A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems

Abstract

We present a theoretical analysis of the CORSING ( COmpRessed SolvING ) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s s -term approximation of the weak solution of a PDE with respect to a system of N N trial functions, can be recovered via a Petrov-Galerkin approach using m ≪ N m \ll N test functions. This recovery is guaranteed if the local a a -coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted inf-sup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing.