Constrained Minimization and Multigrid
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1 Constrained Minimization and Multigrid C. Gräser (FU Berlin), R. Kornhuber (FU Berlin), and O. Sander (FU Berlin) Workshop on PDE Constrained Optimization Hamburg, March 27-29, 2008 Matheon
2 Outline Successive line search and multigrid (..., Yserentant 86, Xu 89,...) Self-adjoint elliptic obstacle problems - projected multilevel relaxation (Mandel 84, Kh. 94, Badea 06) - truncated monotone multigrid (Kh. 94) - truncated nonsmooth Newton multigrid and active set strategies (Gräser & Kh. 08) PDE constrained (L 2 or H 1 ) minimization with control constraints - nonsmooth Schur-Newton (multigrid) methods (Gräser & Kh. 06) - global convergence of primal-dual active set strategies (Hintermüller, Ito & Kunisch 03, Gräser 07) Message: Active Sets by Minimization
3 Outline Successive line search and multigrid (..., Yserentant 86, Xu 89,...) Self-adjoint elliptic obstacle problems - projected multilevel relaxation (Mandel 84, Kh. 94, Badea 06) - truncated monotone multigrid (Kh. 94) - truncated nonsmooth Newton multigrid and active set strategies (Gräser & Kh. 08) PDE constrained (L 2 or H 1 ) minimization with control constraints - nonsmooth Schur-Newton (multigrid) methods (Gräser & Kh. 06) - global convergence of primal-dual active set strategies (Hintermüller, Ito & Kunisch 03, Gräser 07) Message: Active Sets by Minimization
4 Spectral Properties of Self-Adjoint Elliptic Bilinear Forms quadratic minimization: Poisson equation: u H : J (u) J (v) v H H = H0(Ω), 1 J (v) = 1 2a(v, v) l(v), a(v, w) = ( v, w), l(v) = (f, v) Proposition: eigenfunctions e l, eigenvalues µ l : a(e l, v) = µ l (e l, v) v H, l = 1, 2,... e l a-orthogonal, µ l for l interpretation: a-orthogonal e l = (e l ) scale of frequencies basic idea of multigrid: almost a-orthogonal λ l = (λ l ) scale of frequencies
5 Spectral Properties of Self-Adjoint Elliptic Bilinear Forms quadratic minimization: Poisson equation: u H : J (u) J (v) v H H = H0(Ω), 1 J (v) = 1 2a(v, v) l(v), a(v, w) = ( v, w), l(v) = (f, v) Proposition: eigenfunctions e l, eigenvalues µ l : a(e l, v) = µ l (e l, v) v H, l = 1, 2,... e l a-orthogonal, µ l for l interpretation: a-orthogonal e l = (e l ) scale of frequencies basic idea of multigrid: almost a-orthogonal λ l = (λ l ) scale of frequencies
6 Spectral Properties of Self-Adjoint Elliptic Bilinear Forms quadratic minimization: Poisson equation: u H : J (u) J (v) v H H = H0(Ω), 1 J (v) = 1 2a(v, v) l(v), a(v, w) = ( v, w), l(v) = (f, v) Proposition: eigenfunctions e l, eigenvalues µ l : a(e l, v) = µ l (e l, v) v H, l = 1, 2,... e l a-orthogonal, µ l for l interpretation: a-orthogonal e l = (e l ) scale of frequencies basic idea of multigrid: almost a-orthogonal λ l = (λ l ) scale of frequencies
7 Discrete Quadratic Minimization triangulations: T 0,..., T j (successive refinement) nodes: N 0 N 1 N j nested finite element spaces: S 0 S 1 S j nodal basis: S k = span{ λ (k) p p N k }, dimension N k = n k Ritz-Galerkin discretization: u j S j : J (u j ) J (v) v S j goal: iterative solvers with O(n j ) complexity and mesh-independent convergence rates
8 Discrete Quadratic Minimization triangulations: T 0,..., T j (successive refinement) nodes: N 0 N 1 N j nested finite element spaces: S 0 S 1 S j nodal basis: S k = span{ λ (k) p p N k }, dimension N k = n k Ritz-Galerkin discretization: u j S j : J (u j ) J (v) v S j goal: iterative solvers with O(n j ) complexity and mesh-independent convergence rates
9 Successive Subspace Correction (..., Yserentant 86, Xu 92,... ) select search directions: λ l S j, l = 1,..., m, subspaces: V l = span {λ l } Algorithm (successive line search) given: w 0 := u ν S j. for l = 1,..., m: { solve: v l V l : J (w l 1 + v l ) J (w l 1 + v) v V l update: } w l := w l 1 + v l new iterate: u ν+1 := w m = u ν + m l=1 a-orthogonal search directions (λ l ) = exact solver! v l
10 Successive Subspace Correction (..., Yserentant 86, Xu 92,... ) select search directions: λ l S j, l = 1,..., m, subspaces: V l = span {λ l } Algorithm (successive line search) given: w 0 := u ν S j. for l = 1,..., m: { solve: v l V l : J (w l 1 + v l ) J (w l 1 + v) v V l update: } w l := w l 1 + v l new iterate: u ν+1 := w m = u ν + m l=1 a-orthogonal search directions (λ l ) = exact solver! v l
11 Subspace Correction and Multigrid Gauß Seidel relaxation: select λ l(p) := λ (j) p, p N j Theorem: exponential decay of convergence rates: ρ j 1 O(n 2 j ) multilevel relaxation: select λ l(p,k) := λ (k) p, p N k, k = 0,..., j optimal complexity: O(n j ) (classical multigrid V (1, 0) cycle, Gauß Seidel smoother) Theorem: (..., Yserentant 86, Oswald 91, Bramble,Pasciak,Wang & Xu 91, Dahmen & Kunoth 92, Bornemann & Yserentant 93,...) mesh-independent convergence rates: ρ j ρ < 1 ((λ l ) scale of frequencies)
12 Subspace Correction and Multigrid Gauß Seidel relaxation: select λ l(p) := λ (j) p, p N j Theorem: exponential decay of convergence rates: ρ j 1 O(n 2 j ) multilevel relaxation: select λ l(p,k) := λ (k) p, p N k, k = 0,..., j optimal complexity: O(n j ) (classical multigrid V (1, 0) cycle, Gauß Seidel smoother) Theorem: (..., Yserentant 86, Oswald 91, Bramble,Pasciak,Wang & Xu 91, Dahmen & Kunoth 92, Bornemann & Yserentant 93,...) mesh-independent convergence rates: ρ j ρ < 1 ((λ l ) scale of frequencies)
13 Multigrid on Complicated Domains (Kh. & Yserentant 94) computational domain Ω not resolved by coarse grid T k for k < j difficulty: reduced coarse-grid correction = reduced convergence speed remedy: truncated search directions λ (k) p (interpolation of λ (k) p to Ω) hope: preserved coarse-grid correction = preserved convergence speed
14 Multigrid on Complicated Domains (Kh. & Yserentant 94) computational domain Ω not resolved by coarse grid T k for k < j difficulty: reduced coarse-grid correction = reduced convergence speed remedy: truncated search directions λ (k) p (interpolation of λ (k) p to Ω) hope: preserved coarse-grid correction = preserved convergence speed
15 A Fractal Domain f 1, V (1, 0) cycle, symmetric Gauß-Seidel smoothing Initial triangulation T 0 Approximate solution on Ω 0 convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 std trc (0, 1)
16 A Fractal Domain f 1, V (1, 0) cycle, symmetric Gauß-Seidel smoothing Initial triangulation T 0 Approximate solution on Ω 1 convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 std trc (0, 1)
17 A Fractal Domain f 1, V (1, 0) cycle, symmetric Gauß-Seidel smoothing Initial triangulation T 0 Approximate solution on Ω 2 convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 std trc (0, 1)
18 A Fractal Domain f 1, V (1, 0) cycle, symmetric Gauß-Seidel smoothing Initial triangulation T 0 Approximate solution on Ω 3 convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 std trc (0, 1)
19 A Fractal Domain f 1, V (1, 0) cycle, symmetric Gauß-Seidel smoothing Initial triangulation T 0 Approximate solution on Ω 4 convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 std trc (0, 1)
20 A Fractal Domain f 1, V (1, 0) cycle, symmetric Gauß-Seidel smoothing Initial triangulation T 0 Approximate solution on Ω 5 convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 std trc (0, 1)
21 A Fractal Domain f 1, V (1, 0) cycle, symmetric Gauß-Seidel smoothing Initial triangulation T 0 Approximate solution on Ω 5 convergence rates: j=2 j=3 j=4 j=5 j=6 j=7 std trc (0, 1)
22 constrained minimization: Self-Adjoint Elliptic Obstacle Problems u j K j : J (u j ) J (v) v K j = {v S j v(p) [α(p), β(p)] p N j } Algorithm (successive line search) given: w 0 := u ν S j. for l = 1,..., m: { defect constraints: D l = ( w l 1 + K j ) V l solve: v l D l : J (w l 1 + v l ) J (w l 1 + v) v D l update: } w l := w l 1 + v l new iterate: u ν+1 := w m = u ν + m l=1 v l
23 Projected Multilevel Relaxation PMLR projected Gauß Seidel relaxation: select λ l(p) := λ (j) p, p N j Theorem: exponential decay of convergence rates: ρ j 1 O(n 2 j ) projected multilevel relaxation: select λ l(p,k) := λ (k) p, p N k, k = 0,..., j suboptimal complexity: O(n j log(n j )) (additional prolongations to check v l D l ) Theorem: (..., Badea, Tai and Wang 03, Badea 06) polylogarithmic convergence rates in 2D: ρ j 1 O(j + 1) 5
24 Monotone Iterations (Kh. 94) two-stage selection: a) λ l = λ (j) p l, l = 1,..., n j b) arbitrary λ l, l = n j + 1,..., m two-stage (energy-)monotone iteration: a) leading projected Gauß Seidel step: u ν j uν j = GS(u ν j ) b) (energy-) monotone coarse-grid correction Theorem: Monotone iterations converge globally goal: improve convergence speed of PMLR by adapting λ (k) p to the active set N j
25 Truncated Monotone Multigrid TMMG (Kh. 94) complicated reduced computational domain: active set N j not resolved by T k a) use truncated search directions λ (k) p (interpolation of λ (k) p to N j ) b) replace unknown active set N j by its actual approximation N j (u j) optimal complexity: O(n j ) by monotone restriction of defect constraints Theorem: global convergence asymptotic polylogarithmic convergence rates ρ j 1 O(1 + j) 2 no global mesh-independent convergence rates (next-neighbor inactivation by projected Gauß Seidel u ν j = GS(u ν j ))
26 Truncated Nonsmooth Newton Multigrid TNMG (Gräser & Kh. 08) modifications of TMMG to obtain TNMG b) ignore defect constraints (v l V l instead of v l D l = ( w l 1 + K j ) V l ) c) enforce monotonicity of coarse-grid correction by projection and damping alternative derivation of TNMG: a) nonsmooth Newton linearization of GS(u j ) u j = 0 at u ν j b) linear truncated multigrid step for the resulting linear system c) projection and damping Theorem: global convergence asymptotic polylogarithmic convergence rates ρ j 1 O(1 + j) 2 no global mesh-independent convergence rates (next-neighbor inactivation by projected Gauß Seidel u ν j = GS(u ν j ))
27 Truncated Nonsmooth Newton Multigrid TNMG (Gräser & Kh. 08) modifications of TMMG to obtain TNMG b) ignore defect constraints (v l V l instead of v l D l = ( w l 1 + K j ) V l ) c) enforce monotonicity of coarse-grid correction by projection and damping alternative derivation of TNMG: a) nonsmooth Newton linearization of GS(u) u = 0 at u ν j b) linear truncated multigrid step for the resulting linear system c) projection and damping Theorem: global convergence asymptotic polylogarithmic convergence rates ρ j 1 O(1 + j) 2 no global mesh-independent convergence rates (next-neighbor inactivation by projected Gauß Seidel u ν j = GS(u ν j ))
28 Multigrid versus Primal Dual Active Set (Hintermüller, Ito & Kunisch 03) primal dual TNMG TMMG inactivation proj. Jacobi proj. Gauß Seidel proj. Gauß Seidel linear solve exact linear MG step loc. proj. MG step convergence maximum principle monotonicity monotonicity common lack of robustness: no global mesh-independent convergence rates due to local inactivation hybrid approach HMG: a) global inactivation by PMLR (not by GS!): u j ν = PMLR(u ν j ) b) linear truncated multigrid step c) projection and damping Theorem: (Gräser & Kh. 08) global convergence, asymptotic polylogarithmic convergence
29 Spiral Active Set (Gräser & Kh 08) lower obstacle: u j ϕ j uniform refinement: j = 9 with unknowns PMLR TMMG TNMG active set N j asymptotic convergence rates
30 Iteration Histories 10 5 PMLR TMMG TNMG PMLR TMMG TNMG nested iteration starting from the obstacle
31 Degenerate Problem (Gräser & Kh 08) lower obstacle: u j ϕ j = I Sj ϕ, ϕ = f uniform refinement: j = 9 with unknowns PMLR TNMG HMG active set N j asymptotic convergence rates
32 Iteration Histories 10 5 PMLR TNMG HMG PMLR TNMG HMG nested iteration starting from the obstacle
33 Two-Body Contact in Linear Elasticity Kh. & Krause 01, Wohlmuth & Krause 03, Sander 08 coarse mesh (2 372 dof) refined mesh ( dof) stress (planar cut) implemented in Dune
34 Two-Body Contact in Linear Elasticity Kh. & Krause 01, Wohlmuth & Krause 03, Sander 08 coarse mesh (2 372 dof) refined mesh ( dof) stress (planar cut) implemented in Dune
35 Comparison with Linear Multigrid coarse-grid solver: interior point method Ipopt Waechter & Biegler 04 adaptive refinement: hierarchical error estimate, j = 4 1e MMG Linear MG MMGP MMG Linear MG MMGP Error Error e 04 1e 04 1e 06 1e 06 1e Iteration 1e Iteration initial iterate: u 0 j = 0 nested iteration linear multigrid convergence rates (ρ = 0.56)
36 PDE-Constrained Minimization with Control Constraints minimize J (y, u) = 1 2 a 1(y, y) a 2(u, u) l(y, u) pde constraints ( y, v) = u, v v S j box constraints: u K j = {u S j u 1} L 2 control problem: a 1 (y, v) = y, v, a 2 (u, v) = ε u, v (lumped L 2 -scalar product: diagonal) spatial Cahn-Hilliard problem (deep-quench limit): a 1 (y, z) = τ( y, z) + y, 1 z, 1, a 2 (u, v) = γ( u, v) + u, 1 v, 1
37 Set-Valued Saddle-Point Problems ( F B T ) ( w ) ( f ) B C λ g w = (y, u) T, F = A + χ R n 1 K j A R n,n s.p.d., B R m,n, C R m,m symmetric, positive semi-definite generalization: F = ϕ, ϕ : R n R strictly convex, l.s.c., proper, coercive: F 1 single valued, Lipschitz
38 Non-linear Schur Complement H(λ) = 0, H(λ) = BF 1 (f B T λ) + Cλ + g Proposition Let ϕ : R n R denote the polar functional of ϕ with F = ϕ J (λ) = ϕ (f B T λ) (Cλ, λ) + (g, λ) is Fréchet differentiable H = J H(λ) = 0 is equivalent to unconstrained (!) convex minimization λ R m : J (λ) J (v) v R m
39 Gradient-Related Descent Methods /..., Ortega & Rheinboldt 70,... ) λ ν+1 = λ ν + ρ ν d ν, d ν = H 1 ν J (λ ν ), H ν s.p.d. Assumption on d ν : γ v 2 (H ν v, v) Γ v 2 ν N Assumption on ρ ν : J (λ ν + ρ ν d ν ) J (λ ν ) c( J (λ ν ), d ν ) 2 / d ν 2 Theorem: The iteration is globally convergent.
40 Damping Strategies Armijo s strategy: select σ, β (0, 1) accept ρ, if J (λ ν + ρd ν ) J (λ ν ) + σρ( J ((λ ν, d ν )), else set ρ := βρ and try again drawback: evaluation of J is expensive inexact damping: approximate the solution of (H(λ ν + ρd ν ), d ν ) = 0 by bisection computational cost: evaluation of F 1 in each bisection step numerical experiments: usually: 1 step, exceptions: up to 8 steps both strategies satisfy the assumption on ρ
41 Inexact Evaluation of H 1 ν Proposition: Let d ν d ν = H 1 ν J (λ ν ). Then the accuracy conditions d ν d ν 1 ν d ν, (H(λ ν ), d ν ) < 0 ν N preserve convergence.
42 Selection of H ν : Nonsmooth Schur-Newton Methods Newton-like methods: λ ν+1 = λ ν ρ ν H 1 ν H(λ ν ) Newton methods: H ν = H (λ ν ) non-smooth Newton methods: H ν B H(λ ν ) H(λ ν ) Proposition: Let rank B = n and w ν = F 1 (f B T )λ ν. Then H ν := S(w ν ) = B 1 B T + C H(λ ν ) where  = T AT + I T, with T ii = 0, if i N (w ν ) else T ii = 1 L 2 -control: Newton-like (rank B = m < n) Cahn-Hilliard: Newton (rank B = n)
43 Selection of H ν : Nonsmooth Schur-Newton Methods Newton-like methods: λ ν+1 = λ ν ρ ν H 1 ν H(λ ν ) Newton methods: H ν = H (λ ν ) non-smooth Newton methods: H ν B H(λ ν ) H(λ ν ) Proposition: Let rank B = n and w ν = F 1 (f B T )λ ν. Then H ν := S(w ν ) = B 1 B T + C H(λ ν ) where  = T AT + I T, with T ii = 0, if i N (w ν ) else T ii = 1 L 2 -control: Newton-like (rank B = m < n) Cahn-Hilliard: Newton (rank B = n)
44 Convergence Results for Nonsmooth Schur-Newton Methods Theorem: global convergence for appropriate damping finite termination for non-degenerate problems (locally superlinear, quadratic) inexact versions converge globally, if - obstacle problem: computation of N j (wν ) - linear saddle point problem: approximation up to sufficient accuracy linear convergence, if S(w) is s.p.d. for all w R n L 2 -control: global linear convergence Cahn-Hilliard: global convergence
45 Convergence Results for Nonsmooth Schur-Newton Methods Theorem: global convergence for appropriate damping finite termination for non-degenerate problems (locally superlinear, quadratic) inexact versions converge globally, if - obstacle problem: computation of N j (wν ) - linear saddle point problem: approximation up to sufficient accuracy linear convergence, if S(w) is s.p.d. for all w R n L 2 -control: global linear convergence Cahn-Hilliard: global convergence
46 Interpretations preconditioned Uzawa iteration: evaluation of F 1 : obstacle problem (up to N j (wν )) exact (L 2 -control), multigrid or projected Gauß-Seidel (Cahn-Hilliard) (inexact) evaluation of S(w ν ) 1 : linear saddle point problem exact, multigrid (Vanka 86, Zulehner & Schöberl 03) generalization and globalization of primal-dual active set strategies: (Gräser 07) L 2 -control: nonsmooth Schur-Newton primal-dual active set (Hintermüller, Ito, Kunisch 03)
47 Interpretations preconditioned Uzawa iteration: evaluation of F 1 : obstacle problem (up to N j (wν )) exact (L 2 -control), multigrid or projected Gauß-Seidel (Cahn-Hilliard) (inexact) evaluation of S(w ν ) 1 : linear saddle point problem exact, multigrid (Vanka 86, Zulehner & Schöberl 03) generalization and globalization of primal-dual active set strategies: (Gräser 07) L 2 -control: nonsmooth Schur-Newton primal-dual active set (Hintermüller, Ito, Kunisch 03)
48 L 2 -Control: Iteration Histories for Varying ε uniform refinement: j = 7 with unknowns error error Newton iteration steps ε = Newton iteration steps 20 ε = 10 8
49 L 2 -Control: Mesh Dependence for Varying ε number of iteration steps to roundoff error 10 Newton 100 Newton 8 80 iteration steps 6 4 iteration steps refinement level refinement level ε = 10 4 ε = 10 8
50 Cahn-Hilliard: Iteration Histories for Varying u 0 uniform refinement: j = 9 with unknowns work unit: linear V (3, 3) cycle for the saddle point problem error error Newton like Inexact Gauss Seidel work units bad initial iterate: u 0 = Newton like Inexact Gauss Seidel work units good initial iterate: previous time step averaged convergence rate ρ = 0.58
51 Cahn-Hilliard: Iteration Histories for Varying u 0 uniform refinement: j = 9 with unknowns work unit: linear V (3, 3) cycle for the saddle point problem error error Newton like Inexact Gauss Seidel work units bad initial iterate: u 0 = Newton like Inexact Gauss Seidel work units good initial iterate: previous time step averaged convergence rate ρ = 0.58
52 Cahn-Hilliard: Mesh-Dependence for Varying u 0 number of iteration steps to roundoff error Newton like Inexact Newton like Inexact iteration steps iteration steps refinement level bad initial iterate: u 0 = refinement level good initial iterate: previous time step
53 bad initial iterate (inexact version): Where Does the CPU Time Go? Inexact # tests % Armijo % obstacle % linear work units good initial iterate by nested iteration: table starts at row 9 (no damping) numerical bottleneck: damping linear saddle point solver
54 Conclusion Obstacle and PDE Constrained Optimal Control Problems globally convergent inexact active set strategies by minimization techniques numerical experiments: mesh-independence and linear multigrid convergence speed (nested iteration) Outlook proof of multigrid convergence rates for the hybrid active set methods proof of mesh-independent convergence rates for Schur-Newton methods nonsmooth Schur-Newton methods in function space (Gräser & Schiela 08) applications (biomechanics, hydrology, phase-field models,... )
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