Inverse Problems: A Deterministic Approach using Physics-Based Reduced Models

TL;DR

This paper discusses a deterministic approach to inverse problems using physics-based reduced models, emphasizing optimal performance, practical algorithms, and sensor placement, as alternatives to Bayesian methods.

Contribution

It introduces a deterministic framework for inverse problems with provably near-optimal algorithms and discusses sensor placement strategies, offering an alternative to Bayesian inversion.

Findings

Algorithms achieve performance close to the theoretical optimal

Reduced models enable efficient and accurate inverse problem solutions

Sensor placement strategies improve measurement efficiency

Abstract

These lecture notes summarize various summer schools that I have given on the topic of solving inverse problems (state and parameter estimation) by combining optimally measurement observations and parametrized PDE models. After defining a notion of optimal performance in terms of the smallest reconstruction error that any reconstruction algorithm can achieve, the notes present practical numerical algorithms based on nonlinear reduced models for which one can prove that they can deliver a performance close to optimal. We also discuss algorithms for sensor placement with the approach. The proposed concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic notions of accuracy quantification.