Adaptive wavelet methods: Quantitative improvements and extensions - PowerPoint PPT Presentation

Adaptive wavelet methods: Quantitative improvements and extensions Rob Stevenson Korteweg-de Vries Institute for Mathematics University of Amsterdam Collaborators: Christoph Schwab (ETH, Z¨ urich), Nabi Chegini (Univ. of Tafresh, Iran)

Contents • Adaptive wavelet methods for solving well-posed operator equations with symmetric, coercive Fr´ echet derivatives • An efficient approximate residual evaluation for 1st order systems • Adaptive wavelet methods for solving general well-posed operator equations: Nonlinear least squares • Time evolution problems: Simultaneous space-time variational formulations of parabolic problems and (N)SE 1/38

Well-posed op. eqs. For X (real) sep. Hilbert space, let • F : X ⊃ dom( F ) → X ′ , • F cont. Fr´ echet diff. in neighb. of a sol u of F ( u ) = 0 , • DF ( u ) ∈ L is( X , X ′ ) , DF ( u ) = DF ( u ) ′ > 0 , (so linearized eq. is SPD). 2/38

Well-posed op. eqs. For X (real) sep. Hilbert space, let • F : X ⊃ dom( F ) → X ′ , • F cont. Fr´ echet diff. in neighb. of a sol u of F ( u ) = 0 , • DF ( u ) ∈ L is( X , X ′ ) , DF ( u ) = DF ( u ) ′ > 0 , (so linearized eq. is SPD). Ex. � R d , d ≤ 3 , X = H 1 Ω grad u · grad v + u 3 v − fv dx • Ω ⊂ I 0 (Ω) , F ( u )( v ) = � � � � ( u ( y ) − u ( x ))( v ( y ) − v ( x )) 1 • F ( u )( v ) = dy − v ( x ) f ( x ) dx , | x − y | 3 4 π ∂ Ω ∂ Ω 1 R 3 , X = H Ω ⊂ I 2 ( ∂ Ω) / R (hypersingular boundary integral equation). 2/38

Reformulation as a countable set of coupled scalar eqs Let Ψ = { ψ λ : λ ∈ ∇} Riesz basis for X , i.e., synthesis operator , � F : c �→ c ⊤ Ψ := c λ ψ λ ∈ L is( ℓ 2 ( ∇ ) , X ) , λ ∈∇ 3/38

Reformulation as a countable set of coupled scalar eqs Let Ψ = { ψ λ : λ ∈ ∇} Riesz basis for X , i.e., synthesis operator , � F : c �→ c ⊤ Ψ := c λ ψ λ ∈ L is( ℓ 2 ( ∇ ) , X ) , λ ∈∇ and so its adjoint, the analysis operator , F ′ : g �→ g (Ψ) := [ g ( ψ λ )] λ ∈∇ ∈ L is( X ′ , ℓ 2 ( ∇ )) . 3/38

Reformulation as a countable set of coupled scalar eqs Let Ψ = { ψ λ : λ ∈ ∇} Riesz basis for X , i.e., synthesis operator , � F : c �→ c ⊤ Ψ := c λ ψ λ ∈ L is( ℓ 2 ( ∇ ) , X ) , λ ∈∇ and so its adjoint, the analysis operator , F ′ : g �→ g (Ψ) := [ g ( ψ λ )] λ ∈∇ ∈ L is( X ′ , ℓ 2 ( ∇ )) . Then with F = F ′ F F : ℓ 2 (Λ) ⊃ dom( F ) → ℓ 2 (Λ) , equiv. form. F ( u ) = 0 , where u := F − 1 u . 3/38

Reformulation as a countable set of coupled scalar eqs Let Ψ = { ψ λ : λ ∈ ∇} Riesz basis for X , i.e., synthesis operator , � F : c �→ c ⊤ Ψ := c λ ψ λ ∈ L is( ℓ 2 ( ∇ ) , X ) , λ ∈∇ and so its adjoint, the analysis operator , F ′ : g �→ g (Ψ) := [ g ( ψ λ )] λ ∈∇ ∈ L is( X ′ , ℓ 2 ( ∇ )) . Then with F = F ′ F F : ℓ 2 (Λ) ⊃ dom( F ) → ℓ 2 (Λ) , equiv. form. F ( u ) = 0 , where u := F − 1 u . Norm on ℓ 2 ( ∇ ) will be denoted as � · � . � u − w � � � u − F w � X . 3/38

Adaptive wavelet Galerkin method In its original form introduced by [Cohen, Dahmen, DeVore ’01, 02] Alg ( awgm). % Let U ⊂ ℓ 2 (Λ) be a neigh. of u , µ ∈ (0 , 1] , finite Λ 0 ⊂ ∇ . for i = 0 , 1 , . . . do solve u i ∈ U with supp u i ⊆ Λ i s.t. F ( u i ) | Λ i = 0 determine a smallest Λ i +1 ⊃ Λ i s.t. � F ( u i ) | Λ i +1 � ≥ µ � F ( u i ) � endfor 4/38

Adaptive wavelet Galerkin method In its original form introduced by [Cohen, Dahmen, DeVore ’01, 02] Alg ( awgm). % Let U ⊂ ℓ 2 (Λ) be a neigh. of u , µ ∈ (0 , 1] , finite Λ 0 ⊂ ∇ . for i = 0 , 1 , . . . do solve u i ∈ U with supp u i ⊆ Λ i s.t. F ( u i ) | Λ i = 0 determine a smallest Λ i +1 ⊃ Λ i s.t. � F ( u i ) | Λ i +1 � ≥ µ � F ( u i ) � endfor Thm ( convergence ) . ∃ α < 1 s.t. when U and inf v ∈ ℓ 2 (Λ 0 ) � u − v � suff. small, � u − u i � � α i � u − u 0 � . For affine F , use ||| u − u i +1 ||| 2 = ||| u − u i ||| 2 − ||| u i +1 − u i ||| 2 , and saturation ||| u i +1 − u i ||| � ||| u − u i ||| by ‘bulk chasing’. Perturb arg. for non-affine. 4/38

Adaptive wavelet Galerkin method In its original form introduced by [Cohen, Dahmen, DeVore ’01, 02] Alg ( awgm). % Let U ⊂ ℓ 2 (Λ) be a neigh. of u , µ ∈ (0 , 1] , finite Λ 0 ⊂ ∇ . for i = 0 , 1 , . . . do solve u i ∈ U with supp u i ⊆ Λ i s.t. F ( u i ) | Λ i = 0 determine a smallest Λ i +1 ⊃ Λ i s.t. � F ( u i ) | Λ i +1 � ≥ µ � F ( u i ) � endfor Thm ( convergence ) . ∃ α < 1 s.t. when U and inf v ∈ ℓ 2 (Λ 0 ) � u − v � suff. small, � u − u i � � α i � u − u 0 � . For affine F , use ||| u − u i +1 ||| 2 = ||| u − u i ||| 2 − ||| u i +1 − u i ||| 2 , and saturation ||| u i +1 − u i ||| � ||| u − u i ||| by ‘bulk chasing’. Perturb arg. for non-affine. Def ( approx. class ) . For s > 0 , � � A s := N s u ∈ ℓ 2 ( ∇ ): � u � A s := sup { w : # supp w ≤ N } � u − w � < ∞ inf . N ∈ N 4/38

Adaptive wavelet Galerkin method In its original form introduced by [Cohen, Dahmen, DeVore ’01, 02] Alg ( awgm). % Let U ⊂ ℓ 2 (Λ) be a neigh. of u , µ ∈ (0 , 1] , finite Λ 0 ⊂ ∇ . for i = 0 , 1 , . . . do solve u i ∈ U with supp u i ⊆ Λ i s.t. F ( u i ) | Λ i = 0 determine a smallest Λ i +1 ⊃ Λ i s.t. � F ( u i ) | Λ i +1 � ≥ µ � F ( u i ) � endfor Thm ( convergence ) . ∃ α < 1 s.t. when U and inf v ∈ ℓ 2 (Λ 0 ) � u − v � suff. small, � u − u i � � α i � u − u 0 � . For affine F , use ||| u − u i +1 ||| 2 = ||| u − u i ||| 2 − ||| u i +1 − u i ||| 2 , and saturation ||| u i +1 − u i ||| � ||| u − u i ||| by ‘bulk chasing’. Perturb arg. for non-affine. Def ( approx. class ) . For s > 0 , � � A s := N s u ∈ ℓ 2 ( ∇ ): � u � A s := sup { w : # supp w ≤ N } � u − w � < ∞ inf . N ∈ N Thm ( optimal rate ) . If µ is suff. small, then if u ∈ A s , (# supp u i ) s � u − u i � � 1 . 4/38

Practical awgm Thm. With approx. eval. of F ( u i ) with rel. tolerance δ > 0 (suff. small but fixed), awgm also converges with optimal rate. 5/38

Practical awgm Thm. With approx. eval. of F ( u i ) with rel. tolerance δ > 0 (suff. small but fixed), awgm also converges with optimal rate. Thm ( optimal comput. compl. ) . If cost to compute such approx. residuals is O ( � u − u i � − 1 /s + # supp u i ) , then ( cost to compute u i ) s � u − u i � � 1 . 5/38

Practical awgm Thm. With approx. eval. of F ( u i ) with rel. tolerance δ > 0 (suff. small but fixed), awgm also converges with optimal rate. Thm ( optimal comput. compl. ) . If cost to compute such approx. residuals is O ( � u − u i � − 1 /s + # supp u i ) , then ( cost to compute u i ) s � u − u i � � 1 . This cost condition has been verified for large class of linear PDEs and singular integral eqs using compactly supported wavelets that are sufficiently smooth and have sufficiently many vanishing moments . 5/38

Practical awgm Thm. With approx. eval. of F ( u i ) with rel. tolerance δ > 0 (suff. small but fixed), awgm also converges with optimal rate. Thm ( optimal comput. compl. ) . If cost to compute such approx. residuals is O ( � u − u i � − 1 /s + # supp u i ) , then ( cost to compute u i ) s � u − u i � � 1 . This cost condition has been verified for large class of linear PDEs and singular integral eqs using compactly supported wavelets that are sufficiently smooth and have sufficiently many vanishing moments . Such bases for the common Sob. spaces are available on general polygonal domains and consist of piecewise polynomial wavelets. Wavelet ψ λ on ‘level’ | λ | ∈ I N has diam supp ψ λ � 2 −| λ | . 5/38

Usual residual evaluation ([CDD01]) For F ( u ) = Au − f , approximate both F ′ A F u i and F ′ f separately within absolute tolerance 1 2 δ � u − u i � . 6/38

Usual residual evaluation ([CDD01]) For F ( u ) = Au − f , approximate both F ′ A F u i and F ′ f separately within absolute tolerance 1 2 δ � u − u i � . � � � � Ex (Poisson) . Terms read as Ω grad Ψ · grad Ψ u i and Ω f Ψ . Assuming ˜ d vanishing moments, rhs approximation based on � | fψ λ | ≤ � ψ λ � L 2 (Ω) inf � f − p � L 2 (supp ψ λ ) . p ∈ P ˜ Ω d − 1 6/38

Usual residual evaluation ([CDD01]) For F ( u ) = Au − f , approximate both F ′ A F u i and F ′ f separately within absolute tolerance 1 2 δ � u − u i � . � � � � Ex (Poisson) . Terms read as Ω grad Ψ · grad Ψ u i and Ω f Ψ . Assuming ˜ d vanishing moments, rhs approximation based on � | fψ λ | ≤ � ψ λ � L 2 (Ω) inf � f − p � L 2 (supp ψ λ ) . p ∈ P ˜ Ω d − 1 Similar arg. shows that stiffness is ‘near-sparse’. Restricting it to fixed ‘band’ gives right complexity, but not suff. accuracy. u ∈ A s means that vector is ‘near-sparse’. One has � u i � A s � � u � A s . Approximate j th column of stiffness with accuracy proportional to | ( u i ) j | . Realizes cost condition. Quantitatively expensive. 6/38