Adaptive Wavelet Methods on Unbounded Domains

In this paper, we introduce an adaptive wavelet method for operator equations on unbounded domains. We use wavelet bases on ℝn to equivalently express the operator equation in terms of a well-conditioned discrete problem on sequence spaces. By realizing an approximate adaptive operator application also for unbounded domains, we obtain a scheme that is convergent at an asymptotically optimal rate. We show the quantitative performance of the scheme by various numerical experiments.

[1]  Claudio Canuto,et al.  The wavelet element method. Part I: Construction and analysis. , 1997 .

[2]  Ronald A. DeVore,et al.  Multiscale, Nonlinear and Adaptive Approximation , 2009 .

[3]  Winfried Sickel,et al.  Best m-term approximation and Lizorkin-Triebel spaces , 2011, J. Approx. Theory.

[4]  Michael Christian Lehn,et al.  FLENS - a flexible library for efficient numerical solutions , 2008 .

[5]  Erwan Faou,et al.  Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets , 2009, SIAM J. Sci. Comput..

[6]  Tammo Jan Dijkema,et al.  Adaptive tensor product wavelet methods for solving PDEs , 2009 .

[7]  George C. Donovan,et al.  Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets , 1996 .

[8]  Steffen Dereich,et al.  Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction , 2011, 1101.1369.

[9]  R. Schneider,et al.  Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme , 1995 .

[10]  Wolfgang Dahmen,et al.  Fast computation tools for adaptive wavelet schemes , 2005 .

[11]  Rob Stevenson,et al.  An Adaptive Wavelet Method for Solving High-Dimensional Elliptic PDEs , 2009 .

[12]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Nonlinear Variational Problems , 2003, SIAM J. Numer. Anal..

[13]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[14]  Christian Bender,et al.  Dual pricing of multi-exercise options under volume constraints , 2011, Finance Stochastics.

[15]  G. Teschke,et al.  A compressive Landweber iteration for solving ill-posed inverse problems , 2008 .

[16]  Pál-Andrej Nitsche,et al.  Best N Term Approximation Spaces for Tensor Product Wavelet Bases , 2006 .

[17]  Erich Novak,et al.  Optimal approximation of elliptic problems by linear and nonlinear mappings IV: Errors in L2 and other norms , 2004, J. Complex..

[18]  Michael Griebel,et al.  Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems , 1995, Adv. Comput. Math..

[19]  Stephan Dahlke,et al.  Adaptive wavelet methods and sparsity reconstruction for inverse heat conduction problems , 2010, Adv. Comput. Math..

[20]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

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

[22]  Rob P. Stevenson,et al.  Computation of differential operators in wavelet coordinates , 2005, Math. Comput..

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

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

[25]  Winfried Sickel,et al.  Best m-Term Approximation and Sobolev–Besov Spaces of Dominating Mixed Smoothness—the Case of Compact Embeddings , 2012 .

[26]  Daniel Rudolf,et al.  Error bounds for computing the expectation by Markov chain Monte Carlo , 2009, Monte Carlo Methods Appl..

[27]  T. Goodman.  A Class of Orthogonal Refinable Functions and Wavelets , 2003 .

[28]  Rob P. Stevenson,et al.  Space-time adaptive wavelet methods for parabolic evolution problems , 2009, Math. Comput..

[29]  G. Teschke,et al.  Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints , 2010 .

[30]  Wolfgang Dahmen,et al.  Fast high-dimensional approximation with sparse occupancy trees , 2011, J. Comput. Appl. Math..

[31]  Karsten Urban,et al.  Wavelet Methods for Elliptic Partial Differential Equations , 2008 .

[32]  Rob Stevenson,et al.  On the Compressibility of Operators in Wavelet Coordinates , 2004, SIAM J. Math. Anal..

[33]  Winfried Sickel,et al.  Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross , 2009, J. Approx. Theory.

[34]  Pál-Andrej Nitsche,et al.  Sparse Approximation of Singularity Functions , 2003 .

[35]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[36]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[37]  D. Hardin,et al.  Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets , 1999 .

[38]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[39]  Rob Stevenson,et al.  Adaptive wavelet algorithms for elliptic PDE's on product domains , 2008, Math. Comput..

[40]  Rob Stevenson,et al.  Adaptive wavelet methods for solving operator equations: An overview , 2009 .

[41]  K. Bredies,et al.  Regularization with non-convex separable constraints , 2009 .

[42]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[43]  Anita Tabacco,et al.  Ondine Biortogonali: teoria e applicazioni , 1999 .

[44]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[45]  G. Plonka.  The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data , 2009 .

[46]  Claudio Canuto,et al.  An anisotropic functional setting for convection-diffusion problems , 2001, J. Num. Math..

[47]  Peter Oswald,et al.  Criteria for Hierarchical Bases in Sobolev Spaces , 2000 .

[48]  Wolfgang Dahmen,et al.  Adaptive methods for boundary integral equations: Complexity and convergence estimates , 2007, Math. Comput..

[49]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[50]  Albert Cohen,et al.  Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition , 2000, Numerische Mathematik.

[51]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[52]  Rob P. Stevenson,et al.  An optimal adaptive wavelet method without coarsening of the iterands , 2006, Math. Comput..

[53]  Lars Grasedyck,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig a Projection Method to Solve Linear Systems in Tensor Format a Projection Method to Solve Linear Systems in Tensor Format , 2022 .

[54]  Stefano Berrone,et al.  An Adaptive WEM Algorithm for Solving Elliptic Boundary Value Problems in Fairly General Domains , 2006, SIAM J. Sci. Comput..