Path-OED for infinite-dimensional Bayesian linear inverse problems governed by PDEs

TL;DR

This paper develops a rigorous mathematical and computational framework for optimal sensor path design in infinite-dimensional Bayesian linear inverse problems governed by PDEs, with applications demonstrated through computational experiments.

Contribution

It introduces a novel infinite-dimensional path-OED framework with discretized methods for efficient sensor path optimization in PDE-governed inverse problems.

Findings

Framework is flexible and scalable for various inverse problems.

Efficient methods for optimal sensor path computation are developed.

Computational experiments validate the approach's effectiveness.

Abstract

We consider infinite-dimensional Bayesian linear inverse problems governed by time-dependent partial differential equations (PDEs) and develop a mathematical and computational framework for optimal design of mobile sensor paths in this setting. The proposed path optimal experimental design (path-OED) framework is established rigorously in a function space setting and elaborated for the case of Bayesian c-optimality, which quantifies the posterior variance in a linear functional of the inverse parameter. The latter is motivated by goal-oriented formulations, where we seek to minimize the uncertainty in a scalar prediction of interest. To facilitate computations, we complement the proposed infinite-dimensional framework with discretized formulations, in suitably weighted finite-dimensional inner product spaces, and derive efficient methods for finding optimal sensor paths. The resulting computational framework is flexible, scalable, and can be adapted to a broad range of linear inverse problems and design criteria. We also present extensive computational experiments, for a model inverse problem constrained by an advection-diffusion equation, to demonstrate the effectiveness of the proposed approach.