Compressive algorithms. adaptive solutions of PDEs and variational problems

M. Fornasier, RICAM, Linz, Austria

Wednesday, December 17
at 09.15, HG D 16.2

Solutions of certain PDEs and variational problems may be characterized by a few significant degrees of freedom, and one may want to take advantage of this feature in order to design efficient numerical solutions. Examples of such situations are ubiquitous: adaptive solution of PDEs, singular PDEs for image processing, crack modelling and free-discontinuity problems, viscosity solutions of Hamilton-Jacobi equations, digital signal coding/decoding, and compressed sensing. In the first part of the talk we review the role of variational principles, in particular L1-minimization, as a method for sparsifying solutions in several contexts. Then we address particular applications and numerical methods.

We present the analysis of a superlinear-convergent algorithm for L1 minimization based on an iteratively re-weighted least-squares method. An analogous algorithm is then applied for the efficient solution of a system of singular PDEs for image re-colorization in a relevant real-life problem of art restoration.

This introduces us to other algorithms for performing efficiently L1 minimization, based on projected gradient methods and subspace-correction/domain-decomposition methods.

The second part of the talk addresses the issue of embedding compressibility in numerical simulation, and in particular the use of adaptive strategies for the solution of elliptic partial differential equations discretized by means of redundant frames. We discuss the construction of wavelet frames on bounded domains and the optimal performances of adaptive solvers in this context. We conclude our presentation with a brief discussion on active cooperations and ongoing projects of the candidate.