On dimension reduction in Gaussian filters

TL;DR

This paper extends a priori dimension reduction techniques from stationary to non-stationary inverse problems, enabling significant computational savings in sequential filtering of dynamical systems by identifying low-dimensional subspaces offline.

Contribution

It introduces a novel offline-online dimension reduction framework for non-stationary inverse problems, improving efficiency of filtering algorithms like Kalman filters.

Findings

Reduces problem dimensionality by orders of magnitude.

Achieves up to two orders of magnitude in computational savings.

Demonstrates effectiveness on various numerical examples.

Abstract

A priori dimension reduction is a widely adopted technique for reducing the computational complexity of stationary inverse problems. In this setting, the solution of an inverse problem is parameterized by a low-dimensional basis that is often obtained from the truncated Karhunen-Loeve expansion of the prior distribution. For high-dimensional inverse problems equipped with smoothing priors, this technique can lead to drastic reductions in parameter dimension and significant computational savings. In this paper, we extend the concept of a priori dimension reduction to non-stationary inverse problems, in which the goal is to sequentially infer the state of a dynamical system. Our approach proceeds in an offline-online fashion. We first identify a low-dimensional subspace in the state space before solving the inverse problem (the offline phase), using either the method of "snapshots" or regularized covariance estimation. Then this subspace is used to reduce the computational complexity of various filtering algorithms - including the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within a novel subspace-constrained Bayesian prediction-and-update procedure (the online phase). We demonstrate the performance of our new dimension reduction approach on various numerical examples. In some test cases, our approach reduces the dimensionality of the original problem by orders of magnitude and yields up to two orders of magnitude in computational savings.