Two Applications of Adaptive Wavelet Methods
Angela Kunoth · International series of numerical mathematics · 2003
In this survey two different applications are discussed where the particular features of wavelet bases, namely, the Riesz basis property together with their locality are exploited. The first example concerns explicitly given information and discusses the problem of fitting nonuniformly distributed data to approximate surfaces. The second application, in which the information is contained implicitly , is concerned with an optimal control problem with constraints appearing in form of a linear elliptic partial differential equation. Both applications have in common that the formulation of the solution method is based on minimizing a quadratic functional and that the concept of adaptivity in a coarse—to—fine fashion plays a central role. An appropriate refinement criterion together with a thresholding technique guarantees to avoid a large number of small and unnecessary expansion coefficients of the target objects.