## Friday, September 11, 2015

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

Simone just sent me the following:

Dear Igor,

I am a Ph.D. student at MOX - Politecnico di Milano and an avid reader of your blog Nuit Blanche. :)

I'm writing to bring to your attention a recent work of me, Stefano Micheletti, Fabio Nobile and Simona Perotto about a method for the numerical approximation of PDEs based on compressed sensing, that we named CORSING (acronym for COmpRessed SolvING).

We present the CORSING method and an extensive set of numerical examples in [1], whereas a theoretical analysis of the method is carried out in [2].

[1] Compressed solving: a numerical approximation technique for elliptic PDEs based on compressed sensing
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Computers & Mathematics with Applications 70(6), pp. 1306-1335, 2015.

preprint: http://www.researchgate.net/publication/281125194

[2] A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems
MOX Report 42/2015 - Dipartimento di Matematica, Politecnico di Milano.

It would be amazing to share this work with the compressed sensing community through your excellent blog, in order to receive helpful suggestions and comments!

Best wishes,

Simone Brugiapaglia
Thanks Simone !

Here are the papers:

Compressed solving: A numerical approximation technique for elliptic PDEs based on Compressed Sensing by S. Brugiapaglia, , S. Micheletti , S. Perotto

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 ℓ0ℓ0- and ℓ1ℓ1-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.

A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems by S. Brugiapaglia, F. Nobile, S. Micheletti and S. Perotto

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-term approximation of the weak solution of a PDE with respect to an orthonormal system of N trial functions, can be recovered via a Petrov-Galerkin approach using m N orthonormal test functions. This recovery is guaranteed if the local 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.

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