# Ensemble Kalman filter for multiscale inverse problems

## TL;DR

This paper introduces a new ensemble Kalman filter-based algorithm for efficiently solving inverse problems involving multiscale elliptic PDEs, leveraging homogenization and finite element methods.

## Contribution

It presents a novel multiscale inverse problem solver that combines numerical homogenization with ensemble Kalman filtering, including convergence analysis and Bayesian interpretation.

## Key findings

- The method accurately recovers oscillatory tensors from measurements.
- Convergence of the approximate solution and posterior distribution is established.
- Numerical experiments demonstrate computational efficiency and robustness.

## Abstract

We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analysed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is provided, and convergence of the approximate conditional posterior distribution is proved with respect to the Wasserstein distance. A numerical experiment validates our methodology, with a particular emphasis on modelling error and computational cost.