An optimal adaptive wavelet method discrete problems Linear - PowerPoint PPT Presentation

Continuous and An optimal adaptive wavelet method discrete problems Linear operator equations for strongly elliptic operator equations Discretization A convergent adaptive Galerkin method Galerkin approximation Tsogtgerel Gantumur Convergence (Utrecht University) Complexity analysis Nonlinear approximation Optimal complexity Workshop on Fast Numerical Solution of PDE’s 20-22 December 2005 Utrecht 1

Motivation and overview [Cohen, Dahmen, DeVore ’01, ’02] Continuous and discrete problems ◮ Using a wavalet basis Ψ , transform Au = g to a Linear operator equations Discretization matrix-vector system Au = g A convergent adaptive Galerkin ◮ Solve it by an iterative method method Galerkin approximation ◮ They apply to A symmetric positive definite Convergence Complexity ◮ If A is perturbed by a compact operator? analysis Nonlinear approximation ◮ Normal equation: A T Au = A T g Optimal complexity ◮ Condition number squared, application of A T A expensive ◮ We modified [Gantumur, Harbrecht, Stevenson ’05] 2

Sobolev spaces Continuous and discrete problems Linear operator equations Discretization A convergent ◮ Let Ω be an n -dimensional domain adaptive Galerkin method ◮ H s := ( L 2 (Ω) , H 1 0 (Ω)) s , 2 for 0 ≤ s ≤ 1 Galerkin approximation Convergence ◮ H s := H s (Ω) ∩ H 1 0 (Ω) for s > 1 Complexity analysis ◮ H s := ( H − s ) ′ for s < 0 Nonlinear approximation Optimal complexity 3

Model problem ◮ A : H 1 → H − 1 linear, self-adjoint, H 1 -elliptic: Continuous and discrete problems Linear operator equations � Au , u � ≥ c � u � 2 u ∈ H 1 (= H 1 0 (Ω)) Discretization 1 A convergent ◮ B : H 1 − σ → H − 1 linear ( σ > 0 ) adaptive Galerkin method ◮ L := A + B : H 1 → H − 1 Galerkin approximation Convergence Complexity Find u ∈ H 1 s.t. ( g ∈ H − 1 ) Lu = g analysis Nonlinear approximation ◮ Regularity: g ∈ H − 1 + σ ⇒ � u � 1 + σ � � g � − 1 + σ Optimal complexity ◮ Example: Helmholtz equation � ∇ u · ∇ v − κ 2 uv � Lu , v � = Ω 4

Riesz bases Continuous and discrete problems Linear operator equations Ψ = { ψ λ : λ ∈ ∇} is a Riesz basis for H 1 Discretization – each v ∈ H 1 has a unique expansion A convergent adaptive Galerkin method Galerkin approximation ψ λ , v �| 2 ≤ C � v � 2 � � ˜ � |� ˜ c � v � 2 v = ψ λ , v � ψ λ s.t. 1 ≤ Convergence 1 Complexity λ ∈∇ λ ∈∇ analysis Nonlinear approximation Optimal complexity ψ λ } ⊂ H − 1 is the dual basis: � ˜ ◮ ˜ Ψ = { ˜ ψ λ , ψ µ � = δ λµ ◮ For v ∈ H 1 , we have v = { v λ } := {� ˜ ψ λ , v �} ∈ ℓ 2 ( ∇ ) 5

Wavelet basis ◮ Ψ = { ψ λ } Riesz basis for H 1 Continuous and discrete problems ◮ Nested index sets ∇ 0 ⊂ ∇ 1 ⊂ . . . ⊂ ∇ j ⊂ . . . ⊂ ∇ , Linear operator equations Discretization ◮ S j = span { ψ λ : λ ∈ ∇ j } ⊂ H 1 A convergent adaptive Galerkin method ◮ diam ( supp ψ λ ) = O ( 2 − j ) if λ ∈ ∇ j \ ∇ j − 1 Galerkin approximation Convergence ◮ All polynomials of degree d − 1 , P d − 1 ⊂ S 0 Complexity analysis Nonlinear approximation Optimal complexity v ∈ H s ( 1 ≤ s ≤ d ) v j ∈S j � v − v j � 1 ≤ C · 2 − j ( s − 1 ) / n � v � s inf ◮ If λ ∈ ∇ \ ∇ 0 , we have � P ˜ d − 1 , ψ λ � L 2 = 0 6

Equivalent discrete problem Continuous and [CDD01, CDD02] discrete problems ◮ Wavelet basis Ψ = { ψ λ : λ ∈ ∇} Linear operator equations Discretization ◮ Stiffness L = � L ψ λ , ψ µ � λ,µ and load g = � g , ψ λ � λ A convergent adaptive Galerkin method Linear equation in ℓ 2 ( ∇ ) Galerkin approximation Convergence Complexity analysis Lu = g , L : ℓ 2 ( ∇ ) → ℓ 2 ( ∇ ) invertible and g ∈ ℓ 2 ( ∇ ) Nonlinear approximation Optimal complexity ◮ u = � λ u λ ψ λ is the solution of Lu = g ◮ � u − v � ℓ 2 � � u − v � 1 with v = � λ v λ ψ λ ◮ A good approx. of u induces a good approx. of u 7

Galerkin solutions ◮ A := � A ψ λ , ψ µ � λ,µ SPD, recall L = A + B Continuous and 1 ◮ | 2 is a norm on ℓ 2 discrete problems | | · | | | := � A · , ·� Linear operator equations Discretization ◮ Λ ⊂ ∇ A convergent ◮ L Λ := P Λ L | ℓ 2 (Λ) : ℓ 2 (Λ) → ℓ 2 (Λ) , and g Λ := P Λ g ∈ ℓ 2 (Λ) adaptive Galerkin method Galerkin approximation Convergence Lemma Complexity analysis Nonlinear approximation ∃ j 0 : If Λ ⊃ ∇ j with j ≥ j 0 , a unique solution u Λ ∈ ℓ 2 (Λ) to Optimal complexity L Λ u Λ = g Λ exists, and | ≤ [ 1 + O ( 2 − j σ/ n )] | | | u − u Λ | | v ∈ ℓ 2 (Λ) | | | u − v | | | inf Ref: [Schatz ’74] 8

Quasi-orthogonality Continuous and discrete problems Linear operator equations ◮ j ≥ j 0 Discretization ◮ ∇ j ⊂ Λ 0 ⊂ Λ 1 A convergent adaptive Galerkin method ◮ L Λ i u i = g Λ i , i = 0 , 1 Galerkin approximation Convergence Complexity | 2 − | | 2 − | analysis � | 2 � � | | | u − u 0 | | | | u − u 1 | | | | u 1 − u 0 | | � Nonlinear approximation Optimal complexity | 2 + | ≤ O ( 2 − j σ/ n ) | 2 � � | | | u − u 0 | | | | u − u 1 | | Ref: [Mekchay, Nochetto ’04] 9

A sketch of a proof Continuous and discrete problems | 2 = | | 2 + | | 2 + 2 � A ( u − u 1 ) , u 1 − u 0 � Linear operator equations | | | u − u 0 | | | | u − u 1 | | | | u 1 − u 0 | | Discretization A convergent adaptive Galerkin method Galerkin approximation � L ( u − u 1 ) , u 1 − u 0 � = 0 Convergence Complexity analysis Nonlinear approximation Optimal complexity � A ( u − u 1 ) , u 1 − u 0 � = −� B ( u − u 1 ) , u 1 − u 0 � = −� B ( u − u 1 ) , u 1 − u 0 � ≤ � B � 1 − σ →− 1 � u − u 1 � 1 − σ � u 1 − u 0 � 1 � u − u 1 � 1 − σ ≤ O ( 2 − j σ/ n ) � u − u 1 � 1 (Aubin-Nitsche) 10

Error reduction Continuous and | 2 ≤ [ 1 + O ( 2 − j σ/ n )] | 2 − | | 2 � � discrete problems | | | u − u 1 | | | | | u − u 0 | | | | u 1 − u 0 | | Linear operator equations Discretization A convergent adaptive Galerkin method Lemma Galerkin approximation Convergence Let µ ∈ ( 0 , 1 ) , and Λ 1 be s.t. Complexity analysis � P Λ 1 ( g − Lu 0 ) � ≥ µ � g − Lu 0 � Nonlinear approximation Optimal complexity Then we have | ≤ [ 1 − κ ( A ) − 1 µ 2 + O ( 2 − j σ/ n )] 1 2 | | | | u − u 1 | | | | u − u 0 | | | Ref: [CDD01] 11

Exact algorithm Continuous and discrete problems Linear operator equations Discretization SOLVE [ ε ] → u k A convergent k := 0 ; Λ 0 := ∇ j adaptive Galerkin method do Galerkin approximation Convergence Solve L Λ k u k = g Λ k Complexity r k := g − Lu k analysis Nonlinear approximation determine a set Λ k + 1 ⊃ Λ k , with minimal Optimal complexity cardinality, such that � P Λ k + 1 r k � ≥ µ � r k � k := k + 1 while � r k � > ε 12

Approximate Iterations Continuous and discrete problems Approximate right-hand side Linear operator equations Discretization A convergent RHS [ ε ] → g ε with � g − g ε � ℓ 2 ≤ ε adaptive Galerkin method Galerkin approximation Approximate application of the matrix Convergence Complexity analysis APPLY [ v , ε ] → w ε with � Lv − w ε � ℓ 2 ≤ ε Nonlinear approximation Optimal complexity Approximate residual RES [ v , ε ] := RHS [ ε/ 2 ] − APPLY [ v , ε/ 2 ] 13

Best N -term approximation Continuous and discrete problems Linear operator equations Given u = ( u λ ) λ ∈ ℓ 2 , approximate u using N nonzero Discretization coeffs A convergent adaptive Galerkin method Galerkin approximation � ℵ N := ℓ 2 (Λ) Convergence Complexity Λ ⊂∇ :#Λ= N analysis Nonlinear approximation Optimal complexity ◮ 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 14

Nonlinear vs. linear approximation Nonlinear approximation Continuous and If u ∈ B 1 + ns 2 + s for some s ∈ ( 0 , d − 1 ( L τ ) with 1 τ = 1 n ) discrete problems τ Linear operator equations Discretization ε N = � u N − u � ≤ O ( N − s ) A convergent adaptive Galerkin method Galerkin approximation Convergence Linear approximation Complexity If u ∈ H 1 + ns for some s ∈ ( 0 , d − 1 analysis n ] , uniform refinement Nonlinear approximation Optimal complexity ε j = � u j − u � ≤ O ( N − s j ) ◮ H 1 + ns is a proper subset of B 1 + ns ( L τ ) τ ◮ [Dahlke, DeVore]: u ∈ B 1 + ns ( L τ ) \ H 1 + ns "often" τ 15

Approximation spaces ◮ Approximation space Continuous and A s := { v ∈ ℓ 2 : � v − v N � ℓ 2 ≤ O ( N − s ) } discrete problems Linear operator equations Discretization ◮ Quasi-norm | v | A s := sup N ∈ N N s � v − v N � ℓ 2 A convergent 2 + s for some s ∈ ( 0 , d − 1 ◮ u ∈ B 1 + ns ( L τ ) with 1 τ = 1 adaptive Galerkin n ) τ method ⇒ u ∈ A s Galerkin approximation Convergence Complexity Assumption analysis Nonlinear approximation Optimal complexity u ∈ A s for some s ∈ ( 0 , d − 1 n ) Best approximation � u − v � ≤ ε satisfies # supp v ≤ ε − 1 / s | u | 1 / s A s 16