Adaptive wavelet–Galerkin methods: Algorithm & Applications Rob Stevenson Korteweg-de Vries Institute for Mathematics University of Amsterdam joint work with Christoph Schwab and Rafaela Guberovic (Z¨ urich), Monique Dauge (Rennes), Stephan Dahlke and Ulrich Friedrich (Marburg), Nabi Chegini and Tammo Jan Dijkema (A’dam)
Contents • Adaptive wavelet-Galerkin method • Application to least-squares formulations • Applications: – Elliptic eqs with iso- and anisotropic wavelets – Parabolic problems – Navier-Stokes – First Order Systems Least Squares (FOSLS) 1/40
(Nonlinear) operator equations F : X → X ′ , solve F ( u ) = 0 . We assume that a sol. u exists, and that ( c 1 ) F is continuously Fr´ echet differentiable in a neighborhood of u , ( c 2 ) DF ( u )( v )( w ) is inner product, with “energy”-norm � · � u � � · � X . ❀ Solution u locally unique. 2/40
(Nonlinear) operator equations F : X → X ′ , solve F ( u ) = 0 . We assume that a sol. u exists, and that ( c 1 ) F is continuously Fr´ echet differentiable in a neighborhood of u , ( c 2 ) DF ( u )( v )( w ) is inner product, with “energy”-norm � · � u � � · � X . ❀ Solution u locally unique. Ex 1. X = H 1 0 (Ω) , F ( u )( w ) = �∇ u, ∇ w � + � u 3 , w � − � f, w � , so DF ( u )( v )( w ) = �∇ v, ∇ w � + � 3 u 2 v, w � . 2/40
(Nonlinear) operator equations F : X → X ′ , solve F ( u ) = 0 . We assume that a sol. u exists, and that ( c 1 ) F is continuously Fr´ echet differentiable in a neighborhood of u , ( c 2 ) DF ( u )( v )( w ) is inner product, with “energy”-norm � · � u � � · � X . ❀ Solution u locally unique. Ex 1. X = H 1 0 (Ω) , F ( u )( w ) = �∇ u, ∇ w � + � u 3 , w � − � f, w � , so DF ( u )( v )( w ) = �∇ v, ∇ w � + � 3 u 2 v, w � . Galerkin : S Λ ⊂ X closed, find u Λ ∈ S Λ s.t. F ( u Λ )( v Λ ) = 0 ( v Λ ∈ S Λ ). Lem 1. If d ( u, S Λ ) suff. small, then Galerkin has unique sol near u , and � u − u Λ � X � d ( u, S Λ ) . 2/40
Adaptive Galerkin method Let Ψ = { ψ λ : λ ∈ ∇} with clos X spanΨ = X . For Λ ⊂ ∇ , S Λ := clos X span { ψ λ : λ ∈ Λ } . 3/40
Adaptive Galerkin method Let Ψ = { ψ λ : λ ∈ ∇} with clos X spanΨ = X . For Λ ⊂ ∇ , S Λ := clos X span { ψ λ : λ ∈ Λ } . Goal: Whenever for some s > 0 , u ∈ A s := { v ∈ X : � v � A s := sup N s { Λ ⊂∇ :#Λ= N } d ( v, S Λ ) < ∞} , inf N ∈ I N i.e., s is rate of best N -term approximation , to find Λ 0 ⊂ Λ 1 ⊂ · · · , and w Λ i ∈ S Λ i , with � u − w Λ i � X � (#Λ i ) − s . 3/40
Adaptive Galerkin method Let Ψ = { ψ λ : λ ∈ ∇} with clos X spanΨ = X . For Λ ⊂ ∇ , S Λ := clos X span { ψ λ : λ ∈ Λ } . Goal: Whenever for some s > 0 , u ∈ A s := { v ∈ X : � v � A s := sup N s { Λ ⊂∇ :#Λ= N } d ( v, S Λ ) < ∞} , inf N ∈ I N i.e., s is rate of best N -term approximation , to find Λ 0 ⊂ Λ 1 ⊂ · · · , and w Λ i ∈ S Λ i , with � u − w Λ i � X � (#Λ i ) − s . Let Q Λ : X → X be some projector onto S Λ , unif. bound. in Λ ⊂ ∇ . 3/40
Adaptive Galerkin method Alg 1 ( agm). % Let 0 < µ 0 < 1 , δ, γ > 0 be suff. small constants. % Let Λ 0 ⊂ ∇ , w Λ 0 ∈ S Λ 0 with � u − w Λ 0 � X suff. small. for i = 0 , 1 , . . . do Compute r i ∈ X ′ such that � r i − F ( w Λ i ) � X ′ ≤ δ � r i � X ′ . Determine a near-smallest Λ i +1 ⊃ Λ i with � Q ′ Λ i +1 r i � X ′ ≥ µ 0 � r i � X ′ Compute w Λ i +1 ∈ S Λ i +1 with � w Λ i +1 − u Λ i +1 � X ≤ γ � r i � X ′ . enfor Thm 2 (St. ’11) . For some α < 1 , � u − w Λ i +1 � u ≤ α � u − w Λ i � u . If, for whatever s > 0 , u ∈ A s , then � u − w Λ i � X � (#Λ i ) − s . 4/40
Rephrasing in coordinates Let Ψ be Riesz basis (wavelets) for X , i.e. analysis and so synthesis operators F ∈ L ( X ′ , ℓ 2 ( ∇ )) : g �→ [ g ( ψ λ )] λ ∈∇ , F ′ ∈ L ( ℓ 2 ( ∇ ) , X ) : v �→ v ⊤ Ψ := � v λ ψ λ . λ ∈∇ are boundedly invertible. Not.: F ′ v = v . One has v ∈ A s ⇐ ⇒ � v − v N � � N − s . 5/40
Rephrasing in coordinates Let Ψ be Riesz basis (wavelets) for X , i.e. analysis and so synthesis operators F ∈ L ( X ′ , ℓ 2 ( ∇ )) : g �→ [ g ( ψ λ )] λ ∈∇ , F ′ ∈ L ( ℓ 2 ( ∇ ) , X ) : v �→ v ⊤ Ψ := � v λ ψ λ . λ ∈∇ are boundedly invertible. Not.: F ′ v = v . One has v ∈ A s ⇐ ⇒ � v − v N � � N − s . With F := F F F ′ , F ( u ) = 0 ⇐ ⇒ F ( u ) = 0 . I Λ : ℓ 2 (Λ) → ℓ 2 ( ∇ ) extension with zeros, I ′ Λ : ℓ 2 ( ∇ ) → ℓ 2 (Λ) restriction. With F Λ := I ′ Λ F I Λ , F ( u Λ )( v Λ ) = 0 ( v Λ ∈ S Λ ) ⇐ ⇒ F Λ ( u Λ ) = 0 . 5/40
Rephrasing in coordinates Let Ψ be Riesz basis (wavelets) for X , i.e. analysis and so synthesis operators F ∈ L ( X ′ , ℓ 2 ( ∇ )) : g �→ [ g ( ψ λ )] λ ∈∇ , F ′ ∈ L ( ℓ 2 ( ∇ ) , X ) : v �→ v ⊤ Ψ := � v λ ψ λ . λ ∈∇ are boundedly invertible. Not.: F ′ v = v . One has v ∈ A s ⇐ ⇒ � v − v N � � N − s . With F := F F F ′ , F ( u ) = 0 ⇐ ⇒ F ( u ) = 0 . I Λ : ℓ 2 (Λ) → ℓ 2 ( ∇ ) extension with zeros, I ′ Λ : ℓ 2 ( ∇ ) → ℓ 2 (Λ) restriction. With F Λ := I ′ Λ F I Λ , F ( u Λ )( v Λ ) = 0 ( v Λ ∈ S Λ ) ⇐ ⇒ F Λ ( u Λ ) = 0 . Λ ( F ′ ) − 1 (biorth. proj.), so that Q ′ Take Q Λ := F ′ I Λ I ′ Λ := F − 1 I Λ I ′ Λ F , and equip X with � ( F ′ ) − 1 · � ℓ 2 ( ∇ ) . 5/40
Adaptive (wavelet) Galerkin method Alg 3 ( awgm). % Let 0 < µ 0 < 1 , δ, γ > 0 be suff. small constants. % Let Λ 0 ⊂ ∇ , with d ( u, S Λ 0 ) suff. small., and let w Λ 0 ∈ ℓ 2 (Λ 0 ) % with F Λ 0 ( w Λ 0 ) suff. small. for i = 0 , 1 , . . . do Compute r i ∈ ℓ 2 ( ∇ ) such that � r i − F ( w Λ i ) � ≤ δ � r i � . Determine a near-smallest Λ i +1 ⊃ Λ i with � r i | Λ i +1 � ≥ µ 0 � r i � Compute w Λ i +1 ∈ ℓ 2 (Λ i +1 ) with � F Λ i +1 ( w Λ i +1 ) � ≤ γ � r i � . enfor 6/40
Historical notes For affine (linear) problems, i.e. F ( u ) = Lu − f , current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries. 7/40
Historical notes For affine (linear) problems, i.e. F ( u ) = Lu − f , current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries. In 2002, CDD2. Basic convergent iterative scheme on bi-infinite system + coarsening. In 2003, CDD3. Extension of CDD2 to nonlinear equations. 7/40
Historical notes For affine (linear) problems, i.e. F ( u ) = Lu − f , current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries. In 2002, CDD2. Basic convergent iterative scheme on bi-infinite system + coarsening. In 2003, CDD3. Extension of CDD2 to nonlinear equations. In 2007, Gantumur, Harbrecht & St. showed optimality of CDD1 without coarsening. Quantitively more efficient than CDD2, for which coarsening is required. Spin-off: optimally convergent adaptive fem. 7/40
Historical notes For affine (linear) problems, i.e. F ( u ) = Lu − f , current alg. by Cohen, Dahmen, DeVore in 2000 (CDD1). As an additional ingredient it contained coarsening: Recurrent clean-up step of current approx. by deleting small entries. In 2002, CDD2. Basic convergent iterative scheme on bi-infinite system + coarsening. In 2003, CDD3. Extension of CDD2 to nonlinear equations. In 2007, Gantumur, Harbrecht & St. showed optimality of CDD1 without coarsening. Quantitively more efficient than CDD2, for which coarsening is required. Spin-off: optimally convergent adaptive fem. In 2003, Y. Xu and Q. Zhou extended CDD1 to nonlinear equations and removed coarsening. Optimality proof hinges on equivalence of sequence norms and Besov norms which can be avoided. 7/40
General well-posed nonlinear eqs: Least-squares approach G : X → Y ′ , solve G ( u ) = 0 . We assume that a sol. u exists, and that ( c ′ 1 ) G is two times continuously Fr´ echet differentiable near u , ( c ′ L ( X , Y ′ ) ∈ 2 ) DG ( u ) is homeomorphism onto its range, i.e., � DG ( u )( v ) � Y ′ � � v � X ( v ∈ X ). 8/40
General well-posed nonlinear eqs: Least-squares approach G : X → Y ′ , solve G ( u ) = 0 . We assume that a sol. u exists, and that ( c ′ 1 ) G is two times continuously Fr´ echet differentiable near u , ( c ′ L ( X , Y ′ ) ∈ 2 ) DG ( u ) is homeomorphism onto its range, i.e., � DG ( u )( v ) � Y ′ � � v � X ( v ∈ X ). With Q ( v ) := 1 2 � G ( v ) � 2 Y ′ , u = argmin v ∈X Q ( v ) , and so � DG ( u )( v ) , G ( u ) � Y ′ = 0 ( v ∈ X ). F ( u )( v ) := DQ ( u )( v ) = F : X → X ′ and ( c 1 ) F is continuously Fr´ echet differentiable in a neighborhood of u , ( c 2 ) DF ( u )( v )( w ) = � DG ( u )( v ) , DG ( u )( w ) � Y ′ is inner product, with “energy”-norm � · � u � � · � X . 8/40
Recommend
More recommend
