Multi-type Sensor Placement for PDE-based Bayesian Inverse Problems

We address optimal placement of multi-type sensors for Bayesian inverse problems governed by partial differential equations (PDEs).

The proposed framework allows for sensors with different accuracies and observation types.

We formulate the optimal experimental design (OED) problem as a knapsack-constrained binary optimization problem for maximizing expected information gain (EIG).

To approximately solve the resulting optimization problems, we propose a stochastic cost-benefit greedy algorithm, which admits theoretical guarantees for monotone submodular set functions.

Specifically, these guarantees apply in the case of linear Gaussian inverse problems with uncorrelated measurement errors, where the EIG admits a convenient closed-form expression.

For nonlinear inverse problems, we develop a non-intrusive approach that uses the Bayesian approximation error framework to define an observation model with an error-corrected global linear model.

We show that the corresponding approximate EIG is a lower bound for the exact EIG and thus provides a principled surrogate objective for the OED problem.

The effectiveness of the proposed methods is demonstrated in two model inverse problems governed by PDEs.