A-optimal encoding weights for nonlinear inverse problems, with applications to the Helmholtz inverse problem

TL;DR

This paper introduces A-optimal encoding weights for nonlinear inverse PDE problems, reducing computational costs and uncertainty, with a focus on Helmholtz problems and potential for real-time applications.

Contribution

It develops a Bayesian framework for computing A-optimal encoding weights applicable to nonlinear inverse problems, including Helmholtz, with an emphasis on infinite-dimensional formulation and efficient discretization.

Findings

Framework achieves cost-independent discretization complexity

Derivation of adjoint-based gradient expressions for optimization

Potential for real-time inverse problem monitoring

Abstract

The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint-based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost.