An Efficient Adaptive Wavelet-Collocation Method Using Lifted Interpolating Wavelets

- Journal title : Transactions of the Korean Society of Mechanical Engineers A
- Volume 24, Issue 8, 2000, pp.2100-2107
- Publisher : The Korean Society of Mechanical Engineers
- DOI : 10.22634/KSME-A.2000.24.8.2100

Title & Authors

An Efficient Adaptive Wavelet-Collocation Method Using Lifted Interpolating Wavelets

Kim, Yun-Yeong; Kim, Jae-Eun;

Kim, Yun-Yeong; Kim, Jae-Eun;

Abstract

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.

Keywords

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

Language

Korean

References

1.

Kim, Y. Y. and Yoon, G.H., 1999, Multi-Resolution, Multi-Scale Topology Optimization - A New Paradigm, to Appear in Int. J. Solids Structures

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

8.

Donoho, D.L., 1992, Interpolating Wavelet Transforms, Technical Report, Department of statistics, Stanford University

9.

Daubechies, I., 1998, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, SIAM Philadelphia

10.

Deslauriers, G. and Dubuc, S., 1989, Systematic Iterative Interpolation Processes, Constructive Approximation, Vol. 5, pp. 49-68

11.

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

12.

Sweldens, Wim, 1998, The Lifting Scheme: A Construction of Second Generation Wavelets, SIAM J. MATH. ANAL, Vol. 29(2), pp. 511-546

13.

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

14.

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.

15.

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

16.

Dahmen, W., 1997, Wavelet and Multiscale Methods for Operator Equations, Acta Numerica, Cambridge University Press, Vol. 6, pp. 55-228