Compressed solving: a numerical approximation technique for PDEs based on compressed sensing

43/2014

Compressed solving: a numerical approximation technique for PDEs based on compressed sensing

Wednesday 22nd October 2014

Brugiapaglia, S.; Micheletti, S.; Perotto, S.

We introduce a new numerical method denoted by CORSING (COmpRessed SolvING) to approximate one-dimensional advection-diffusion-reaction problems, motivated by the recent developments in the sparse representation field, and particularly in Compressed Sensing. The object of CORSING is to lighten the computational cost characterizing a Petrov-Galerkin discretization by reducing the dimension of the test space with respect to the trial space. This choice yields an underdetermined linear system which is solved by exploiting optimization procedures, standard in Compressed Sensing, such as the l0- and l1-minimization. A Matlab implementation of the method assesses the robustness and reliability of the proposed strategy, as well as its effectiveness in reducing the computational cost of the corresponding full-sized Petrov- Galerkin problem. Finally, a preliminary extension of CORSING to the two-dimensional setting is checked on the classical Poisson problem.