Wavelet Methods for PDE-Constrained Control Problems:
Optimal Preconditioners, Fast Iterative Solvers and Adaptivity
Angela Kunoth
Institut fur Angewandte Mathematik & Institut fur Numerische Simulation
Universitat Bonn, Germany
Central goal: Development of efficient solution algorithms with optimal linear complexity
Central issues: Iterative solvers, multilevel preconditioning and adaptivity
Problem classes:
(I) Optimal control problems constrained by linear elliptic PDEs with distributed or Neumann
boundary control ; single operator equation as constraint
(II) . . . . . . with Dirichlet boundary control ; saddle point problem as constraint
(III) Problem (I) with additional inequality constraints on the control
Supported by SFB 611 (Deutsche Forschungsgemeinschaft) 1
(I) PDE-Constrained Optimal Control Problem with Distributed Control
given y∗, f
ω > 0
minimize J(y, u) = 12‖y − y∗‖2H1−s(Ω)
+ ω2‖u‖2
(H1−t(Ω))′
subject to −∆y + y = f + u in Ω ⊂ Rd
∂y∂n
= 0 on ∂Ω
0 ≤ s ≤ 1 smoothness parameter for state y
0 ≤ t smoothness parameter for control u
A : H1(Ω)→ (H1(Ω))′ 〈Av,w〉 :=R
Ω(∇v · ∇w + vw)dx
; weak formulation nontrivial solution for y∗ 6≡ A−1f
minimize J(y, u) = 12‖y − y∗‖2H1−s(Ω)
+ ω2‖u‖2
(H1−t(Ω))′
subject to Ay = f + u
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 2
(II) PDE-Constrained Optimal Control Problem with Dirichlet Boundary Control
given y∗, f
ω > 0
ΩΓyΓ
minimize J(y, u) = 12‖Ty − y∗‖2
Hs− 1
2 (Γy)+ ω
2‖u‖2
Ht(Γ)
subject to −∆y + y = f in Ω ⊂ Rd
y = u on Γ control boundary
∂y∂n
= 0 on ∂Ω \ Γ
s, t ∈ [ 12, 32] smoothness parameters for state and control
〈Av,w〉 :=R
Ω(∇v · ∇w + vw)dx 〈Bv, q〉 :=R
Γ v q dΓ bilinear form on H1(Ω)× (H1/2(Γ))′
; weak formulation (appending essential Dirichlet b.c. by Lagrange multipliers)
minimize J(y, u) = 12‖y − y∗‖2Hs(Γy)
+ ω2‖u‖2
Ht(Γ)
subject to L` y
p
´
:=“
A BT
B 0
” “
yp
”
=“
fu
”
. . . allows also combination with fictitious domain method and changing boundary Γ . . .
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 3
(III) Distributed Control Problem and Control Inequality Constraints
Problem (I) with additional inequality constraints on control
minimize J(y, u) = 12‖y − y∗‖2H1−s(Ω)
+ ω2‖u‖2
(H1−t(Ω))′
subject to Ay = f + u
and u ≤ u ≤ u
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 4
PDE-constrained optimal control problems ; requires repeated solution of PDE constraint
Ay = f + u or L`yp
´
=`fu
´
; requires fast solver as core ingredient
Numerical Solution of a Single Elliptic PDE
Elliptic PDE Ay = f s.th. ‖Av‖Y ′ ∼ ‖v‖Y ; find y ∈ Y : 〈v,Ay〉 = 〈v, f〉 for all v ∈ Y
Conventional discretization on a uniform grid: Yh ⊂ Y dimYh <∞ ; Ah yh = fh
Obstructions:
Large linear systems of equations ; iterative solver
High desired accuracy ; small h ; larger problem ; worse condition cond2(Ah) ∼ h−2
LBB condition for saddle point problems
Resolution of singularities in data and/or geometry ; small h
Ingredients for Efficient Numerical Solution:
(i) Multilevel preconditioner Ch
multigrid methods, BPX preconditioner, wavelet discretization ; cond2(ChAh) ∼ 1
(ii) Nested iteration
(iii) Additionally: Adaptive refinement
a–posteriori error estimation ; local grid refinement ; convergence/convergence rates?
First goal: Realize discretization error accuracy ε with minimal amount of work O(N(ε))
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 5
A-priori Estimates for Finite Elements
Quality measure: Approximation in norm ‖y − yh‖L2(Ω) ≤ ε
A–priori error estimates: Ω ⊂ Rd dimYh = N ∼ h−d uniform grid
‖y − yh‖L2(Ω) <∼ hr ‖y‖Hr(Ω) yh ∈ Yh 0 ≤ r ≤ rmax
⇐⇒ ‖y − yN‖L2(Ω) <∼ N−r/d ‖y‖Hr(Ω)
‖y − yN‖H1(Ω)
<∼ N−(r−1)/d ‖y‖Hr(Ω)
N degrees of freedom ←→ accuracy O(N−r/d)
Approximation rate determined by
(i) approximation order rmax of Yh
(ii) space dimension d
(iii) amount of smoothness of y in L2
Target: Realize discretization error accuracy ε ∼ h2 ∼ 2−2J for grid with spacing h ∼ 2−J
Problem complexity: For h ∼ 2−J a total of N ∼ 2Jd unknowns
Optimal complexity for iterative solver: Minimal amount of work is O(N)
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 6
Ingredients for Efficient Numerical Solution: (i) Multilevel Preconditioner
Asymptotically optimal preconditioner: Ch such that cond2(ChAh) ∼ 1
and setup and application of Ch in optimal linear complexity O(N)
Schwarz iterative schemes based on subspace corrections ; multilevel schemes:
• multiplicative schemes ; multigrid methods Brandt, Braess, Bramble, Hackbusch . . .
• additive schemes ; BPX preconditioner; wavelet discretizationBramble, Pasciak, Xu, Yserentant, Oswald, Dahmen, Kunoth . . .
Relevant idea from Approximation Theory: Multilevel characterization of function spaces
and norm equivalences
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 7
Multilevel Characterization of Function Spaces
Multiresolution Yj0 ⊂ Yj0+1 ⊂ . . . ⊂ Yj ⊂ Yj+1 ⊂ . . . Y, closY
“
S∞j=j0
Yj
”
= Y
Linear (orthogonal) projectors Qj : Y → Yj s.th. QjQℓ = Qj for j ≤ ℓ ; Qj −Qj−1 projector
Theorem: [Dahmen, Kunoth ’92], [Oswald ’92]
(S) Φj uniformly stable basis for Yj : ‖c‖ℓ2 ∼ ‖cT Φj‖L2
(J) Jackson estimate infvj∈Yj
‖v − vj‖L2<∼ 2−sj‖v‖Hs v ∈ Hs 0 < s ≤ d
(B) Bernstein inequality ‖vj‖Hs <∼ 2sj‖vj‖L2
vj ∈ Yj s < t
=⇒ Norm equivalence
‖v‖2Hs ∼
∞X
j=j0
22sj‖(Qj −Qj−1)v‖2L2s ∈ (−σ, σ)
0 < σ := mind, t, 0 < σ := mind, t,
Proof: (J) and discrete Hardy inequality ; upper estimate for ‖ · ‖Hs .
(B), ‖Qj‖L2<∼ 1 and Whitney estimate ; lower estimate
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 8
Norm Equivalence for Optimal Preconditioning
Theorem: [Jaffard ’92], [Dahmen, Kunoth ’92], [Oswald ’92]
Y = Hs C−1J := Aj0Qj0 +
JX
j=j0
22sj(Qj −Qj−1)
is optimal preconditioner for AJ : YJ → YJ : cond2(C1/2J AJC
1/2J ) ∼ 1 as J →∞
Proof: Isomorphism ‖Av‖Y ′ ∼ ‖v‖Y on YJ combined with norm equivalence for Y = Hs
Realization of C−1J :
• Any s ∈ (−σ, σ): Explicit representation of (Qj −Qj−1)v ; wavelet basis together with
diagonal Ds := (2sj)j=j0...J ; Fast Wavelet Preconditioner (FWT) realizes preconditioning in
optimal linear complexity [Jaffard ’92], [Dahmen, Kunoth ’92]
• s > 0 : Replace CJ = Aj0Qj0 +J
X
j=j0
2−2sj(Qj −Qj−1) by spectrally equivalent preconditioner
C−1J := Aj0Qj0 +
JX
j=j0
2−2sjQj
; BPX preconditioner is also optimal preconditioner
developed by [Bramble, Pasciak, Xu ’90], optimality proved by [Dahmen, Kunoth ’92], [Oswald ’92]
Hierarchical basis preconditioner by [Yserentant ’89] not optimal
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 9
Building Blocks: (Biorthogonal Spline–) Wavelets
H Hilbert space with ‖ · ‖H H′ dual space for H with 〈·, ·〉
Ψ := ψλ : λ ∈ II ⊂ H Wavelets II (infinite) index set
(NE) Ψ Riesz basis for H
v ∈ H: v = vT Ψ :=X
λ∈II
vλ ψλ such that ‖v‖H ∼ ‖v‖ℓ2(II)
(L) Locality diam (suppψλ) ∼ 2−|λ| |λ| resolution
ψλ centered around 2−|λ|k
(CP) Vanishing moments
0 1
ψ2,2
ψ2,1
[Dahmen, Kunoth, Urban ’99] [Dahmen, Schneider ’99; Kunoth, Sahner ’05] [Harbrecht, Schneider ’00]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 10
Numerical Results with Fast Wavelet Transform: Spectral Condition Numbers
Elliptic partial differential operator on Ω = (0, 1)d with FWT preconditioning
−∆ + 1 (−∆ + 1)CK
j 0 1 0 1
3 229 22.3 256 27.1
4 244 23.9 263 27.9
5 255 25.0 289 30.6
6 262 25.7 301 31.9
8 271 26.6 319 33.9
10 276 27.1 330 35.0
12 278 27.3 337 35.8
space dimension d = 1
−∆ + 1 (−∆ + 1)CK
j 0 1 4 5 0 1 3 4
3 519 78.2 76.0 49.5 256 27.8 17.3 9.64
4 627 129 128 124 308 33.4 20.9 11.8
5 646 149 149 147 372 40.4 25.3 14.3
6 664 165 165 165 416 45.1 28.2 16.0
8 681 179 179 179 480 52.1 32.6 18.4
space dimension d = 2
−∆ + 1 (−∆ + 1)CK
j 0 9 0 1 4
3 1103 269 256 28.5 18.3
4 1917 1913 520 57.8 37.1
5 2228 2222 557 62.0 39.8
6 2459 2443 572 63.6 40.9
space dimension d = 3
Uniformly bounded and absolutely small spectral condition numbers cond2(AJ ) [Burstedde ’05]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 11
Ingredients for Efficient Numerical Solution: (ii) Nested Iteration
Recall goal: realize discretization error accuracy εJ ∼ h2 ∼ 2−2J for grid with spacing h ∼ 2−J
with minimal amount of work O(N) N ∼ 2Jd unknowns
Naive strategy:
• Iterate only on highest level J and iterate until discretization error accuracy
needs O(J) = O(− log εJ ) iterations to achieve prescribed discretization error accuracy
εJ ∼ 2−2J
• Each application of optimally conditioned AJ requires O(NJ ) arithmetic operations
; a total of O(J NJ ) arithmetic operations iterating on finest level only
Theorem:
Starting with coarsest level j0, solve Ajyj = fj on each level j up to discretization error
accuracy εj and prolongate result from level j to next level j + 1 as initial guess
; Optimal preconditioner + nested iteration yields method of optimal complexity O(NJ )
to reach discretization error accuracy on finest level J
Proof: For mupliplicative Schwarz schemes: known and known as full multigrid
For additive preconditioners: optimal condition of Aj ; fixed amount of iterations on each level to reach discretization error accuracy on that level;
spaces nested and Nj ∼ 2dj and geometric series argument [Dahmen, Kunoth, Schneider ’99]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 12
Application to (I) PDE-Constrained Optimal Control Problem with Distributed Control
Control problem in wavelet coordinates
minimize J(y, u) = 12‖D−s (y − y∗)‖2 + ω
2‖Dt u‖2 0 ≤ s ≤ 1, 0 ≤ t
subject to Ay = f + u A : ℓ2 → ℓ2 automorphism ‖ · ‖ := ‖ · ‖ℓ2
Necessary and Sufficient Conditions
Lagr(y,u,p) := J(y,u) +˙
p, Ay − (f + D−t u)¸
δLagr = 0 ;
Ay = f + D−t u
AT p = −D−2s (y − y∗)
ωu = D−t p
⇐⇒ Qu = g
Q : ℓ2 → ℓ2 automorphism
whereQ := D−tA−T D−2sA−1D−t + ωI symmetric positive definite
g := D−tA−T D−2s(y∗ −A−1f)
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 13
Nested-Iteration-Inexact-Conjugate-Gradient Algorithm NIICG
Essential idea: Conjugate gradient (cg) iteration on Qu = g (outer loop) and cg iteration for primal
and dual systems (inner iterations); combined with nested iteration [Burstedde, Kunoth ’05]
Numerical Examples for Distributed Control Problem
d = 2: y∗ =q
|x− ( 13, 13)T | ω = 1 f ≡ 1 s = t = 0 runtime 177 sec
j ‖rj‖ #O #E #A #R ǫR(y) ǫP (y) ǫR(u) ǫP (u)
3 1.60e-04 3.73e-04 1.29e-05 3.51e-05
4 7.87e-06 6 4 1 18 1.41e-04 2.17e-04 5.11e-06 5.90e-06
5 3.89e-06 5 4 1 19 1.67e-05 8.34e-05 1.77e-06 1.79e-06
6 2.02e-06 4 4 2 17 1.43e-05 4.30e-05 7.68e-07 7.69e-07
7 1.13e-06 2 7 3 16 1.30e-05 2.39e-05 7.19e-07 7.19e-07
8 4.76e-07 5 3 1 15 1.42e-06 9.90e-06 1.85e-07 1.85e-07
9 2.25e-07 3 5 2 16 1.12e-06 4.51e-06 1.33e-07 1.33e-07
10 1.32e-07 3 5 2 16 9.48e-07 9.48e-07 1.10e-07 1.10e-07
d = 3: y∗ =q
|x− ( 13, 13, 13)T | ω = 1 f ≡ 1 s = t = 0 runtime 3502 sec
j ‖rj‖ #O #E #A #R ǫR(y) ǫP (y) ǫR(u) ǫP (u)
3 1.41e-04 2.92e-04 1.13e-05 2.36e-05
4 6.09e-06 10 9 1 49 1.27e-04 1.78e-04 3.46e-06 3.79e-06
5 3.25e-06 10 7 1 58 1.11e-05 6.14e-05 9.47e-07 9.53e-07
6 1.71e-06 7 6 1 57 1.00e-05 2.86e-05 5.03e-07 5.03e-07
7 8.80e-07 6 6 1 53 9.19e-06 9.19e-06 3.72e-07 3.72e-07
3.2GHz Pentium IV computer (family 15, model 4, stepping 1, with 1MB L2 Cache) [Burstedde ’05]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 14
Application to (II) PDE-Constrained Optimal Control Problem with Dirichlet Boundary Control
Optimality conditions ; system of saddle point problems ; additional inner iterations
[Kunoth ’01]
L“y
p
”
=“f
u
”
ωRHt(Γ)u = µ
LT“z
µ
”
=“−TT R
Hs− 1
2 (Γy)(Ty − yΓy )
0
”
⇐⇒ NU = F :⇐⇒
0
@
L E
E LT
1
A
0
B
B
B
B
B
@
y
p
z
u
1
C
C
C
C
C
A
:=
0
B
B
B
B
B
@
A BT 0 0
B 0 0 −ω−1R−1Ht(Γ)
TT RH
s− 12 (Γ)
T 0 AT BT
0 0 B 0
1
C
C
C
C
C
A
0
B
B
B
B
B
@
y
p
z
u
1
C
C
C
C
C
A
=
0
B
B
B
B
B
@
f
0
−TT yΓy
0
1
C
C
C
C
C
A
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 15
Control Problem with Dirichlet Boundary Control: Solution Graphics
[Pabel ’05]
00.25
0.50.75
1
0
0.250.5
0.751
0
0.25
0.5
0.75
1
00.25
0.50.75
0
0.250.5
0.75
Ω = 100
Ω = 10
Ω = 5
Ω = 2
Ω = 1
Ω = 0.5
Ω = 0.4
Ω = 0.3
Ω = 0.2
Ω = 0.1
Ω = 0.01
00.25
0.50.75
1
0
0.250.5
0.751
0
0.25
0.5
0.75
1
00.25
0.50.75
0
0.250.5
0.75
s = 0.0
s = 0.1
s = 0.2
s = 0.3
s = 0.4
s = 0.5
s = 0.7
s = 0.9
s = 1.0
s = 1.5
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 16
Numerical Results for Control Problem with Dirichlet Boundary Control
All–In–One Solver: cg method & nested iteration on NT NU = NT F
Best results for inexact gradient iteration on u and Uzawa with conjugate directions for each of saddle
point problems & nested iteration
d = 2: yΓy ≡ 1 f ≡ 1 ω = 1 s = 12
L2(Γy) t = 12
H1/2(Γ) [Pabel ’05]
P with D−11 P O with D−1
a
J ‖r(kJ )J
‖ ‖y−y(kJ )J
‖ kJ#Int-It
kJ
#Int-ItkJ
‖r(kJ )J
‖ ‖y−y(kJ )J
‖ kJ#Int-It
kJ
#Int-ItkJ
4 1.6105e-02 7.7490e-00 0 – – 3.9807e-02 8.5283e-02 4 1.5 2.5
5 1.6105e-02 7.7506e-00 0 – – 1.3970e-02 2.5233e-02 2 1 1.5
6 6.3219e-03 1.7544e-02 2 1 1 1.4421e-02 1.2301e-02 1 1 1
7 5.8100e-03 3.3873e-02 0 – – 6.1354e-03 4.6339e-03 2 1 1.5
8 1.6378e-03 3.4958e-03 2 1 1 2.2413e-03 1.7083e-03 2 1.5 1.5
9 1.8247e-03 7.4741e-03 0 – – 9.2961e-04 8.8526e-04 2 1.5 1
10 4.3880e-04 9.2663e-04 2 1 1.5 9.2954e-04 8.7262e-03 0 – –
11 4.6181e-04 1.8486e-03 0 – – 4.4790e-04 3.8321e-04 2 1.5 1.5
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 17
Distributed Control Problem with Inequality Constraints
Treatment of control inequality constraints:
projected gradient method — interior point (IP) method — primal-dual active set (PDAS) method
u(x) ≡ −0.5 y∗(x) = x1(1− x1) · x2(1− x2)− 32[x1(1− x1) + x2(1− x2)]
f(x) :=
8
<
:
0.5 + 2[x1(1− x1) + x2(1− x2)] for x ∈ K•,
16x1(1− x1)x2(1− x2) + 2[x1(1− x1) + x2(1− x2)] otherwise
; exact solution y(x) = x1(1− x1) · x2(1− x2)
j J(y(k), u(k)) ε(u(k)j
) ε(y(k)j
) k ∅ (#CG) run time
4 6.963009e + 01 1.099181e − 02 1.125383e − 02 2 9 31.1186
5 6.577754e + 01 1.910503e − 03 5.154380e − 03 3 45 40.7679
6 6.391144e + 01 7.092608e − 04 2.361333e − 03 3 92 181.723
7 6.305831e + 01 2.237459e − 04 1.115190e − 03 3 148 1484.53
8 6.274058e + 01 8.386212e − 05 5.381711e − 04 3 186 11545.4
9 6.268009e + 01 2.803506e − 05 2.625252e − 04 3 230 100938
<∼ 2−3/2 j <
∼ 2−j ∼ const
PDAS [Hoffmann & K ’07]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 18
So far: uniform grids . . . . . . further reduction of complexity by adaptive refinement
Recall: A-priori Estimates for Finite Elements
Quality measure: Approximation in norm ‖y − yh‖L2(Ω) ≤ ε
A–priori error estimates: Ω ⊂ Rd dimYh = N ∼ h−d uniform grid
‖y − yh‖L2(Ω) <∼ hr ‖y‖Hr(Ω) yh ∈ Yh 0 ≤ r ≤ rmax
⇐⇒ ‖y − yN‖L2(Ω) <∼ N−r/d ‖y‖Hr(Ω)
‖y − yN‖H1(Ω)
<∼ N−(r−1)/d ‖y‖Hr(Ω)
N degrees of freedom ←→ accuracy O(N−r/d)
Approximation rate determined by
(i) approximation order rmax of Yh
(ii) space dimension d
(iii) amount of smoothness of y in L2
‘Wish list’ for adaptive method
• uses no a–priori knowledge
• realizes rate under less smoothness assumptions
• universally usable: automatic realization of order
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 19
Paradigm for One Stationary PDE [Cohen, Dahmen, DeVore ’99–’01]
(i) Well–posed variational problem: given f ∈ y′, find y ∈ Y : 〈v,Ay〉 = 〈v, f〉 ∀v ∈ Y Ay = f
(MP) ‖Av‖Y ′ ∼ ‖v‖Y for all v ∈ Y
(ii) Wavelet basis for Y : Ψ = ψλ : λ ∈ II ⊂ Y
(NE) c ‖v‖ℓ2 ≤ ‖P
λ∈II vλψλ‖Y ≤ C ‖v‖ℓ2 for all v = (vλ) ∈ ℓ2
Ay := (〈ψλ, Ay〉)λ∈II f = (〈ψλ, f〉)λ∈II
;
Theorem Ay = f ⇐⇒ Ay = f A : ℓ2 → ℓ2 and Ay = f well-posed in ℓ2
(MP) + (NE) =⇒ (c2 cA)−1 ‖w‖ℓ2 ≤ ‖Av‖ℓ2 ≤ C2CA ‖w‖ℓ2 w ∈ ℓ2
(iii) (Idealized) iteration
yn+1 = yn + Cn(f −Ayn) n = 0, 1, 2, . . . ‖yn+1 − y‖ℓ2 ≤ ρ ‖yn − y‖ℓ2 ρ < 1
(iv) Approximate realization through adaptive evaluation of Ayn (CP)
Solve [ε,C,A, f ]→ y(ε)
(i) Initialize ‖y − v‖ℓ2 ≤ δ η = δ
(ii) perturbed update v + rη := v + Res [η,C,A, f ,v]→ v η/2→ η
until ‖rη‖ℓ2 “small enough”
(iii) Coarse [cδ,v]→ v δ → δ/2 η = δ go to (ii) until δ ≤ ε
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 20
Complexity Analysis
Based on benchmark:
decay rate s for (wavelet-)best N term approximation As := v ∈ ℓ2 : ‖v − vN‖ℓ2 <∼ N−s
Work/accuracy balance of best N term approximation:
Target accuracy ε (∼ N−s) ←→ Work ε−1/s (∼ N)
Convergence and Complexity
(Idealized) iteration yn+1 = yn + Cn(f −Ayn) update via Res [η,C,A, f ,v]→ rη
Theorem [CDD] Output rη of Res [η,C,F, f ,v] satisfies ‖rη‖As <∼ ‖v‖As + ‖u‖As
and # supp rη , #flops <∼ η−1/s (‖v‖
1/sAs + ‖u‖
1/sAs + 1)
=⇒ for every variational problem satisfying (MP) perturbed scheme Solve has properties:
(I) For every target accuracy ε > 0 Solve produces after finitely many steps
approximate solution uε such that ‖u− uε‖ℓ2 ≤ ε
(II) Exact solution u ∈ As =⇒ suppuε, # flops ∼ ε−1/s ∼ N
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 21
; construct Res [η,C,F, f ,v] such that output rη satisfies ‖rη‖As <∼ ‖v‖As + ‖u‖As
and #supp rη , #flops <∼ η−1/s (‖v‖
1/sAs + ‖u‖
1/sAs + 1)
F(u) = Au ; Properties of Res derived from s– compressibility of A
Core Ingredient of Solve : Compressible Operators
A s∗–compressible: for every 0 < s < s∗ there exists Aj
with ≤ αj2j nonzero entries per row and column such that
‖A−Aj‖ ≤ αj2−sj j ∈ N0
X
j∈N0
αj <∞
supp v <∞ ; v[j] := v2j best 2j approximations wj := Ajv[0] + Aj−1v[1] + · · ·+ A0v[j]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 22
Generalization to (I) PDE-Constrained Optimal Control Problem with Distributed Control
Control problem in wavelet coordinates
minimize J(y, u) = 12‖D−s (y − y∗)‖2 + ω
2‖Dt u‖2 0 ≤ s ≤ 1, 0 ≤ t
subject to Ay = f + u A : ℓ2 → ℓ2 automorphism ‖ · ‖ := ‖ · ‖ℓ2
Necessary and Sufficient Conditions
Lagr(y,u,p) := J(y,u) +˙
p, Ay − (f + D−t u)¸
δLagr = 0 ;
Ay = f + D−t u
AT p = −D−2s (y − y∗)
ωu = D−t p
⇐⇒ Qu = g
Q : ℓ2 → ℓ2 automorphism
whereQ := D−tA−T D−2sA−1D−t + ωI symmetric positive definite
g := D−tA−T D−2s(y∗ −A−1f)
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 23
Convergence and Complexity Analysis
Essential idea: Res for Solve [. . . ,Q, . . .] reduced to Res for Solve [. . . ,A, . . .]
and system of Euler equations ←→ condensed system
Theorem [Dahmen, Kunoth ’05]
For any target accuracy ε > 0 Solve [ε,Q,g]→ uε converges in finitely many steps
‖u− uε‖ ≤ ε ‖y − yε‖ <∼ ε ‖p− pε‖ <
∼ ε uε,yε,pε finitely supported
u,y,p ∈ As =⇒
(# suppuε) + (# suppyε) + (# supppε) <∼ ε−1/s
“
‖u‖1/sAs + ‖y‖
1/sAs + ‖p‖
1/sAs
”
‖uε‖As + ‖yε‖As + ‖pε‖As <∼ ‖u‖As + ‖y‖As + ‖p‖As
#flops ∼ ε−1/s
Extension to PDE-constrained control problems with Dirichlet boundary control
; System of saddle point problems ; additional inner iterations [Kunoth ’05]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 24
Numerical Example for Distributed Control Problem (1D)
min J(y, u) J(y, u) = 12‖y − y∗‖2H1(Ω)
+ 12‖u‖2
L2(Ω)(s = 0 t = 1)
under constraints
8
<
:
−y′′ + y = f + u in Ω := (0, 1)dydn
= 0 at 0, 1
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
right hand side f
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
target state y*
right hand side f target state y∗
Wavelet coefficients: State y adjoint state p control u [Burstedde, Kunoth ’05]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 25
Numerical Example for Distributed Control Problem (2D)
min J(y, u) J(y, u) = 12‖y − y∗‖2
H1/2(Ω)+ 1
2‖u‖2
L2(Ω)y∗ = h2 ⊗ h2 (s = 1/2 t = 0)
under constraints
8
<
:
−∆y + y = D3/2(h1 ⊗ h1) + u in Ω := (0, 1)2
dydn
= 0 on ∂Ω
Optimal rate in energy norm (r = 2 d = 2) is r−1d
= 12
j ‖rj‖ #O #E #A #R S Nad ǫP (y) ǫP (u)
3 1.31e-02 2.19e-04
4 3.09e-04 1 6 1 11 54.0% 156 1.02e-02 2.19e-04
5 3.55e-04 1 6 2 11 49.0% 534 5.08e-03 2.19e-04
6 1.80e-04 4 4 1 20 51.6% 2182 2.55e-03 2.19e-04
7 1.22e-04 6 6 1 21 43.1% 7169 1.31e-03 2.19e-04
8 5.61e-05 8 8 1 23 36.0% 23745 6.73e-04 2.19e-04
9 2.22e-05 10 9 1 23 30.6% 80525 3.33e-04 1.55e-04
10 1.15e-05 12 9 2 24 27.6% 289790 1.25e-04 1.07e-04
σ ≈ 0.55 [Burstedde ’05]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 26
Numerical Example for Distributed Control Problem (2D) Target y∗
0 0.2 0.4 0.6 0.8 x1 00.20.40.60.8x2
0.40.60.8
1
[Burstedde ’05]
type e = (1, 0) type e = (0, 1)
4
5
6
7
1.69e-03
0.00e+00 4
5
6
7
1.69e-03
0.00e+00
4
5
6
7
2.08e-01
0.00e+00 4
5
6
7
2.08e-01
0.00e+00
4
5
6
7
4.34e-03
0.00e+00 4
5
6
7
4.34e-03
0.00e+00
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 27
Summary
• Fast iterative solution based on multiscale ideas of first order optimality conditions for
PDE-constrained control problems
• Uniformly bounded condition numbers of system matrices + nested iteration
=⇒ solver produces discretization error accuracy in optimal linear complexity O(NJ )
• Application to control problems constrained by linear elliptic PDEs with distributed or Dirichlet
boundary control
– Modelling of objective functional [Burstedde, Kunoth ’05]
– Uniformly bounded condition numbers of system matrices
– A–posteriori error estimators for coupled system of operator equations
– automated adaptive refinement for each of state y, control u and adjoint state p
– convergence proof and algorithmic efficiency: first optimal complexity estimates
method has optimal work/accuracy rate
Extensions, Generalizations and Outlook
• Goal-oriented error estimation [Dahmen, Kunoth, Vorloeper ’05]
• Attractive for control problems with linear parabolic PDE constraints
; PDEs coupled globally in time – space-time approach in weak form
• Control problem with nonlinear elliptic PDE as constraints: SQP methods . . .Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 28
Numerical Results: BPX Preconditioner and Nested Iteration
Laplacian on sphere ∆S y = f f(x) = 2x exact solution y(x) = x
stopping criterion ‖rj‖ℓ2 ≤ 2−2j (∼ h2) piecewise linear finite elements
its: iterations on each level to reach stopping criterion [Maes, Kunoth, Bultheel ’05]
j cond ‖rj‖ℓ2its
1 3.1 2.4897e-05 12
2 3.7 1.6766e-05 9
3 4.6 4.7350e-06 11
4 5.5 4.5474e-06 11
5 6.2 1.6705e-06 12
6 6.7 1.0193e-06 12
7 7.0 6.2720e-07 12
8 7.4 1.6451e-07 13
Biharmonic equation on sphere ∆2S y = f C1 cubic conforming finite elements
j cond ‖rj‖ℓ2its
1 52.0 2.2290e-03 0
2 66.7 5.1424e-04 2
3 78.4 4.2928e-04 1
4 87.7 3.1846e-04 3
5 95.2 1.6570e-04 3
6 100.6 7.9261e-05 5
7 105.5 4.0583e-05 4
Optimality of BPX preconditioner [Kunoth ’94; Oswald ’94]
Numerical results [Maes, Kunoth, Bultheel ’05]
Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 29