An Efficient Adaptive Wavelet-Collocation Method Using Lifted Interpolating Wavelets

수정된 보간 웨이블렛응 이용한 적응 웨이블렛-콜로케이션 기법

  • Published : 2000.08.01


The wavelet theory is relatively a new development and now acquires popularity and much interest in many areas including mathematics and engineering. This work presents an adaptive wavelet method for a numerical solution of partial differential equations in a collocation sense. Due to the multi-resolution nature of wavelets, an adaptive strategy can be easily realized it is easy to add or delete the wavelet coefficients as resolution levels progress. Typical wavelet-collocation methods use interpolating wavelets having no vanishing moment, but we propose a new wavelet-collocation method on modified interpolating wavelets having 2 vanishing moments. The use of the modified interpolating wavelets obtained by the lifting scheme requires a smaller number of wavelet coefficients as well as a smaller condition number of system matrices. The latter property makes a preconditioned conjugate gradient solver more useful for efficient analysis.


Wavelet;Multi-Resolution Analysis;interpolating Wavelet;Lifting Scheme;Boundary Wavelet;Preconditioned Conjugate Gradient Method;Adaptive Algorithm;Thresholding Parameter


  1. Dahlke, S., Dahmen, W., Hochmuth, R. and Schneider, R., 1997, Stable Multiscale Bases and Local Error Estimation for Elliptic Problems, Applied Numerical Mathematics, Vol. 23, pp. 21-47
  2. Dahmen, W., 1997, Wavelet and Multiscale Methods for Operator Equations, Acta Numerica, Cambridge University Press, Vol. 6, pp. 55-228
  3. Sweldens, Wim, 1998, The Lifting Scheme: A Construction of Second Generation Wavelets, SIAM J. MATH. ANAL, Vol. 29(2), pp. 511-546
  4. Sweldens, Wim and Schroder, P., 1995, Building Your Wavelets at Home, Technical Report 1995:5, Industrial Mathematics Initiative, Department of Mathematics, University of South Carolina
  5. Cohen, A. and Masson, R., 1997, Wavelet Adaptive Methods for Elliptic Equations-Preconditioning and Adaptivity, Preprint, Lan, University Pierre et Marie Curie, Paris, to appear in SIAM J. Sci. Comp.
  6. Daubechies, I., 1998, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, SIAM Philadelphia
  7. Deslauriers, G. and Dubuc, S., 1989, Systematic Iterative Interpolation Processes, Constructive Approximation, Vol. 5, pp. 49-68
  8. Beylkin, G. and Saito, N., 1993, Multiresolution Representations Using the Auto-Correlation Functions of Compactly Supported Wavelets, IEEE Trans. Singal Processing Dec., Vol. 41, pp. 3584-3590
  9. Glowinski, R., Pan, T.W., R.O.Wells Jr., R.O., and Zhou, X., 1994, Wavelet and Finite Element Solutions for the Neumann Problem Using Fictitious Domains, Computational Mathematics Laboratory, Technical Report, Rice University
  10. Diaz, A.R., 1999, A Wavelet-Galerkin Scheme for Analysis of Large-Scale Problems on Simple Domains, International Journal for Numerical Methods in Engineering, Vol. 44, pp. 1599-1616<1599::AID-NME556>3.0.CO;2-P
  11. Bertoluzza, S. and Naldi, G., 1996, A Wavelet Collocation Method for the Numerical Solution of Partial Differential Equations, Appl. Comput. Harmon. Anal, Vol. 3, pp. 1-9
  12. Bertoluzza, S., 1997, An Adaptive Collocation Method Based on Interpolating Wavelets, Multiscale Wavelet Methods for Partial Differential Equations, Academic Press, San Diego, pp. 109-135
  13. Donoho, D.L., 1992, Interpolating Wavelet Transforms, Technical Report, Department of statistics, Stanford University
  14. Kim, Y. Y. and Yoon, G.H., 1999, Multi-Resolution, Multi-Scale Topology Optimization - A New Paradigm, to Appear in Int. J. Solids Structures
  15. Qian, S., Amaratunga, K., Williams, J. and Weiss, J., 1994, Wavelet-Galerkin Solutions for One-Dimensional Partial Differential Equations, Int. J. Numer. Meth. Eng., Vol. 37, pp. 2703-2716
  16. Cohen, A. and Masson, R., 1997, Wavelet Adaptive Methods for Second Order Elliptic Problems - Boundary Conditions and Domain Decomposition, Preprint, Lan, Universite Pierre et Marie Curie, Paris