Tikhonov Regularization within Ensemble Kalman Inversion

Chada, Neil K.; Stuart, Andrew M.; Tong, Xin

doi:10.1137/19m1242331

Cited by 67 publications

(133 citation statements)

References 37 publications

(58 reference statements)

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“…This dynamical system is similar to the noisy EKI (2.7) but has a different noise structure (noise in parameter space not data space) and explicitly accounts for the prior on the right hand side (rather than having it enter through initialization). Inclusion of the Tikhonov regularization term within EKI is introduced and studied in (Chada, Stuart and Tong, 2019). Note that in the linear case (2.9) the two systems (2.5) and (2.13) are identical.…”

Section: The Ensemble Kalman Samplermentioning

confidence: 99%

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Garbuno-Iñigo¹,

Hoffmann²,

Li³

et al. 2020

SIAM J. Appl. Dyn. Syst.

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130

253

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Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of over-damped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker-Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker-Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.

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Section: The Ensemble Kalman Samplermentioning

confidence: 99%

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Garbuno-Iñigo¹,

Hoffmann²,

Li³

et al. 2020

SIAM J. Appl. Dyn. Syst.

Self Cite

130

253

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“…An appropriate modification of EKI, to attack the problem of sampling from the posterior π y given by (2.3), is EKS [22]. Formally this is obtained by adding a prior-related damping term, as in [9], and a Θ-dependent noise to obtain…”

Section: Calibrate -Eki and Eksmentioning

confidence: 99%

Calibrate, emulate, sample

Cleary

Garbuno-Iñigo

Lan

et al. 2021

Journal of Computational Physics

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173

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Many parameter estimation problems arising in applications are best cast in the framework of Bayesian inversion. This allows not only for an estimate of the parameters, but also for the quantification of uncertainties in the estimates. Often in such problems the parameterto-data map is very expensive to evaluate, and computing derivatives of the map, or derivative-adjoints, may not be feasible. Additionally, in many applications only noisy evaluations of the map may be available. We propose an approach to Bayesian inversion in such settings that builds on the derivative-free optimization capabilities of ensemble Kalman inversion methods. The overarching approach is to first use ensemble Kalman sampling (EKS) to calibrate the unknown parameters to fit the data; second, to use the output of the EKS to emulate the parameter-to-data map; third, to sample from an approximate Bayesian posterior distribution in which the parameter-to-data map is replaced by its emulator. This results in a principled approach to approximate Bayesian inference that requires only a small number of evaluations of the (possibly noisy approximation of the) parameter-to-data map. It does not require derivatives of this map, but instead leverages the documented power of ensemble Kalman methods. Furthermore, the EKS has the desirable property that it evolves the parameter ensembles towards the regions in which the bulk of the parameter posterior mass is located, thereby locating them well for the emulation phase of the methodology. In essence, the EKS methodology provides a cheap solution to the design problem of where to place points in parameter space to efficiently train an emulator of the parameter-to-data map for the purposes of Bayesian inversion.

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“…As discussed in Section 2, the EnKF computes a solution to the inverse problem as mean of the ensembles in the large time behavior. Since the kinetic equation (13) formally holds in the limit of a large number of ensembles, here we analyze approximations to the solution of the inverse problem provided by the first moment m(t) of the kinetic distribution, see (10).…”

Section: Moment Equations and Linear Stability Analysismentioning

confidence: 99%

Kinetic methods for inverse problems

Herty¹,

Visconti²

2019

Kinetic &Amp; Related Models

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The Ensemble Kalman Filter method can be used as an iterative numerical scheme for parameter identification or nonlinear filtering problems. We study the limit of infinitely large ensemble size and derive the corresponding mean-field limit of the ensemble method. The solution of the inverse problem is provided by the expected value of the distribution of the ensembles and the kinetic equation allows, in simple cases, to analyze stability of these solutions. Further, we present a slight but stable modification of the method which leads to a Fokker-Planck-type kinetic equation. The kinetic methods proposed here are able to solve the problem with a reduced computational complexity in the limit of a large ensemble size. We illustrate the properties and the ability of the kinetic model to provide solution to inverse problems by using examples from the literature.Mathematics Subject Classification (2010) 35Q84, 65N21, 93E11, 65N75

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Tikhonov Regularization within Ensemble Kalman Inversion

Cited by 67 publications

References 37 publications

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Calibrate, emulate, sample

Kinetic methods for inverse problems

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