Abstract

Krylov subspace methods are popular numerical linear algebra tools that can be successfully employed to regularize linear large-scale inverse problems, such as those arising in image deblurring and computed tomography. Though they are commonly used as purely iterative regularization methods (where the number of iterations acts as a regularization parameter), they can be also employed in a hybrid fashion, i.e., to solve Tikhonov regularized problems (where both the number of iterations and and the Tikhonov parameter play the role of regularizations parameters, which can be chosen adaptively). Krylov subspace methods can naturally handle unconstrained penalized least squares problems. The goal of this talk is to 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).