Computation of operators in wavelet coordinates Tsogtgerel Gantumur and Rob Stevenson Department of Mathematics Utrecht University The Netherlands Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.1/19
Contents Settings: linear operator equation, wavelet basis Optimal complexity of CDD2 algorithm Differential operators Compressibility Computability Boundary integral operators Compressibility Computability Conclusion Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.2/19
Linear operator equations Let Ω be an n -dimensional domain or smooth manifold H t ⊂ H t (Ω) be a subspace, and H − t be its dual space Consider the problem of finding u from Lu = g where L : H t → H − t is a self-adjoint elliptic operator of order 2 t and g ∈ H − t is a linear functional Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.3/19
Differential operators Partial differential operators of order 2 t � � ∂ α v, a αβ ∂ β u � , � v, Lu � = | α | , | β |≤ t Example: The reaction-diffusion equation ( t = 1 ) � ∇ v · ∇ u + κ 2 vu, � v, Lu � = Ω Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.4/19
Singular integral operators Boundary integral operators � [ Lu ]( x ) = K ( x, y ) u ( y ) d Ω y Ω with the kernel K ( x, y ) singular at x = y Example: The single layer operator for the Laplace BVP in 3 -d domain ( t = − 1 2 ) 1 K ( x, y ) = 4 π | x − y | Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.5/19
Wavelet bases Multiresolution: S 0 ⊂ S 1 ⊂ . . . ⊂ H t dim S j = O (2 jn ) (dyadic refinements) S j contains all piecewise pols of degree d − 1 S j is globally C r -smooth γ := r + 3 2 Ψ = { ψ λ : λ ∈ Λ } is a Riesz basis for H t span { ψ λ : | λ | ≤ j } = S j diam(supp ψ λ ) � 2 −| λ | Ψ has ˜ d vanishing moments Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.6/19
Nonlinear approximation u = ( u λ ) λ ∈ ℓ 2 s.t. u = � λ u λ ψ λ u N best approximation of u with #supp u N ≤ N with 1 τ = 1 2 + s for some s < d − t If u ∈ B t + ns τ,τ n λ [ u N ] λ ψ λ − u � H t � � u N − u � � N − s ε N = � � u is given as the solution to Lu = g . [Dahlke, DeVore]: u ∈ B t + ns under mild requirements τ,τ Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.7/19
Equivalent problem in ℓ 2 Wavelet basis Ψ = { ψ λ : λ ∈ Λ } Stiffness L = � Lψ λ , ψ µ � λ,µ and load g = � g, ψ λ � λ Linear equation in ℓ 2 (Λ) Lu = g L : ℓ 2 (Λ) → ℓ 2 (Λ) SPD and g ∈ ℓ 2 (Λ) u = � λ u λ ψ λ is the solution of Lu = g Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.8/19
Richardson iterations in ℓ 2 [Cohen, Dahmen, DeVore ’02] u (0) = 0 u ( i +1) = u ( i ) + α [ g − Lu ( i ) ] i = 0 , 1 , . . . g and Lu ( i ) are infinitely supported Approximate them by finitely supported sequences Algorithm SOLVE [ N, L , g ] → u [ N ] ( N operations) #supp u [ N ] � N and ε [ N ] = � u [ N ] − u � → 0 as N → ∞ ε [ N ] speed of convergence? Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.9/19
Complexity of SOLVE Matrix L is called q ∗ -computable, when for each N one can construct an infinite matrix L ∗ N s.t. for any q < q ∗ , � L ∗ N − L � � N − q having in each column O ( N ) non-zero entries whose computation takes O ( N ) operations [CDD’02]: Suppose that [ s < d − t � u N − u � � N − s n ] L is q ∗ -computable with q ∗ > s [ q ∗ ≥ d − t n is suff. ] Some condition on g then u [ N ] = SOLVE [ N, L , g ] satisfies � u [ N ] − u � � N − s Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.10/19
Compressibility of diff. ops. [Stevenson ’04]: Suppose L is a diff. op. L, L ′ : H t + σ → H − t + σ are bounded with σ ≥ d − t Ψ are piecewise polynomial wavelets that are smooth: γ ≥ d − d − t n have vanishing moments: ˜ d ≥ d − 2 t then we can construct L N by dropping entries from L , s.t. with some q ∗ ≥ d − t ( > s ) n for any q < q ∗ , � L N − L � � N − q L N has � N non-zeros in each column Need to spend O ( N ) ops. for each column of L N Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.11/19
Quadrature for diff. ops. Θ a αβ ∂ α ψ λ ∂ β ψ µ � L λ,µ = � Θ = supp ψ λ ∩ supp ψ µ αβ a αβ are piecewise smooth Ψ are piecewise pols of order e (degree e − 1 ) Internal, positive, interpolatory quadratures Composite quadrature of rank W : Subdivide Θ into W subdomains Apply quadrature of order p to each subdomain Fixed order p , variable rank W work = O ( W ) ❀ Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.12/19
Computability of diff. ops. [T.G., Stevenson ’04] Fix quadrature order p > q ∗ n + e − 1 − t Fix θ and ̺ satisfying q ∗ n p ≤ θ ≤ ̺ < 1 − e − 1 − t p L ∗ N — computed approximation of L N Work W N,λ,µ � max { 1 , N θ 2 −|| λ |−| µ || n̺ } for [ L ∗ N ] λ,µ Then ( ⇒ ∀ q < q ∗ : � L − L ∗ N � � N − q ∗ � L N − L ∗ N � � N − q ) Work for each column of L ∗ N is O ( N ) Therefore L is q ∗ -computable with q ∗ ≥ d − t n Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.13/19
Compressibility of b.i.o. [Stevenson ’04]: Suppose L is a b.i.o. Ω is sufficiently smooth K ( x, y ) satisfies the Calderon-Zygmund estimate Ψ are sufficiently smooth and have sufficiently many vanishing moments Then we can construct L N by dropping entries from L , s.t. with some q ∗ ≥ d − t ( > s ) n for any q < q ∗ , � L N − L � � N − q L N has � N (log 2 N ) − 2 n − 1 non-zeros in each column Need to spend O ( N ) ops. for each column of L N Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.14/19
Quadrature for b.i.o. Θ = supp ψ λ , Θ ′ = supp ψ µ � � L λ,µ = Θ ′ K ( x, y ) ψ λ ψ µ Θ Far field: dist(Θ , Θ ′ ) � 2 − min {| λ | , | µ |} work = O ( p 2 n ) Uniform mesh, variable order p ❀ Near field: dist(Θ , Θ ′ ) � 2 − min {| λ | , | µ |} Adaptive (non-uniform) mesh, variable order p ❀ work = O ( p 2 n (1 + || λ | − | µ || )) [Schneider ’95], [von Petersdorff, Schwab ’97], [Lage, Schwab ’99], [Harbrecht ’01] Singular integrals: dist(Θ , Θ ′ ) = 0 Duffy’s transformation [Duffy ’82], [Sauter ’96], [vPS’97] Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.15/19
Computability of b.i.o. [T.G. ’04] L ∗ N — computed approximation of L N With sufficiently large, fixed θ and ̺ Choose the quadrature order for [ L ∗ p = θ log 2 N + ̺ || λ | − | µ || + const. N ] λ,µ Then ( ⇒ ∀ q < q ∗ : � L − L ∗ N � � N − q ∗ � L N − L ∗ N � � N − q ) Work for each column of L ∗ N is O ( N ) Therefore L is q ∗ -computable with q ∗ ≥ d − t n Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.16/19
Conclusion Given d and t , for wavelets that are sufficiently smooth have sufficiently many vanishing moments on sufficiently smooth manifolds any internal, positive, interpolatory quadrature formula yields computa- tional schemes for the infinite stiffness matrix L , such that the CDD2 al- gorithm converges at the same rate as that of best N -term approximation. Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.17/19
References A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods II - Beyond the elliptic case. Found. Comput. Math. , 2(3):203–245, 2002. R.P . Stevenson. On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal. , 35(5):1110–1132, 2004. T. Gantumur and R.P . Stevenson. Computation of differential operators in wavelet coordinates. Technical Report 1306, Utrecht University, August 2004. Submitted. T. Gantumur. Computation of singular integral operators in wavelet coordinates. Technical Report, Utrecht University, 2004. To appear. This work was supported by the Netherlands Organization for Scientific Research and by the EC-IHP project “Breaking Complexity”. Tsogtgerel Gantumur - “Computation of operators in wavelet coordinates” - IHP Mid-Term Meeting. Pavia. Italy. Dec 2004. – p.18/19
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