Randomization and reweighted $\ell_1$-minimization for A-optimal design of linear inverse problems

Abstract: We consider optimal design of PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We focus on the A-optimal design criterion, defined as the average posterior variance and quantified by the trace of the posterior covariance operator. We propose using structure exploiting randomized methods to compute the A-optimal objective function and its gradient, and provide a detailed analysis of the error for the proposed estimators. To ensure sparse and binary design vectors, we develop a novel reweighted $\ell_1$-minimization algorithm. We also introduce a modified A-optimal criterion and present randomized estimators for its efficient computation. We present numerical results illustrating the proposed methods on a model contaminant source identification problem, where the inverse problem seeks to recover the initial state of a contaminant plume, using discrete measurements of the contaminant in space and time.