Operator Reconstruction in Wavelet Bases and Its Use in Partial Differential Equations

<div class="line" id="line-5"> We give a survey on recent development on wavelet-based numerical solution of time-dependent partial di&fflig;erential equa-tions. The fundamental idea is to use wavelet to give sparse representations of the solution operators involved. Thus it leads to a fast algorithm for e&ffilig;cient approximation of the solution to the PDE. We demonstrate the general scheme by considering the anisotropic di&fflig;usion equation arising in thin &filig;lm image processing. Among other examples are advection-di&fflig;usion equations arising in CF D. Numerical results are presented.</div>