Submodularity of the expected information gain in infinite-dimensional linear inverse problems

Abstract: We consider infinite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor data, and focus on the problem of finding sensor placements that maximize the expected information gain (EIG). This study is motivated by optimal sensor placement for linear inverse problems constrained by partial differential equations (PDEs). We consider measurement models where each sensor collects a single-snapshot measurement. This covers sensor placement for inverse problems governed by linear steady PDEs or evolution equations with final-in-time observations. It is well-known that in the finite-dimensional (discretized) formulations of such inverse problems, EIG is a monotone submodular function. This also entails a theoretical guarantee for greedy sensor placement in the discretized setting. We extend the result on submodularity of the EIG to the infinite-dimensional setting, proving that the approximation guarantee of greedy sensor placement remains valid in the infinite-dimensional limit. We also discuss computational considerations and present strategies that exploit problem structure and submodularity to yield an efficient implementation of the greedy procedure.