Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion
Type
ArticleKAUST Department
Computer, Electrical and Mathematical Science and Engineering (CEMSE) DivisionApplied Mathematics and Computational Science Program
Date
2022-03-10Embargo End Date
2023-03-10Permanent link to this record
http://hdl.handle.net/10754/672919
Metadata
Show full item recordAbstract
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-posedness and convergence of various loss functions, but with the addition of noisy observations for the limiting stochastic differential equations (i.e. stochastic TEKI). 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 maximum a posteriori 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.Citation
Weissmann, S., Chada, N. K., Schillings, C., & Tong, X. T. (2022). Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion. Inverse Problems, 38(4), 045009. https://doi.org/10.1088/1361-6420/ac5729Sponsors
CS and SW are grateful to the DFG RTG1953 ‘Statistical Modeling of Complex Systems and Processes’ for funding of this research. NKC is supported by KAUST baseline funding. XTT is supported by the National University of Singapore grant R-146-000-292-114. The authors acknowledge support by the state of Baden-Württemberg through bwHPC.Publisher
IOP PublishingJournal
Inverse ProblemsarXiv
2110.09142Additional Links
https://iopscience.iop.org/article/10.1088/1361-6420/ac5729ae974a485f413a2113503eed53cd6c53
10.1088/1361-6420/ac5729