Compressed solving: A numerical approximation technique for elliptic PDEs based on Compressed Sensing

Simone Brugiapaglia, Stefano Micheletti, Simona Perotto

Abstract: We introduce a new numerical method denoted by CORSING (COmpRessed SolvING) to approximate advection–diffusion 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<math><msub is="true"><mrow is="true"><mi is="true">ℓ</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub></math>- and ℓ1<math><msub is="true"><mrow is="true"><mi is="true">ℓ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub></math>-minimization. A Matlab® implementation of the method assesses the robustness and reliability of the proposed strategy, as well as its effectivity in reducing the computational cost of the corresponding full-sized Petrov–Galerkin problem.