Compressive Algorithms. Adaptive Solutions of PDEs and Variational Problems

Abstract. This paper is concerned with an overview of the main concepts and a few significant applications of a class of adaptive iterative algorithms which allow for dimensionality reductions when used to solve large scale problems. We call Compressive Algorithms this class of numerical methods. The introduction of this paper presents an historical excursus on the developments of the main ideas behind compressive algorithms and stresses the common features of diverse applications. The first part of the paper is addressed to the optimal performances of such algorithms when compared with known benchmarks in the numerical solution of elliptic partial differential equations. In the second part we address the solution of inverse problems both with sparsity and compressibility constraints. We stress how compressive algorithms can stem from variational principles. We illustrate the main results and applications by a few significant numerical examples. We conclude by pointing out future developments.