PDEs. A recent attempt has been made by Jawerth and Sweldens [29] to which the reader is referred for more
The paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling comparative information. The currently existing algofunction. This approach permits to compute the decomposition of rithms can be classified in different ways. First of all, we a function into a lacunary wavelet basis, i.e., a basis constituted can distinguish between Galerkin or Petrov-Galerkin of a subset of all basis functions up to a certain scale, without schemes, collocation schemes, and algebraic methods. By modification. The construction is then extended to operatorthe latter we mean algorithms which start from a classical adapted biorthogonal wavelets. This is relevant for the solution of certain nonlinear evolutionary PDEs where a priori information discretization, e.g., by finite differences in space. Wavelets about the significant coefficients is available. We pursue the apare then used in the following stages to speed up the linear proach described in (J. Fro ¨hlich and K. Schneider, Europ. J. Mech. algebra. On the other hand, the former two schemes em-B/Fluids 13, 439, 1994) which is based on the explicit computation ploy wavelets or wavelet-like functions directly for the of the scalewise contributions of the approximated function to the discretization of the solution and the operators which then values at points of hierarchical grids. Here, we present an improved construction employing the cardinal function of the multiresolution. induces the subsequent linear algebra. The new method is applied to the Helmholtz equation and illustrated Another classification can be made according to whether by comparative numerical results. It is then extended for the solution a wavelet representation is used for the efficient represenof a nonlinear parabolic PDE with semi-implicit discretization in time tation of an operator, the compressed representation of and self-adaptive wavelet discretization in space. Results with full the solution or for both. The first family comprises the adaptivity of the spatial wavelet discretization are presented for a one-dimensional flame front as well as for a two-dimensional ''decomposition schemes.'' They are based on the fact that problem. ᮊ 1997 Academic Press
wavelets are well localized in physical space and in Fourier space. Hence, an operator which does not too much perturb this localization has, for a given precision, a sparse We subsume the cited methods (and others not mentioned 174