Iterated Kalman Methodology For Inverse Problems

TL;DR

This paper introduces a new stochastic dynamical system for inverse problems, demonstrating exponential convergence with regularization and showing that the unscented Kalman inversion outperforms ensemble Kalman inversion in this framework.

Contribution

It presents a novel stochastic dynamical system embedding the parameter-to-data map and compares UKI and EKI, showing improved inversion with UKI.

Findings

Exponential convergence of the filtering mean to regularized least squares solution.

UKI yields better inversion results than EKI on the new dynamical system.

The approach handles black-box forward maps and high-dimensional parameters effectively.

Abstract

This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Application of ensemble Kalman filters, for example use of the ensemble Kalman inversion (EKI) algorithm, has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI). The paper contains two main contributions. Firstly, we identify a novel stochastic dynamical system in which the parameter-to-data map is embedded. We present theory in the linear case to show exponential convergence of the mean of the filtering distribution to the solution of a regularized least squares problem. This is in contrast to previous work in which the EKI has been employed where the dynamical system used leads to algebraic convergence to an unregularized problem. Secondly, we show that the application of the UKI to this novel stochastic dynamical system yields improved inversion results, in comparison with the application of EKI to the same novel stochastic dynamical system.