Krylov Subspace Methods for Regularisation and Sparse Reconstruction

Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretised, they lead to ill-conditioned linear systems, often of huge dimensions: regularisation consists in replacing the original system by a nearby problem with better numerical properties, in order to find a meaningful approximation of its solution. After briefly addressing some classical regularisation strategies (such as Tikhonov method) and surveying some standard iterative regularisation methods (such as many Krylov methods), this talk will introduce the recent and promising class of the Krylov-Tikhonov iterative regularisation methods. In particular, strategies for choosing the regularisation parameter and the regularisation matrix will be emphasized. Also, this talk will present a common framework that exploits a flexible version of well-known Krylov methods such as CGLS and GMRES to handle nonnegativity constraints and regularization terms expressed with respect to the 1-norm, resulting in an efficient way to enforce sparse reconstructions of the solution. Numerical experiments and comparisons with other well-known methods for the computation of nonnegative and sparse solutions will be presented. These results have been obtained working jointly with James Nagy (Emory University), Paolo Novati (University of Trieste), Yves Wiaux (Heriot-Watt University), and Julianne Chung (Virginia Polytechnic Institute and State University).