Robust optimal design of large-scale Bayesian nonlinear inverse problems

TL;DR

This paper introduces a scalable, robust optimal experimental design framework for nonlinear Bayesian inverse problems governed by PDEs, accounting for uncertainties and ensuring optimality under variations.

Contribution

It develops a new worst-case scenario framework that is scalable, efficient, and suitable for infinite-dimensional PDE-constrained inverse problems, with analytical gradient evaluation.

Findings

Effective sensor placement in elliptic PDE inverse problems.

Framework remains robust under model, prior, and error uncertainties.

Efficient approximation of expected information gain achieved.

Abstract

We consider robust optimal experimental design (ROED) for nonlinear Bayesian inverse problems governed by partial differential equations (PDEs). An optimal design is one that maximizes some utility quantifying the quality of the solution of an inverse problem. However, the optimal design is dependent on elements of the inverse problem such as the simulation model, the prior, or the measurement error model. ROED aims to produce an optimal design that is aware of the additional uncertainties encoded in the inverse problem and remains optimal even after variations in them. We follow a worst-case scenario approach to develop a new framework for robust optimal design of nonlinear Bayesian inverse problems. The proposed framework a) is scalable and designed for infinite-dimensional Bayesian nonlinear inverse problems constrained by PDEs; b) develops efficient approximations of the utility, namely, the expected information gain; c) employs eigenvalue sensitivity techniques to develop analytical forms and efficient evaluation methods of the gradient of the utility with respect to the uncertainties we wish to be robust against; and d) employs a probabilistic optimization paradigm that properly defines and efficiently solves the resulting combinatorial max-min optimization problem. The effectiveness of the proposed approach is illustrated for optimal sensor placement problem in an inverse problem governed by an elliptic PDE.