Computation of operators in wavelet coordinates

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1 Computation of operators in wavelet coordinates Tsogtgerel Gantumur and Rob Stevenson Department of Mathematics Utrecht University Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.1/22

2 Overview Linear operator equation Lu = g with L : H H Riesz basis Ψ = {ψ λ } of H, e.g. u = λ u λψ λ Infinite dimensional matrix-vector system Lu = g, with u = (u λ ) λ and L : l 2 l 2 Convergent iterations such as u (i+1) = u (i) + α[g Lu (i) ] We can approximate Lu (i) by a finitely supported vector How cheap can we compute this approximation? The answer will depend on L and Ψ Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.2/22

3 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 2t and g H t is a linear functional Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.3/22

4 Differential operators Partial differential operators of order 2t v, Lu = α v, a αβ β u, α, β t Example: The reaction-diffusion equation (t = 1) v, Lu = v u + κ 2 vu, Ω Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.4/22

5 Singular integral operators Boundary integral operators [Lu](x) = K(x, y)u(y)dω y with the kernel K(x, y) singular at x = y Ω Example: The single layer operator for the Laplace BVP in 3-d domain (t = 1 2 ) K(x, y) = 1 4π x y Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.5/22

6 Multiresolution analysis S 0 S 1... H t and S0 S 1... H t dim S j, dim S j = O(2 jn ) (dyadic refinements) S j contains all piecewise pols of degree d 1 S j contains all piecewise pols of degree d 1 S j is globally C r -smooth Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.6/22

7 Wavelet bases Ψ = {ψ λ : λ Λ} is a Riesz basis for H t each v H t has a unique expansion v = λ Λ v λ ψ λ s.t. c v v H t C v For every index λ Λ, there is a number λ IN 0 called the level of ψ λ span{ψ λ : λ j} = S j ψ λ, v = 0 for any v S λ 1 diam(supp ψ λ ) = O(2 λ ) Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.7/22

8 Typical wavelets ψ λ ψ µ x ψ λ is a piecewise polynomial of degree d 1 x k ψ λ (x)dx = 0 for k < d ( d vanishing moments) Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.8/22

9 Galerkin methods Wavelet basis Ψ j := {ψ λ : λ j} of S j Stiffness L (j) = Lψ λ, ψ µ λ, µ j load g (j) = g, ψ λ λ j Linear equation in IR N j (N j := dim S j ) L (j) u (j) = g (j) L (j) : IR N j IR N j SPD and g (j) IR N j u (j) = λ [u (j)] λ ψ λ approximates the solution of Lu = g Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.9/22

10 Galerkin approximation If u H s for some s [t, d] N j = dim S j = O(2 jn ) ε (j) O(N s t n j ) ε (j) := u (j) u H t O(2 j(s t) ) Solve L (j) u (j) = g (j) with CG complexity O(N j ) Similar estimates for FEM Better convergence? Adaptive methods? Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.10/22

11 Nonlinear approximation Given u = (u λ ) λ l 2 Approximate u using N coeffs u λ λ Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.11/22

12 Nonlinear approximation Given u = (u λ ) λ l 2 Approximate u using N coeffs u λ λ (arranged) Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.11/22

13 Nonlinear approximation Given u = (u λ ) λ l 2 Approximate u using N coeffs u λ } {{ } N biggest u N best approximation of u with #supp u N N Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.11/22

14 Nonlinear approximation Given u = (u λ ) λ l 2 Approximate u using N coeffs [u N u] λ N zeroes u N best approximation of u with #supp u N N Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.11/22

15 Nonlinear vs. linear approximation If u B s τ,τ with 1 τ = s t n for some s < d ε N = u N u O(N s t n ) If u H s for some s d, uniform refinement ε (j) = u (j) u O(N s t n j ) B s τ,τ is bigger than H s Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.12/22

16 Besov vs. Sobolev regularity s d 1 τ = s t n B s τ,τ t τ [Dahlke, DeVore]: u B d τ,τ with 1 τ = d t n "often" Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.13/22

17 Equivalent problem in l 2 [Cohen, Dahmen, DeVore 02] 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 λψ λ is the solution of Lu = g Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.14/22

18 Richardson iterations in l 2 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] O(N) and ε [N] = u [N] u 0 as N ε [N] speed of convergence? Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.15/22

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 O(N q ) having in each column O(N) non-zero entries whose computation takes O(N) operations [CDD 02]: Suppose that u N u O(N s ) [s < d t n ] L is q -computable with q > s then for suitable g, u [N] = SOLVE[N, L, g] satisfies u [N] u O(N s ) Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.16/22

20 Computability [L λ,µ ] λ Λ µ-th column L λ,µ Approximate by N entries? λ Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.17/22

21 Computability [L λ,µ ] λ Λ µ-th column arranged by modulus L λ,µ N biggest entries? λ (arranged) Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.17/22

22 Computability [L λ,µ ] λ Λ µ-th column L λ,µ }{{} N Compute the N biggest entries Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.17/22

23 Computability The µ-th column of the difference [L N L] λ,µ } {{ } N Need to locate the biggest entries a priori Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.17/22

24 Compressibility L is called q -compressible, when L is q -computable assuming each entry of L is available at unit cost [CDD 01], [Stevenson 04]: Suppose {ψ λ } are piecewise polynomial wavelets that are sufficiently smooth and have sufficiently many vanishing moments L is either differential or singular integral operator then L is q -compressible for some q d t n (> s) Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.18/22

25 Computability Distribute computational works over the entries W Require: shaded area = O(N) N Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.19/22

26 Computability Distribute computational works over the entries N θ p(x) = N θ x ϱ when θ ϱ < 1 N N 0 N θ x ϱ dx = 1 1 ϱ N 1+θ ϱ = O(N) Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.20/22

27 Computability [T.G., Stevenson 04], [T.G. 04]: Suppose {ψ λ } are piecewise polynomial wavelets that are sufficiently smooth and have sufficiently many vanishing moments L is either differential or singular integral operator then L is q -computable for some q d t n (> s) So the adaptive wavelet method has the optimal convergence rate and optimal computational complexity Tsogtgerel Gantumur - Computation of operators in wavelet coordinates - Sixth Minisimposium TIANA. Amsterdam. Oct p.21/22

28 References A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods II - Beyond the elliptic case. Found. Comput. Math., 2(3): , R.P. Stevenson. On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal., 35(5): , T. Gantumur and R.P. Stevenson. Computation of differential operators in wavelet coordinates. Technical Report 1306, Utrecht University, August Submitted. T. Gantumur. Computation of singular integral operators in wavelet coordinates. Technical Report, Utrecht University, 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 - Sixth Minisimposium TIANA. Amsterdam. Oct p.22/22

Adaptive Wavelet Algorithms

Adaptive Wavelet Algorithms for solving operator equations Tsogtgerel Gantumur August 2006 Contents Notations and acronyms vii 1 Introduction 1 1.1 Background.............................. 1 1.2 Thesis