Adaptive wavelet algorithms with truncated residuals Tsogtgerel - PowerPoint PPT Presentation

Adaptive wavelet algorithms with truncated residuals Tsogtgerel Gantumur IHP Breaking Complexity Meeting 14 September 2006 Vienna

Contents Elliptic boundary value problem A convergent adaptive Galerkin method Complexity analysis A method with truncated residuals

Elliptic boundary value problem • H := H 1 0 (Ω) • A : H → H ′ linear, self-adjoint, H -elliptic ( � Av , v � ≥ c � v � 2 v ∈ H ) H ( f ∈ H ′ ) Find u ∈ H s.t. Au = f • Example: Reaction-diffusion equation H = H 1 0 (Ω) � ∇ u · ∇ v + κ 2 uv � Au , v � = Ω

Equivalent discrete problem [Cohen, Dahmen, DeVore ’01, ’02] • Wavelet basis Ψ = { ψ λ : λ ∈ ∇} of H • Stiffness A = � A ψ λ , ψ µ � λ,µ and load f = � f , ψ λ � λ Linear equation in ℓ 2 ( ∇ ) Au = f , A : ℓ 2 ( ∇ ) → ℓ 2 ( ∇ ) SPD and f ∈ ℓ 2 ( ∇ ) • u = � λ u λ ψ λ is the solution of Au = f • � u − v � ℓ 2 � � u − v � H with v = � λ v λ ψ λ

Galerkin solutions 1 • | | | · | | | := � A · , ·� 2 is a norm on ℓ 2 • Λ ⊂ ∇ P Λ := I ∗ • I Λ : ℓ 2 (Λ) → ℓ 2 ( ∇ ) incl., Λ • A Λ := P Λ AI Λ : ℓ 2 (Λ) → ℓ 2 (Λ) SPD • f Λ := P Λ f ∈ ℓ 2 (Λ) Lemma A unique solution u Λ ∈ ℓ 2 (Λ) to A Λ u Λ = f Λ exists, and | | | u − u Λ | | | = v ∈ ℓ 2 (Λ) | | | u − v | | | inf

Galerkin orthogonality • supp w ⊂ Λ , A Λ u Λ = f Λ • � f − Au Λ , v Λ � = 0 for v Λ ∈ ℓ 2 (Λ) | 2 = | | 2 + | | 2 | | | u − w | | | | u − u Λ | | | | u Λ − w | | •

Error reduction | 2 = | | 2 − | | 2 | | | u − u Λ | | | | u − w | | | | u Λ − w | | Lemma [CDD01] Let µ ∈ ( 0 , 1 ) , and Λ be s.t. � P Λ ( f − Aw ) � ≥ µ � f − Aw � Then we have � 1 − κ ( A ) − 1 µ 2 | | | | u − u Λ | | | ≤ | | u − w | | |

Ideal algorithm SOLVE [ ε ] → u k k := 0 ; Λ 0 := ∅ do Solve A Λ k u k = f Λ k r k := f − Au k determine a set Λ k + 1 ⊃ Λ k , with minimal cardinality, such that � P Λ k + 1 r k � ≥ µ � r k � k := k + 1 while � r k � > ε

Approximate Iterations Approximate right-hand side RHS [ ε ] → f ε with � f − f ε � ℓ 2 ≤ ε Approximate application of the matrix APPLY A [ v , ε ] → w ε with � Av − w ε � ℓ 2 ≤ ε Approximate residual RES [ v , ε ] := RHS [ ε/ 2 ] − APPLY A [ v , ε/ 2 ]

Best N -term approximation Given u ∈ H , approximate u using N wavelets �� � Σ N := a λ ψ λ : #Λ ≤ N , a λ ∈ R λ ∈ Λ • Σ N is a nonlinear manifold

Nonlinear vs. linear approximation in H t (Ω) Using wavelets of order d Nonlinear approximation If u ∈ B t + ns 2 + s for some s ∈ ( 0 , d − t ( L p ) with 1 p = 1 n ) p ε N = dist ( u , Σ N ) � N − s Linear approximation If u ∈ H t + ns for some s ∈ ( 0 , d − t n ] , uniform refinement ε j = � u j − u � � N − s j • [Dahlke, DeVore] : u ∈ B t + ns ( L p ) \ H t + ns "often" p

Approximation spaces • Approximation space A s := { v ∈ H : dist ( v , Σ N ) � N − s } • Quasi-norm | v | A s := � v � H + sup N ∈ N N s dist ( v , Σ N ) ( L p ) ⊂ A s with 1 • B t + ns p = 1 2 + s for s ∈ ( 0 , d − t n ) p

Complexity of the problem • U : f �→ ˜ algorithm for solving Au = f u • cost ( U , F ) := sup f ∈ F cost ( U , f ) • e ( U , F ) := sup f ∈ F � U ( f ) − u � H • comp ( ε, F ) := inf { cost ( U , F ) : over all U s.t. e ( U , F ) ≤ ε } r := { v ∈ A s : | v | A s ≤ r } • B s • U ( f ) lin. comb. of N wavs. ⇒ cost ( U , f ) � N Since v ∈ A s ⇔ dist ( v , Σ N ) � N − s | v | A s , we have r )) � r 1 / s ε − 1 / s comp ( ε, A ( B s

Requirements on the subroutines Assume: u ∈ A s for some s ∈ ( 0 , d − t n ) Complexity of RHS RHS [ ε ] → f ε terminates with � f − f ε � ℓ 2 ≤ ε • # supp f ε � ε − 1 / s | u | 1 / s A s • cost � ε − 1 / s | u | 1 / s A s + 1 Complexity of APPLY A For # supp v < ∞ APPLY A [ v , ε ] → w ε terminates with � Av − w ε � ℓ 2 ≤ ε • # supp w ε � ε − 1 / s | v T Ψ | 1 / s A s • cost � ε − 1 / s | v T Ψ | 1 / s A s + # supp v + 1

The subroutine APPLY A • Ψ is piecewise polynomial wavelets that are sufficiently smooth and have sufficiently many vanishing moments • A is either differential or singular integral operator Then we can construct APPLY A satisfying the requirements. Ref: [CDD01], [Stevenson ’04], [Gantumur, Stevenson ’05,’06], [Dahmen, Harbrecht, Schneider ’05]

Optimal expansion Lemma [Gantumur, Harbrecht, Stevenson ’05] r and µ ∈ ( 0 , κ ( A ) − 1 Let u ∈ B s 2 ) . Then the smallest set Λ ⊃ supp w with � P Λ ( f − Aw ) � ≥ µ � f − Aw � satisfies #Λ − # supp w � r 1 / s � f − Aw � − 1 / s

Optimal complexity Theorem [GHS05] SOLVE [ ε ] → w terminates with � f − Aw � ℓ 2 ≤ ε . Whenever r with s ∈ ( 0 , d − t u ∈ B s n ) , we have • # supp w � r 1 / s ε − 1 / s • cost � r 1 / s ε − 1 / s Further result • Can be extended to mildly nonsymmetric and indefinite problems [Gantumur ’06]

Sketch of a proof K � #Λ K + 1 = #Λ k + 1 − #Λ k k = 0 K r 1 / s � � f − Au k � − 1 / s � k = 0 r 1 / s � f − Au K � − 1 / s � r 1 / s ε − 1 / s <

Algorithm with truncated residuals [Harbrecht, Schneider ’02], [Berrone, Kozubek ’04] SOLVE [ ε ] → u k k := 0 ; Λ 0 := ∅ do Solve A Λ k u k = f Λ k r ⋆ k := P Λ ⋆ k ( f − Au k ) determine a set Λ k + 1 ⊃ Λ k , with minimal cardinality, such that � P Λ k + 1 r ⋆ k � ≥ µ � r ⋆ k � k := k + 1 while � r ⋆ k � > ε

Error reduction • r ⋆ k = P Λ ⋆ k ( f − Au k ) truncated residual • r k = f − Au k full residual Suppose Λ ⋆ k = V (Λ k ) is such that � P Λ ⋆ k ( f − Au k ) � ≥ η � f − Au k � then we have � P Λ k + 1 r k � = � P Λ k + 1 r ⋆ k � ≥ µ � r ⋆ k � ≥ µη � r k � → error reduction

Cardinality of expansion ˜ Λ = V (Λ , ¯ Λ) , Λ ⊂ ¯ Λ trees • | | | u ˜ Λ − u Λ | | | ≥ η | | | u ¯ Λ − u Λ | | | • Λ ⊂ ˜ Λ ⊆ V (Λ , ∇ ) • # V (Λ , ∇ ) � #Λ • #(˜ Λ \ Λ) � #(¯ Λ \ Λ) Lemma r and µ ∈ ( 0 , ηκ ( A ) − 1 2 ) . Then with Λ ⋆ = V (Λ , ∇ ) , the Let u ∈ B s smallest tree ˘ Λ ⊃ Λ with Λ r ⋆ � ≥ µ � r ⋆ � � P ˘ satisfies #(˘ Λ \ Λ) � r 1 / s � u − u Λ � − 1 / s

Optimal convergence rate Theorem SOLVE [ ε ] → w terminates with � u − w � ℓ 2 � ε . Whenever u ∈ B s r with s ∈ ( 0 , d − t n ) , we have • # supp w � r 1 / s ε − 1 / s • cost � r 1 / s ε − 1 / s

Activable sets Λ ⋆ = V (Λ , ∇ ) : • #Λ ⋆ � #Λ • � P Λ ⋆ ( f − Au Λ ) � ≥ η � f − Au Λ � [Berrone, Kozubek ’04]: • For λ ∈ Λ , add µ to Λ ⋆ if ψ µ intersects with a contracted support of ψ λ and | µ | = | λ | + 1

FEM error estimators [Verfürth], [Stevenson ’04], [Dahmen, Schneider, Xu ’00], [Bittner, Urban ’05] • f ∈ L 2 (Ω) • S := span { ψ λ : λ ∈ Λ } • T mesh corresponding to S E T ( w ) � � f − Aw � H − 1 (Ω) for w ∈ S if • Λ is a graded tree • Duals ˜ Ψ are compactly supported

Saturation [Verfürth], [Morin, Nochetto, Siebert ’00], [Stevenson ’04], [Mekchay, Nochetto ’04] With S ⋆ = span { ψ λ : λ ∈ Λ ⋆ ⊃ Λ } E T ( w ) � � u S ⋆ − w � H 1 (Ω) for w ∈ S if • f is a piecewise polynomial w.r.t. T • “Bubble functions” are in S ⋆ , i.e., duals ˜ Ψ are compactly supported � P Λ ⋆ ( f − Au Λ ) � � u S ⋆ − u S � H 1 (Ω) � E T ( u S ) � � � f − Au S � H − 1 (Ω) � f − Au Λ � �

Activable sets Λ ⋆ = V (Λ , ∇ ) : • For ∆ ∈ T , add µ to Λ ⋆ if ˜ ψ µ intersects with ∆ and | µ | ≤ | λ | + N

References [GHS05] Ts. Gantumur, H. Harbrecht, R.P . Stevenson. An optimal adaptive wavelet method without coarsening of the iterands. Technical Report 1325, Utrecht University, March 2005. To appear in Math. Comp. . [Gan06] Ts. Gantumur. Adaptive wavelet algorithms for solving operator equations. PhD thesis. Utrecht University. To appear. This work was supported by the Netherlands Organization for Scientific Research and by the EC-IHP project “Breaking Complexity”.