On dimension reduction in Gaussian filters
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Marzouk_On dimension.pdf
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Author(s)
Hakkarainen, Janne
Solonen, Antti
Cui, Tiangang
Marzouk, Youssef M
Date Issued
March 2016
Journal
Inverse Problems
Publisher
IOP Publishing
Citation
Solonen, Antti et al. “On Dimension Reduction in Gaussian Filters.” Inverse Problems 32.4 (2016): 45003.
Version
Author's final manuscript
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–Loève 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.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1088/0266-5611/32/4/045003
