Sampled Tikhonov regularization for large linear inverse problems

Abstract

In this paper we investigate iterative methods that are based on sampling of the data for computing Tikhonov-regularized solutions. We focus on very large inverse problems where access to the entire data set is not possible all-at-once (e.g. for problems with streaming or massive datasets). Row-access methods provide an ideal framework for solving such problems since they only require access to ‘blocks’ of the data at any given time. However, when using these iterative sampling methods to solve inverse problems, the main challenges include a proper choice of the regularization parameter, appropriate sampling strategies, and a convergence analysis. To address these challenges, we describe a family of sampled iterative methods that can incorporate data as they become available (e.g. randomly sampled). We consider two sampled iterative methods where the iterates can be characterized as solutions to a sequence of approximate Tikhonov problems. The first method requires the regularization parameter to be fixed a priori and converges asymptotically to an unregularized solution for randomly sampled data. This is undesirable for inverse problems. Thus, we focus on the second method where the main benefits are that the regularization parameter can be updated during the iterative process and the iterates converge asymptotically to a Tikhonov-regularized solution. We describe adaptive approaches to update the regularization parameter that are based on sampled residuals, and we provide a limited-memory variant for larger problems. Numerical examples, including a large-scale super-resolution imaging example, demonstrate the potential for these methods.