A variational method using Alpert multiwavelets

Abstract

The numerical solution of variational problems is usually achieved by numerical solution of the Euler--Lagrange differential equations or by Rayleigh--Ritz direct methods. In 1975, Chen and Hsiao showed how Walsh functions could be used in a direct method to solve several model variational problems. This Rayleigh--Ritz method was generalized by Sloss and Blyth in 1998. In 2004, Hsiao modified the Walsh function method to provide a Haar wavelet direct method and illustrated how this could be used for the solution of a few model problems. We extend this by applying Alpert multiwavelets to the direct solution of variational problems. Alpert multiwavelets were developed for the numerical solution of integral equations and provide a generalization of Haar wavelets. Alpert multiwavelets have the advantage of being expressible as simple polynomials over disjoint subintervals allowing for ease of computation. The method is applied to an example which allows comparisons with the results for Haar wavelets to be made. Some convergence results are given. References B. K. Alpert. Sparse Representation of Smooth Linear Operators. PhD Thesis, Yale University, 1990. B. K. Alpert, G. Beylkin, D. Gines and L. Vozovoi. Adaptive Solution of Partial Differential Equations in Multiwavelet Bases. Journal of Computational Physics , 182, 149--190, 2002. doi:10.1006/jcph.2002.7160 R. Y. Chang and M. L. Wang. Shifted Legendre direct method for variational problems. J. Optimization Theory Appl. , 39, 299--307, 1983. C. F. Chen and C. H. Hsiao. A Walsh series direct method for solving variational problems. J. Franklin Inst. , 300, 265--280, 1975. I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. , 41(7), 909--996, 1988. C. H. Hsiao. Haar wavelet direct method for solving variational problems. Mathematics and Computers in Simulation , 64, 569--585, 2004. doi:10.1016/j.matcom.2003.11.012 C. Hwang and Y. P. Shih. Laguerre series direct method for variational problems. J. Optimization Theory Appl. , 39, 143--149, 1983. R. S. Schechter. The Variational Method in Engineering . McGraw--Hill, NewYork, 1967. B. G. Sloss and W. F. Blyth. A variational method using Walsh functions. Nonlinear Analysis, Theory, Methods and Applications , 32, 549--561, 1998. D. F. Walnut. An introduction to wavelet analysis . Birkhauser, Boston, 2002.