Estimating and using information in inverse problems

TL;DR

This paper defines a quantitative measure of information in inverse problems based on Bayesian variance, and demonstrates its practical utility in algorithm design, such as mesh discretization, through numerical experiments.

Contribution

It introduces a clear definition of information density in inverse problems and explores its application in improving algorithmic choices like mesh discretization.

Findings

Defined information density based on Bayesian variance.

Showed how information density guides mesh selection.

Validated the approach with numerical experiments.

Abstract

In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts of the function can be inferred accurately and which cannot. For example, it is generally understood that we can identify system parameters accurately only close to detectors, or along ray paths between sources and detectors, because we have "the most information" for these places. Although referenced in many publications, the "information" that is invoked in such contexts is not a well understood and clearly defined quantity. Herein, we present a definition of information density that is based on the variance of coefficients as derived from a Bayesian reformulation of the inverse problem. We then discuss three areas in which this information density can be useful in practical algorithms for the solution of inverse problems, and illustrate the usefulness in one of these areas -- how to choose the discretization mesh for the function to be reconstructed -- using numerical experiments.