Today there are a number of problems in engineering and science, which share a single common computational challenge: the ability to solve and/or model accurately and efficiently a wide range of spatial and temporal scales. Different scales are often not distributed uniformly in space and time and have complex nonlinear dynamics due to different physical feedback mechanisms. In this talk we present a general framework for constructing an adaptive method that takes advantage of the multi-resolution wavelet analysis, a new mathematical concept, which allows one to represent a function in terms of special basis functions, called wavelets.

Wavelets are localized in both space and scale, and as a result functions with localized regions of sharp transition are well compressed using wavelet decomposition. This property allows us to construct an efficient adaptive numerical method, which employs wavelet compression as an integral part of the solution. The adaptation is achieved by retaining only those wavelets whose coefficients are greater than an a priory prescribed threshold. This property of the multi-level wavelet approximation allows local grid refinement up to an arbitrary small scale without a drastic increase in the number of grid points; thus high resolution computations are carried out only in those regions where sharp transitions occur. Wavelet decomposition is used for both grid adaptation and interpolation, while a O(N) hierarchical finite difference scheme, which takes advantage of multi-level wavelet decomposition, is used for derivative calculations. The prowess and computational efficiency of the adaptive wavelet collocation method are demonstrated for the solution of a number of test problems including both evolution-type and elliptic partial differential equations. The results indicate that the computational grid and associated wavelets can very efficiently adapt to the local irregularities of the solution. Furthermore, a solution is obtained on a near optimal grid, i.e. the compression of the solution is performed dynamically as opposed to a posteriori as it is done in data analysis.