Adaptive wavelet algorithms for solving operator equations - PowerPoint PPT Presentation

Adaptive wavelet algorithms for solving operator equations Tsogtgerel Gantumur General Mathematical Colloquium Mathematisch Instituut 21 September 2006

Overview Ideal benchmark: Nonlinear approximation Optimal adaptive wavelet algorithm Numerical illustration

Elliptic operator equation Au = f • u ∈ H (separable Hilbert space), f ∈ H ′ • A : H → H ′ linear, self-adjoint, H -elliptic � Av , v � ≥ c � v � 2 v ∈ H H • Example: Reaction-diffusion equation H = H 1 0 (Ω) � ∇ u · ∇ v + κ 2 uv � Au , v � = Ω

Adaptive wavelet algorithms • Wavelet basis Ψ = ( ψ i ) i ∈ N of H ( � � i a i ψ i � H � � ( a i ) i � ℓ 2 ) u = � • U ε : f �→ ˜ i ∈ E a i ψ i ( E ⊂ N , � ˜ u − u � H ≤ ε ) • Non-adaptive: E = { 1 , 2 , . . . , k } for some k • Adaptive: no (or mild) constraint Computational model 1 Complexity measure: # E as a function of ε

Best N -term approximation Given u ∈ H , approximate u using N wavelets �� � Σ N := a i ψ i : # E ≤ N , a i ∈ R i ∈ E Linear � N � � S N := a i ψ i : a i ∈ R i = 1

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 dist ( u , Σ N ) ≤ cN − s Linear approximation If u ∈ H t + ns for some s ∈ ( 0 , d − t n ] , uniform refinement dist ( u , S N ) ≤ cN − s Poisson on polygon: u ∈ H 1 + 2 s for only s < π α , u ∈ B 1 + ns ( L p ) for ∀ s > 0 p

Approximation spaces • Approximation space A s := { v ∈ H : dist ( v , Σ N ) ≤ cN − 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

Model of computation With unit cost: • Real number model: + , − , . . . in R , function evaluations • multiplication by a scalar, addition in H , e.g., a i ψ i U ε ( f ) lin. comb. of N wavs. ⇒ cost ( U ε , f ) ≥ N • Availability of certain subroutine(s)

Complexity of the problem • U ε : F ∋ f �→ ˜ u algorithm for solving Au = f • cost ( U ε , F ) := sup f ∈ F cost ( U ε , f ) • comp ( ε, F ) := inf { cost ( U ε , F ) : over all U ε } Since v ∈ A s ⇔ dist ( v , Σ N ) ≤ cN − s , we have comp ( ε, A ( A s )) ≥ C ε − 1 / s

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

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

Galerkin orthogonality A E u E = f E • for v E ∈ ℓ 2 ( E ) : 0 = � f − Au E , v E � = � A ( u − u E ) , v E � | 2 = | | 2 + | | 2 | | | u − u E − v E | | | | u − u E | | | | v E | |

Error reduction E 0 ⊂ E 1 ⊂ E 2 ⊂ . . . ⊂ N A E 0 u E 0 = f E 0 , A E 1 u E 1 = f E 1 | 2 = | | 2 − | | 2 | | | u − u E 1 | | | | u − u E 0 | | | | u E 1 − u E 0 | | Lemma [CDD01] Let µ ∈ ( 0 , 1 ) , and E 1 ⊃ E 0 be s.t. � P E 1 ( f − Au E 0 ) � ℓ 2 ≥ µ � f − Au E 0 � ℓ 2 Then we have � 1 − κ ( A ) − 1 µ 2 | | | | u − u E 1 | | | ≤ | | u − u E 0 | | |

Ideal algorithm SOLVE [ ε ] → u k k := 0 ; E 0 := ∅ do Solve A E k u k = f E k r k := f − Au k determine a set E k + 1 ⊃ E k , with minimal cardinality, such that � P E k + 1 r k � ℓ 2 ≥ µ � r k � ℓ 2 k := k + 1 while � r k � > ε

Approximate iterations Assume: u ∈ A s for some s ∈ ( 0 , d − t n ) RHS [ ε ] → f ε with � f − f ε � ℓ 2 ≤ ε • # supp f ε ≤ C ε − 1 / s • cost ≤ C ( ε − 1 / s + 1 ) APPLY A [ v , ε ] → w ε with � Av − w ε � ℓ 2 ≤ ε • # supp w ε ≤ C ε − 1 / s • cost ≤ C ( ε − 1 / s + # supp v + 1 ) RES [ v , ε ] := RHS [ ε/ 2 ] − APPLY A [ v , ε/ 2 ]

The subroutine APPLY A • ( ψ i ) i 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 [Gantumur, Harbrecht, Stevenson ’05] Let µ ∈ ( 0 , κ ( A ) − 1 2 ) . Then the smallest set E ⊃ supp w with � P E ( f − Aw ) � ℓ 2 ≥ µ � f − Aw � ℓ 2 satisfies #( E \ supp w ) ≤ C � u − w � − 1 / s ℓ 2

Optimal complexity Theorem [GHS05] SOLVE [ ε ] → w terminates with � f − Aw � ℓ 2 ≤ ε . Whenever u ∈ A s with s ∈ ( 0 , d − t n ) , we have • # supp w ≤ C ε − 1 / s • cost ≤ C ε − 1 / s 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