWLT
Q
Q
Q
8
QW
QW
QW
QW
QW
QW
QW
QW
8
QW
QW
QW
step 9
1
4
6
2
PWLT
2
PWLT
step 8
2
1
1
1
1
QW
6
1
step 7
step 6
Q
step 12
2
3
4
5
6
7
8
1
PWLT
BR0uS
QR10
BR01
QRD
QRC
AL1
PWRB
2
bR0uS
PWLT
BR0u1
BR10
QRC
odd
subH
PWRB
3
QR10
bR0u1
PWLT
BR0S
odd
subH
AR’
xPWRB
4
bR01
bR10
bR0S
PWLT
BR01
AR’
xPWLT
QW
5
QRD
QRC
odd
bR01
PWLT xPWLT
Q
QW
6
QRC
odd
subV
aR’
xPWLT
Q
QW
subV
aR’
xPWLT
Q
Q
Q
QW
QW
QW
QW
QW
QW
7
AL1
8
PWRB
PWRB xPWRB
Q
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
QRB
subV
aR’
xPWLT
Q
QW
6
odd
bR01
subV
aR’
xPWLT
Q
Q
QW
7
bL01
bL01
AL1
subV
aR’
QW
QW
QW
QW
8
PWRB
PWRB
PWRB
6
step 13
AL
QLA
AL0
8
PWRB
PWRB
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
3
4
5
6
7
8
1
PWLT
BR0v0
RL1
P1d
PA
QLB
BL00
PWRB
2
bR0v0
PWLT
BR0v0
QR11
P1s
QLA
BL00
3
RL1
bR0v0
PWLT
BR0u0
BR1S
AL
QLA
PWRB xPWRB
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
4
p1d
QR11
bR0u0
PWLT
QR0S
BR11
AL0
PWRB
4
p1
AL
bR0uS
PWLT
BR0u1
P0s
BL01
PWRB
4
p1
pA
p0
PWLT
P0
P0
P0
PWRB
5
pA
p1s
bR1S
QR0S
PWLT
BR00
subH
PWRB
5
p1
p1
p0d
bR0u1
PWLT
BR0S
AL1
PWRB
5
p1
p1
p0
p0
PWLT
P1
P1
PWRB
6
QLB
QLA
AL
bR11
bR00
PWLT
AR’
xPWRB
6
AR
AR
pA
p0s
bR0S
PWLT
BR01
PWRB
6
pA
pA
p0
p0
p1
PWLT
P0
PWRB
7
8
bL00
PWRB
bL00
PWRB
QLA
PWRB
AL0
subV
aR’
PWRB PWRB xPWRB
xPWLT
QW
QW
QW
7
8
bL0S
PWRB
bL0S
PWRB
bL01
PWRB
bL01
AL1
PWRB PWRB
bR01
PWLT xPWRB
PWRB xPWRB
QW
7
8
p1
PWRB
p1
PWRB
QLA
AL
QRC
PWRB xPWRB
PWLT xPWLT
QW
xPWLT
Q
QW
QW
QW
QW
step 15
step 14
2
2
QRC
7
1
1
step 11
step 10
1
p0
PWRB
p0
PWRB
p1
PWRB
p0
PWRB
PWLT
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 xPWRB
(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
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
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
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
step 4
step 3
step 2
1
1
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
JD1
HQR2
HQRS
HS
xH
Q
Q
CQX
1
JD2
HQR1
HQR2
HQRS
HS
xH
Q
CQX
1
JD1
HQR2
HQR1
HQR2
HQRS
HS
xH
CQX
2
VQL2
JP
xH
xJ2
Q
Q
Q
HQX
2
VQL1
JD1
HS
xH
xJ2
Q
Q
HQX
2
VQL2
JD2
HQRS
HS
xH
xJ2
Q
3
VQLS
xV
xJ2
Q
Q
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
1
1
1
2
3
4
5
6
7
8
JD1
HQR2
HQR1
HQR2
HQR1
HQR2
HQRS
HKXs
HQR2
1
1
2
3
4
5
6
7
8
JX
HQR1
HQR2
HQR1
HQR2
HQR1
HGX
HKX
JD1
HQR2
1
2
3
4
5
6
7
8
JD2
HQRS
HS
xH
Q
Q
Q
CQX
2
VQLS
xJ
xJ2
Q
Q
Q
Q
HQX
3
VL
xJ2
Q
Q
Q
Q
Q
HQX
4
xV
Q
Q
Q
Q
Q
Q
HQX
5
CQX
VQX
VQX
VQX
VQX
VQX
VQX
JQX
1
2
3
4
5
6
7
8
1
JD2
HQR1
HQR2
HQR1
HQR2
HQRS
HS
xCQX
HQX
2
VQL1
JD1
HQR2
HQRS
HS
xH
xJ2
HQX
HQX
3
VI0
VQL2
JP
xH
xJ2
xJ2
Q
HQX
Q
HQX
4
VAR1
VQLS
xV
xJ2
xJ2
Q
Q
HQX
VQX
JQX
5
VKX
VKXs
VQX
VQX
VQX
JQX
step 10
step 9
step 8
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
HQR2
1
1
2
3
4
5
6
7
8
JBr2
HFW
HW
HQR1
HG0
HQLA
HAL2
HKX
HGX
2
VI0
JD2
HQR1
HQRS
HS
xH
xHQX1
2
V<S
HQR2
HQR1
HQRS
HS
xCQX
V
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 . ℓ T (ℓ, k) k k 1 ≤ k ≤ ℓ - 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
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
QX
QXT
QXT
ctrl
QXT
QXT
QXT
QXX
1
QX
QXT
L
QLS
D2
QXT
QXT
QXX
QXR
2
QXL
Q
L
QLS
D2
Q
Q
QXR
2
QXL
L
QLS
QL2
D1
D2
Q
QXR
QXR
3
QXL
ctrl
QLS
D2
QRS
ctrl
Q
QXR
3
ctrl
QLS
QL2
D1
QR2
QRS
ctrl
QXR
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
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
QX
QXT
QXT
QXT
QXT
QXT
QXT
QXX
1
QX
QXT
QXT
QXT
QXT
QXT
QXT
QXX
2
QXL
Q
Q
Q
Q
Q
Q
QXR
2
QXL
Q
Q
ctrl
Q
Q
Q
3
QXL
Q
Q
P
Q
Q
Q
QXR
3
QXL
Q
ctrl
D1
ctrl
Q
Q
4
QXL
Q
Q
Q
Q
Q
Q
QXR
4
QXL
Q
Q
ctrl
Q
Q
Q
5
QXX
QXB
QXB
QXB
QXB
QXB
QXB
QX
step 4
QXB
QXB
QXB
QX
5
6
7
8
1
KXs
QLS
QL2
QL1
QL2
D1
D2
QXX
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
S
QXB
QX
5
D2
D1
QR2
QR1
QR2
QRS
S
QX
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 1
2
3
4
5
6
7
8
1
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
QXB
step 7 2
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
4
QL2
QL1
D2
QR1
QR2
QR1
QR2
QRS
4
QL1
QL2
D1
QR2
QR1
QR2
QR1
GX
5
QL1
D2
QR1
QR2
QR1
QR2
QRS
KXs
5
QL2
D1
QR2
QR1
QR2
QR1
GX
KX
1
step 10
1
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
<S
X
QR1
G0
1
KX
ARA
BR2
<S
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
QLC
2
QRo2
Ar2
FGW
B>
QLC
BL2
3
SBRD QRe2 QRo1
KX
ARA SBRD QRe2 QRo1
2
ARB sBRD
3
sBRD
QLC
BL2
ALA
KX
5
QRo1
step 16
5
step 15
6
Ar3
4
0
2
a′u (v, p)(w, w) ≥ γ wV
∀(v, p) ∈ B(u, q),
∀w ∈ V.
$
0 1 0 2 3 ℄ * .
S
Q0 ⊂ Q S(q) ∈ g(q) + V
q ∈ Q0
a(S(q), q)(ϕ) = f (ϕ) ∀ϕ ∈ V.
c : Q0 → Z
S
!
c(q) := C(S(q))
" #" %
J = c′ (q)
'
? 1? ?c(q) − C¯ ?2 . Z 2
$
&
c
$
¯ J ∗ c(q) = J ∗ C.
( &
J
c ! ∂ci (q) = Jij = Ci′ (u)(wj ), ∂qj
i = 1 . . . n m , j = 1 . . . np ,
u = S(q) Ci ci Jij
J = c′ (q) wj ∈ gq′ (q) + V
j
a′u (u, q)(wj , ϕ) = −a′qj (u, q)(1, ϕ)
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