Multiscale Modelling and Inverse Problems

TL;DR

This paper explores multiscale inverse problems, focusing on regularization techniques and how homogenization theory can improve parameter estimation across different scales, exemplified by groundwater flow models.

Contribution

It introduces approaches for regularization and homogenization in multiscale inverse problems, highlighting their roles in improving parameter estimation accuracy.

Findings

Regularization choices depend on whether homogenized or full multiscale solutions are needed.

Designing observations can recover homogenized solutions from multiscale data.

Homogenization theory enhances estimation of multiscale parameters.

Abstract

The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters appearing in the model, such as constitutive tensors, initial conditions, boundary conditions, and forcing can be estimated on the basis of observed data. The resulting inverse problems are often ill-posed and some form of regularization is required. These notes discuss parameter estimation in situations where the unknown parameters vary across multiple scales. We illustrate the main ideas using a simple model for groundwater flow. We will highlight various approaches to regularization for inverse problems, including Tikhonov and Bayesian methods. We illustrate three ideas that arise when considering inverse problems in the multiscale context. The first idea is that the choice of space or set in which to seek the solution to the inverse problem is intimately related to whether a homogenized or full multiscale solution is required. This is a choice of regularization. The second idea is that, if a homogenized solution to the inverse problem is what is desired, then this can be recovered from carefully designed observations of the full multiscale system. The third idea is that the theory of homogenization can be used to improve the estimation of homogenized coefficients from multiscale data.