Domain decomposition methods for linear inverse problems with sparsity constraints

Abstract

Quantities of interest appearing in concrete applications often possess sparse expansions with respect to a preassigned frame. Recently, there were introduced sparsity measures which are typically constructed on the basis of weighted ℓ1 norms of frame coefficients. One can model the reconstruction of a sparse vector from noisy linear measurements as the minimization of the functional defined by the sum of the discrepancy with respect to the data and the weighted ℓ1-norm of suitable frame coefficients. Thresholded Landweber iterations were proposed for the solution of the variational problem. Despite its simplicity which makes it very attractive to users, this algorithm converges slowly. In this paper, we investigate methods to accelerate significantly the convergence. We introduce and analyze sequential and parallel iterative algorithms based on alternating subspace corrections for the solution of the linear inverse problem with sparsity constraints. We prove their norm convergence to minimizers of the functional. We compare the computational cost and the behavior of these new algorithms with respect to the thresholded Landweber iterations.