Coping with redundancy: frame-based discretizations of operator equations.

The numerical simulation of differential equations or integral equations usually involves their discretization in a finite-dimensional space. This space is spanned by a basis, which is complete and never redundant. Unfortunately, a basis is hard to construct on domains with even moderate geometrical complexity. High-order approximation schemes on such domains, with associated fast transforms, are virtually non-existent. Frames generalize the notion of a basis, and they hold the promise of delivering fast and high-order approximations on general domains. Their construction is trivial, for example based on periodic Fourier series or wavelets. However, frames are redundant and this induces several challenges when they are used in a discretization. The goal of the project is to explore the fundamental properties of such redundant discretizations, and to leverage recent results and insights on approximations using frames to discretizations of operator equations using frames.

Keywords:frame-based discretizations

Disciplines:Communications, Communications technology