Adaptive solution of multidimensional PDEs via tensor product wavelet decomposition

An adaptive multidimensional algorithm for solving strictly elliptic PDEs is described. The main points of the present approach are: 1) For a given finite (but arbitrary) accuracy ǫ we find scales and locations of ǫ-significant wavelet coefficients by examining the wavelet decompositions of the right-hand side as well as the coefficients of the equation. In doing so we effectively construct subspaces for the solution (or “masks”) for different accuracy thresholds. 2) We use a diagonally preconditioned “constrained” conjugate gradient (CG) method where the preconditioner is that of an extended problem with periodic bondary conditions to compute the actual coefficients of the approximate solution. We use sparse data structures in our implementations in order to take advantage of O(Ns) complexity of the algorithm, where Ns is the number of significant coefficients required for a given accuracy. We present Received: January 26, 2008 c © 2008, Academic Publications Ltd. §Correspondence author 76 A. Averbuch, G. Beylkin, R. Coifman, P. Fischer, M. Israeli numerical examples in one, two and three spatial dimensions. Professor Moshe Israeli passed away on February 18, 2007. This paper is dedicated to his memory. AMS Subject Classification: 35D05

[1]  Y. Meyer.  Wavelets and Operators , 1993 .

[2]  Representation of the Atomic Hartree-Fock Equations in a Wavelet Basis by Means of the BCR Algorithm , 1994 .

[3]  Pat Hanrahan,et al.  Wavelet radiosity , 1993, SIGGRAPH.

[4]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[5]  Amara Lynn Graps,et al.  An introduction to wavelets , 1995 .

[6]  GermanyNumerische Mathematik,et al.  Multilevel Preconditioning , 1992 .

[7]  Amir Averbuch,et al.  Speed versus quality in low bit-rate still image compression , 1999, Signal Process. Image Commun..

[8]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[9]  S. Jaffard.  Wavelet methods for fast resolution of elliptic problems , 1992 .

[10]  Junping Wang,et al.  Convergence analysis of the Schwarz algorithm and multilevel decomposition iterative methods II: nonselfadjoint and indefinite elliptic problems , 1993 .

[11]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[12]  T. Tao.  On the Almost Everywhere Convergence of Wavelet Summation Methods , 1996 .

[13]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Elliptic Problems - Implementation and Numerical Experiments , 2001, SIAM J. Sci. Comput..

[14]  A. Brandt,et al.  Multilevel matrix multiplication and fast solution of integral equations , 1990 .

[15]  Silvia Bertoluzza,et al.  A dynamically adaptive wavelet method for solving partial differential equations , 1994 .

[16]  Amir Averbuch,et al.  Fast adaptive wavelet packet image compression , 2000, IEEE Trans. Image Process..

[17]  G. Beylkin.  On the representation of operators in bases of compactly supported wavelets , 1992 .

[18]  Harry Yserentant,et al.  On the multi-level splitting of finite element spaces , 1986 .

[19]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[20]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[21]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[22]  Xuejun Zhang,et al.  Multilevel Schwarz methods , 1992 .

[23]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[24]  M. Defranceschi,et al.  Numerical Solution of the Schr Odinger Equation in a Wavelet Basis for Hydrogenlike Atoms , 2022 .

[25]  Y. Egorov,et al.  Fourier Integral Operators , 1994 .

[26]  Stanley Osher,et al.  Fast Wavelet Based Algorithms for Linear Evolution Equations , 1994, SIAM J. Sci. Comput..

[27]  François Treves,et al.  Introduction to Pseudodifferential and Fourier Integral Operators , 1980 .

[28]  Y. Meyer,et al.  Wavelets: Calderón-Zygmund and Multilinear Operators , 1997 .