Batch-based Bayesian Optimal Experimental Design in Linear Inverse Problems

Experimental design is central to science and engineering. A ubiquitous challenge is how to maximize the value of information obtained from expensive or constrained experimental settings. Bayesian optimal experimental design (OED) provides a principled framework for addressing such questions. In this paper, we study experimental design problems such as the optimization of sensor locations over a continuous domain in the context of linear Bayesian inverse problems. We focus in particular on batch design, that is, the simultaneous optimization of multiple design variables, which leads to a notoriously difficult non-convex optimization problem. We tackle this challenge using a promising strategy recently proposed in the frequentist setting, which relaxes A-optimal design to the space of finite positive measures. Our main contribution is the rigorous identification of the Bayesian inference problem corresponding to this relaxed A-optimal OED formulation. Moreover, building on recent work, we develop a Wasserstein gradient-flow -based optimization algorithm for the expected utility and introduce novel regularization schemes that guarantee convergence to an empirical measure. These theoretical results are supported by numerical experiments demonstrating both convergence and the effectiveness of the proposed regularization strategy.

💡 Research Summary

This paper addresses the challenging problem of batch experimental design for linear Bayesian inverse problems, where multiple sensor locations (or other design variables) must be chosen simultaneously. In the classical A‑optimal design framework, the goal is to minimize the trace of the posterior covariance (equivalently, the average mean‑square error) after observing data generated by a linear forward operator A with additive Gaussian noise. When the number of sensors B grows, the design problem becomes highly non‑convex and standard gradient‑based methods often get trapped in local minima.

To overcome this difficulty, the authors adopt a recent frequentist strategy that relaxes the design variables from a set of Dirac measures δₓ to the space of finite positive measures μ on the design domain X with total mass B. The relaxed objective is
 U(μ) = Tr