WAVELET-GALERKIN SOLUTION OF BOUNDARY-VALUE-PROBLEMS

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 wavel ets is described here using the language and mathematics of signal pro cessing. 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 different ial equations and image boundary problems to be overcome in image proc essing. Finally, we discuss the fast inversion of matrices arising fro m differential operators by preconditioning the multiscale wavelet mat rix. Wavelet preconditioning is shown to limit the growth of the matri x condition number, such that Krylov subspace iteration methods can ac complish fast inversion of large matrices.