Some Wavelet Algorithms for Partial Differential Equations
Jacques Liandrat · 1996
Abstract The use of wavelets in numerical analysis is at first glance attractive for two basic reasons. First, the wavelet approach and its associated multiresolution analysis provide nice approximation spaces Vj and Wj suitable for the computation of an approximate solution to problems in which small-scale structures are localized in space and whose location may vary in time. Second, the multiresolution spaces lead to fast hierarchical algorithms with O(Nlog N) operation count. These two properties alone are not sufficient to fully exploit the potential of wavelets. On must understand the action of operators and their inverses on wavelets to insure that the implicit systems of equations that are a part of many discrete numerical methods have the required nice properties for efficient inversion. These include diagonal dominance, low condition numbers (or the existence of good preconditioners), and banded structure.