Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion

Weissmann, Simon; Chada, Neil K.; Schillings, Claudia; Tong, Xin

doi:10.1088/1361-6420/ac5729

Cited by 14 publications

(11 citation statements)

References 51 publications

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“…Related Bayesian ensemble Kalman methodologies might be found in [67,68,69,70]. We emphasise that multiple insertion of the same data y without the adjustment (13) of the data likelihood function, and/or over arbitrary numbers of steps, leads to the class of optimization-based Kalman inversion methods: EKI [56], Tikhonovregularized EKI, termed TEKI [71] and unscented Kalman inversion, UKI [72]; see also [73] for recent adaptive methodologies which are variants on TEKI. These variants of the Kalman filter lead to efficient derivative-free optimization approaches to approximating the maximum likelihood estimator or maximum a posteriori estimator in the asymptotic limit as n → ∞.…”

Section: Filtering Methods For Inversionmentioning

confidence: 99%

Efficient Derivative-free Bayesian Inference for Large-Scale Inverse Problems

Huang¹,

Huang²,

Reich³

et al. 2022

Preprint

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We consider Bayesian inference for large scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O(10 4 ) model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system into which the inverse problem is embedded as an observation operator. Theoretical properties of the mean-field model are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proofof-concept linear/nonlinear examples and two large-scale applications: learning of

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Section: Filtering Methods For Inversionmentioning

confidence: 99%

Efficient Derivative-free Bayesian Inference for Large-Scale Inverse Problems

Huang¹,

Huang²,

Reich³

et al. 2022

Preprint

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“…Therefore thinking adaptively allows one to evolve the parameter over the iteration count, now denoted as λ n . The work of Weissmann et al [27] provides these developments in an adaptive fashion.…”

Section: Regularizationmentioning

confidence: 99%

A Review of the EnKF for Parameter Estimation

Chada¹

2023

Inverse Problems - Recent Advances and Applications

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The ensemble Kalman filter is a well-known and celebrated data assimilation algorithm. It is of particular relevance as it used for high-dimensional problems, by updating an ensemble of particles through a sample mean and covariance matrices. In this chapter we present a relatively recent topic which is the application of the EnKF to inverse problems, known as ensemble Kalman Inversion (EKI). EKI is used for parameter estimation, which can be viewed as a black-box optimizer for PDE-constrained inverse problems. We present in this chapter a review of the discussed methodology, while presenting emerging and new areas of research, where numerical experiments are provided on numerous interesting models arising in geosciences and numerical weather prediction.

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“…We mainly consider Tikhonov regularisation which was analysed for the EKI in [12] for example. Recently there has been further analysis on Tikhonov regularisation for the stochastic EKI as well as adaptive Tikhonov strategies to improve the original variant [35]. Considering large ensemble sizes, an analysis of the mean-field limit is presented in [10,14].…”

Section: Literature Overviewmentioning

confidence: 99%

Subsampling in ensemble Kalman inversion

2023

Self Cite

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We consider the Ensemble Kalman Inversion which has been recently introduced as an efficient, gradient-free optimization method to estimate unknown parameters in an inverse setting. In the case of large data sets, the Ensemble Kalman Inversion becomes computationally infeasible as the data misfit needs to be evaluated for each particle in each iteration. Here, randomised algorithms like stochastic gradient descent have been demonstrated to successfully overcome this issue by using only a random subset of the data in each iteration, so-called subsampling techniques. Based on a recent analysis of a continuous-time representation of stochastic gradient methods, we propose, analyse, and apply subsampling-techniques within Ensemble Kalman Inversion. Indeed, we propose two different subsampling techniques: either every particle observes the same data subset (single subsampling) or every particle observes a different data subset (batch subsampling).

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Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion

Cited by 14 publications

References 51 publications

Efficient Derivative-free Bayesian Inference for Large-Scale Inverse Problems

Efficient Derivative-free Bayesian Inference for Large-Scale Inverse Problems

A Review of the EnKF for Parameter Estimation

Subsampling in ensemble Kalman inversion

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