Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions

Abstract Enormous progress has been made in the construction and analysis of adaptive wavelet methods in the recent years. Cohen, Dahmen, and DeVore showed that such methods converge for a wide class of operator equations, both linear and nonlinear. Moreover, they showed that the rate of convergence is asymptotically optimal and that the methods are asymptotically optimally efficient. So far, these methods are based upon biorthogonal wavelets with compactly supported primal and dual functions. Semiorthogonal spline wavelets offer some quantitative advantages, namely small supports and good conditioning of the bases. On the other hand, the corresponding dual functions are globally supported so that they are ruled out for existing wavelet methods for nonlinear variational problems. In this paper, we focus on a core ingredient of adaptive wavelet methods for nonlinear problems, namely the adaptive evaluation of nonlinear functions. We present an efficient adaptive method for approximately evaluating nonlinear functions of wavelet expansions using semiorthogonal spline wavelets. This is achieved by modifying and extending a method for compactly supported biorthogonal wavelets by Dahmen, Schneider, and Xu. In order to do so, we introduce a new adaptive quasi-interpolation scheme, a corresponding prediction and a new decomposition. We give a complete analysis including an investigation of the complexity.

[1]  I. Daubechies,et al.  A STABILITY CRITERION FOR BIORTHOGONAL WAVELET BASES AND THEIR RELATED SUBBAND CODING SCHEME , 1992 .

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

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

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

[5]  Manfred Tasche,et al.  On the Computation of Periodic Spline Wavelets , 1995 .

[6]  Kai Schneider,et al.  Towards an adaptive wavelet-based 3D Navier-Stokes Solver , 1998 .

[7]  Kai Bittner,et al.  Fast Algorithms for Periodic Spline Wavelets on Sparse Grids , 1999, SIAM J. Sci. Comput..

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

[9]  Stefano Grivet-Talocia,et al.  WAVELETS ON THE INTERVAL WITH OPTIMAL LOCALIZATION , 2000 .

[10]  Charles A. Micchelli,et al.  Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two , 1991, Curves and Surfaces.

[11]  Kai Bittner,et al.  Biorthogonal Spline Wavelets on the Interval , 2005 .

[12]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[13]  C. Chui,et al.  On compactly supported spline wavelets and a duality principle , 1992 .

[14]  Wolfgang Dahmen,et al.  Stable multiscale bases and local error estimation for elliptic problems , 1997 .

[15]  I. Daubechies,et al.  Tree Approximation and Optimal Encoding , 2001 .

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

[17]  C. Chui,et al.  A general framework of compactly supported splines and wavelets , 1992 .

[18]  Kai Schneider,et al.  Adaptive Wavelet Methods for the Navier-Stokes Equations , 2001 .

[19]  Zuowei Shen,et al.  Wavelets and pre-wavelets in low dimensions , 1992 .

[20]  E. Quak,et al.  Decomposition and Reconstruction Algorithms for Spline Wavelets on a Bounded Interval , 1994 .

[21]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods : Basic Concepts and Applications to the Stokes Problem , 2002 .

[22]  Reinhard H Stephan Dahlke.  Adaptive Wavelet Methods for Saddle Point Problems , 1999 .

[23]  Tor Berger,et al.  Exact Reconstruction Algorithms for the Discrete Wavelet Transform Using Spline Wavelets , 1995 .

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

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

[26]  Charles A. Micchelli,et al.  Using the refinement equation for the construction of pre-wavelets , 1991, Numerical Algorithms.

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

[28]  Ronald A. DeVore,et al.  Fast computation in adaptive tree approximation , 2004, Numerische Mathematik.

[29]  Winfried Sickel,et al.  Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations , 1996, de Gruyter series in nonlinear analysis and applications.

[30]  Wolfgang Dahmen,et al.  Sparse Evaluation of Compositions of Functions Using Multiscale Expansions , 2003, SIAM J. Math. Anal..

[31]  Larry L. Schumaker,et al.  Numerical Methods in Approximation Theory, Vol. 9 , 1992 .

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

[33]  W. Dahmen,et al.  Wavelets with Complementary Boundary Conditions — Function Spaces on the Cube , 1998 .

[34]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods for Saddle Point Problems - Optimal Convergence Rates , 2002, SIAM J. Numer. Anal..

[35]  Wolfgang Dahmen,et al.  Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation , 2000, Numerische Mathematik.

[36]  C. Chui,et al.  Wavelets on a Bounded Interval , 1992 .

[37]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[38]  Pierre Gilles Lemarié-Rieusset,et al.  Analyse multi-résolution bi-orthogonale sur l’intervalle et applications , 1993 .

[39]  Andrew J. Kurdila,et al.  Scaling Function and Wavelet Preconditioners for Second Order Elliptic Problems , 1997 .

[40]  Silvia Bertoluzza,et al.  On the adaptive computation of integrals of wavelets , 2000 .