Repository: Freie Universit├Ąt Berlin, Math Department

Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion

Weissmann, Simon and Chada, Neil and Schillings, Claudia and Tong, Xin (2022) Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion. . . (Submitted)

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Official URL: https://doi.org/10.48550/arXiv.2110.09142

Abstract

Ensemble Kalman inversion (EKI) is a derivative-free optimizer aimed at solving inverse problems, taking motivation from the celebrated ensemble Kalman filter. The purpose of this article is to consider the introduction of adaptive Tikhonov strategies for EKI. This work builds upon Tikhonov EKI (TEKI) which was proposed for a fixed regularization constant. By adaptively learning the regularization parameter, this procedure is known to improve the recovery of the underlying unknown. For the analysis, we consider a continuous-time setting where we extend known results such as well-posdeness and convergence of various loss functions, but with the addition of noisy observations. Furthermore, we allow a time-varying noise and regularization covariance in our presented convergence result which mimic adaptive regularization schemes. In turn we present three adaptive regularization schemes, which are highlighted from both the deterministic and Bayesian approaches for inverse problems, which include bilevel optimization, the MAP formulation and covariance learning. We numerically test these schemes and the theory on linear and nonlinear partial differential equations, where they outperform the non-adaptive TEKI and EKI.

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > Deterministic and Stochastic PDEs Group
ID Code:2828
Deposited By: Ulrike Eickers
Deposited On:04 May 2022 05:45
Last Modified:04 May 2022 05:45

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