# Randomization and reweighted $\ell_1$-minimization for A-optimal design of linear inverse problems

## TL;DR

This paper introduces randomized methods and reweighted $ ext{l}_1$-minimization techniques to efficiently design optimal experiments for Bayesian linear inverse problems governed by PDEs, focusing on A-optimality and sparse solutions.

## Contribution

It develops structure-exploiting randomized estimators for the A-optimal design criterion and proposes a novel reweighted $ ext{l}_1$-minimization algorithm for sparse, binary design vectors.

## Key findings

- Efficient computation of A-optimal design using randomized estimators.
- Successful application to a contaminant source identification problem.
- Demonstrated sparsity and optimality of the resulting experimental designs.

## Abstract

We consider optimal design of PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We focus on the A-optimal design criterion, defined as the average posterior variance and quantified by the trace of the posterior covariance operator. We propose using structure exploiting randomized methods to compute the A-optimal objective function and its gradient, and provide a detailed analysis of the error for the proposed estimators. To ensure sparse and binary design vectors, we develop a novel reweighted $\ell_1$-minimization algorithm. We also introduce a modified A-optimal criterion and present randomized estimators for its efficient computation. We present numerical results illustrating the proposed methods on a model contaminant source identification problem, where the inverse problem seeks to recover the initial state of a contaminant plume, using discrete measurements of the contaminant in space and time.