Sparse recovery and Compressive sensing in theory and in practice

Abstract: In the 1990's, algorithms for solving linear systems with the number of equations smaller than the number of unknowns, provided that among the unknowns there are only a small number of non-zero ones (however, we do not know which of them are non-zero!) were proposed.

A new stage was opened in the early 2000's by the well-known specialist in signal processing David Donoho and the Fields Medal winner Terence Tao and their students. The results in this area were awarded the 2018 Gauss Prize (given by the International Mathematical Union), they were reported as plenary talks at the International Congress of Mathematicians, etc.

After the works of Donoho, Tao and many other researchers, progress in this area was rapid. This research area was called "compressive sensing" or "compressed sensing" (along with the older name "sparse recovery").

The most well-known applications of these results are in signal processing. Particularly noteworthy are applications of sparse recovery technologies in magnetic resonance imaging (MRI), which reduce the time spend by patients in the MRI machine and improve the quality of the resulting image.

The report will discuss the main ideas of this area and demonstrate a small practical application in the problem of finding jumps in a noisy signal.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods