Summary
In this paper we review the application of wavelets to the solution of partial differential equations. We consider in detail both the single scale and the multiscale Wavelet Galerkin method. The theory of wavelets is described here using the language and mathematics of signal processing. We show a method of adapting wavelets to an interval using an extrapolation technique called Wavelet Extrapolation. Wavelets on an interval allow boundary conditions to be enforced in partial differential equations and image boundary problems to be overcome in image processing. Finally, we discuss the fast inversion of matrices arising from differential operators by preconditioning the multiscale wavelet matrix. Wavelet preconditioning is shown to limit the growth of the matrix condition number, such that Krylov subspace iteration methods can accomplish fast inversion of large matrices.
Similar content being viewed by others
References
Morlet, J., Arens, G., Fourgeau, I. and Giard, D. (1982) “Wave Propagation and Sampling Theory”,Geophysics,47, 203–236.
Morlet, J. (1983), “Sampling theory and wave propagation”,NATO ASI Series., Issues in Acoustic Signal/Image Processing and Recognition, C.H. Chen (ed.), Springer-Verlag.Vol. I, 233–261.
Meyer, Y. (1985), “Principe d'incertitude, bases hilbertiennes et algebres d'operateurs”,Seminaire Bourbaki,No. 662.
Meyer, Y. (1986), “Ondettes et functions splines”,Lectures given at the Univ. of Torino, Italy.
Meyer, Y. (1986), “Ondettes, fonctions splines et analyses graduces”,Seminaire EDP, Ecole Polytechnique, Paris, France.
Daubechies, I. (1988), “Orthonormal bases of compactly supported wavelets”,Comm. Pure and Appl. Math.,41, 909–996.
Daubechies, I. (1992), “Ten Lectures on Wavelets”,CBMS-NSF Regional Conference Series, Capital City Press.
Mallat, S.G. (1986), “Multiresolution approximation and wavelets”,Prepring GRASP Lab., Department of Computer and Information Science, Univ. of Pennsylvania.
Mallat, S.G. (1988), “A theory for multiresolution signal decomposition: the wavelet representation”,Comm. Pure and Appl. Math.,41, 7, 674–693.
Smith, M.J.T. and Barnwell, T.P. (1986), “Exact Reconstruction Techniques for Tree-Structured Subband Coders”,IEEE Trans. ASSP 34, pp. 434–441.
Strang, G. and Truong Nguyen, (1996), “Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA.
Oppenheim, A.V. and Schafer, R.W. (1989), “Discrete-Time Signal Processing”,Prentice Hall Signal Processing Series, Prentice Hall, New Jersey.
Vaidyanathan, P.P. (1993), “Multirate Systems and Filter Banks”,Prentice Hall Signal Processing Series.
Latto, A., Resnikoff, H. and Tenenbaum, E. (1992), “The evaluation of connection coefficients of compactly supported wavelets”, to appear inProc. French—USA workshop on Wavelets and Turbulence, Princeton Univ., June 1991, Springer-Verlag.
Beylkin, G., Coifman, R. and Rokhlin, V. (1991), “Fast wavelet transforms and numerical algorithms I”,Comm. Pure Appl. Math.,44, 141–183.
Beylkin, G. (1993), “On Wavelet-Based Algorithms for Solving Differential Equations”,preprint, University of Colorado, Boulder.
Andersson, L., Hall, N., Jarwerth, B. and Peters, G. (1994), “Wavelets on Closed Subsets of the Real Line”, inTopics in the Theory and Applications of Wavelets, Schumaker and Webb, eds., Academic Press, Boston.
Cohen, A., Daubechies, I., Jarwerth, B. and Vial, P. (1993), “Multiresolution Analysis, Wavelets and Fast Algorithms on an Interval”,C. R. Acad. Sci. Paris I,316, 417–421.
Amaratunga, K. and Williams, J.R. (1995), “Time Integration using Wavelets”,Proceedings SPIE—Wavelet Applications for Dual Use,2491, 894–902. Orlando, FL, April.
Mallat, S.G. (1988), “A Theory for Multiresolution Signal Decomposition: the Wavelet Representation”,Comm. Pure and Appl. Math.,41, 7, 674–693.
Williams, J.R. and Amaratunga, K. (1997), “A Discrete Wavelet Transform Without Edge Effects Using Wavelet Extrapolation”,IESL Technical Report No. 94-07, Intelligent Engineering Systems Laboratory, M.I.T., September 1994,to be published in J. of Fourier Anal. and Appl.
Beylkin, G., Coifman, R. and Rokhlin, V. (1991), “Fast wavelet transforms and numerical algorithms I”,Comm. Pure Appl. Math.,44, 141–183.
Williams, J.R. and Amaratunga, K. (1995), “A Multiscale Wavelet Solver with O(N) Complexity”,J. Comp. Phys.,122, 30–38.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Amaratunga, K., Williams, J.R. Wavelet-Galerkin solution of boundary value problems. Arch Computat Methods Eng 4, 243–285 (1997). https://doi.org/10.1007/BF02913819
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02913819