An optimal adaptive wavelet method without coarsening of the - PowerPoint PPT Presentation

An optimal adaptive wavelet method without coarsening of the iterands Tsogtgerel Gantumur (joint work with H. Harbrecht and R.P . Stevenson) "Recent Progress in Wavelet Analysis and Frame Theory" 23-26 January 2006 Bremen

Overview [Cohen, Dahmen, DeVore ’02], [Stevenson ’04], [Dahlke, Fornasier, Raasch ’04] , [Werner] • Wavelet frame Ψ : Au = f ��� Au = f • Richardson iteration: u ( i + 1 ) := u ( i ) + ω ( f − Au ( i ) ) • Coarsening after K iterations [Cohen, Dahmen, DeVore ’01], [Gantumur, Harbrecht, Stevenson ’05] • Galerkin approximation: u Λ ∈ ℓ 2 (Λ) s.t. � Au Λ , v Λ � = � f , v Λ � ∀ v Λ ∈ ℓ 2 (Λ) • Expand Λ to ˜ Λ s.t. | | | u − u ˜ Λ | | | ≤ ξ | | | u − u Λ | | | with ξ < 1 + Coarsening is not needed for the iterands u ( i ) [GHS05] - Using frames is problematic

Elliptic operator equation • Let H be a separable Hilbert space, H ′ its dual • 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 [CDD01, CDD02] • 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 ( ∇ ) → ℓ 2 (Λ) restr., 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 = ( u λ ) λ ∈ ℓ 2 , approximate u using N nonzero coeffs � ℵ N := ℓ 2 (Λ) Λ ⊂∇ :#Λ= N • ℵ N is a nonlinear manifold • Let u N be a best approximation of u with # supp u N ≤ N • u N can be constructed by picking N largest in modulus coeffs from u

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

Approximation spaces • Approximation space A s := { v ∈ ℓ 2 : � v − v N � ℓ 2 ≤ O ( N − s ) } • Quasi-norm | v | A s := sup N ∈ N N s � v − v N � ℓ 2 • u ∈ B t + ns ( L τ ) with 1 τ = 1 2 + s for some s ∈ ( 0 , d − t n ) ⇒ u ∈ A s τ Assumption u ∈ A s for some s > 0 Best approximation � u − v � ≤ ε satisfies # supp v ≤ ε − 1 / s | u | 1 / s A s

Requirements on the subroutines 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 • flops, memory � 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 | 1 / s A s • flops, memory � ε − 1 / s | v | 1 / s A s + # supp v + 1

The subroutine APPLY A • { ψ λ } are 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 [GHS05] Let µ ∈ ( 0 , κ ( A ) − 1 2 ) . Then the smallest set Λ ⊃ supp w with � P Λ ( f − Aw ) � ≥ µ � f − Aw � satisfies #(Λ \ supp w ) � � u − w � − 1 / s | u | 1 / s A s

Sketch of a proof With ν > 0 , let N be the smallest integer s.t. a best N -term appr. u N of u satisfies � u − u N � ≤ ν � u − w � . Then we have N � � u − w � − 1 / s | u | 1 / s A s If ν s.t. ν 2 ≤ κ ( A ) − 1 − µ 2 then Σ := supp w ∪ supp u N satisfies � P Σ ( f − Aw ) � ≥ µ � f − Aw � By def. of Λ #(Λ \ supp w ) ≤ #(Σ \ supp w ) ≤ N

Adaptive Galerkin method SOLVE [ ε ] → w k k := 0 ; Λ 0 := ∅ do Compute an appr.solution w k of A Λ k u k = f Λ k Compute an appr.residual r k for w k Determine a set Λ k + 1 ⊃ Λ k , with modulo constant factor minimal cardinality, such that � P Λ k + 1 r k � ≥ µ � r k � k := k + 1 while � r k � > ε

Optimal complexity Theorem [GHS05] SOLVE [ ε ] → w terminates with � f − Aw � ℓ 2 ≤ ε . • # supp w � ε − 1 / s | u | 1 / s A s • flops, memory � the same expression Further result • Can be extended to mildly nonsymmetric and indefinite problems [Gantumur ’06]

Numerical illustration • The problem: − ∆ u + u = f on R / Z ( t = 1 ) • u ∈ H 1 + s only for s < 1 u ∈ B 1 + s 2 ; τ,τ for any s > 0 1 0.8 0.6 0.4 0.2 0 ! 0.2 ! 0.4 ! 0.6 ! 0.8 ! 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

Convergence histories • B-spline wavelets of order d=3 with 3 vanishing moments from [Cohen, u ∈ A s for any s < d − t = 3 − 1 ⇒ = 2 Daubechies, Feauveau ’92] n 1 1 10 CDD2 New method 0 10 norm of residual ! 1 10 ! 2 10 2 ! 3 10 1 0 1 2 3 10 10 10 10 wall clock time

References [CDD01] A. Cohen, W. Dahmen, R. DeVore. Adaptive wavelet methods for elliptic operator equations — Convergence rates. Math. Comp. , 70:27–75, 2001. [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. . [Gan05] Ts. Gantumur. An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems. Technical Report 1343, Utrecht University, January 2006. This work was supported by the Netherlands Organization for Scientific Research and by the EC-IHP project “Breaking Complexity”.