Ensemble Kalman inversion: mean-field limit and convergence analysis

Ding, Zhiyan; Li, Qin

doi:10.1007/s11222-020-09976-0

Cited by 38 publications

(43 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Rigorous convergence analysis of EKI addressing questions (i) and (ii) is known to be difficult. Existing work typically assumes that G is linear [32,5,17], and/or assumes the mean field limit (infinite ensemble size, J → ∞) [10]. For example, [12] addresses nonlinear problems, but assumes that the ensemble size is large (J ≥ d u ).…”

Section: 3mentioning

confidence: 99%

“…There are many recent works that carefully describe the mathematics of EKI [5,10], that suggest improvements [11,12,13,14], and that explain how EKI can be used in machine learning [15]. An extension of EKI, such that the EKI ensemble is distributed according to a Bayesian posterior distribution, is discussed in [16] and is called the ensemble Kalman sampler.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Localization in Ensemble Kalman inversion

Tong¹,

Morzfeld²

2022

Preprint

View full text Add to dashboard Cite

Ensemble Kalman inversion (EKI) is a technique for the numerical solution of inverse problems. A great advantage of the EKI's ensemble approach is that derivatives are not required in its implementation. But theoretically speaking, EKI's ensemble size needs to surpass the dimension of the problem. This is because of EKI's "subspace property", i.e., that the EKI solution is a linear combination of the initial ensemble it starts off with. We show that the ensemble can break out of this initial subspace when "localization" is applied. In essence, localization enforces an assumed correlation structure onto the problem, and is heavily used in ensemble Kalman filtering and data assimilation. We describe and analyze how to apply localization to the EKI, and how localization helps the EKI ensemble break out of the initial subspace. Specifically, we show that the localized EKI (LEKI) ensemble will collapse to a single point (as intended) and that the LEKI ensemble mean will converge to the global optimum at a sublinear rate. Under strict assumptions on the localization procedure and observation process, we further show that the data misfit decays uniformly. We illustrate our ideas and theoretical developments with numerical examples with simplified toy problems, a Lorenz model, and an inversion of electromagnetic data, where some of our mathematical assumptions may only be approximately valid.

show abstract

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Localization in Ensemble Kalman inversion

Tong¹,

Morzfeld²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…We compare the MCGAN method with the ensemble Kalman inversion method [47]. We do not compare with PCE since it is infeasible to compute a PCE model for a problem of this high dimensionality.…”

Section: Darcy Flowmentioning

confidence: 99%

“…For the advection-diffusion equation and the pipe flow equations the standard EnKF is utilized while for the Darcy flow the EKI is used. The EnKF implementation is based on [2] and the EKI implementation is based on [47].…”

Section: Appendix C1 Ensemble Kalman Filtermentioning

confidence: 99%

Markov Chain Generative Adversarial Neural Networks for Solving Bayesian Inverse Problems in Physics Applications

Mücke¹,

Sanderse²,

Bohté³

et al. 2021

Preprint

View full text Add to dashboard Cite

In the context of solving inverse problems for physics applications within a Bayesian framework, we present a new approach, Markov Chain Generative Adversarial Neural Networks (MCGANs), to alleviate the computational costs associated with solving the Bayesian inference problem. GANs pose a very suitable framework to aid in the solution of Bayesian inference problems, as they are designed to generate samples from complicated high-dimensional distributions. By training a GAN to sample from a low-dimensional latent space and then embedding it in a Markov Chain Monte Carlo method, we can highly efficiently sample from the posterior, by replacing both the high-dimensional prior and the expensive forward map. We prove that the proposed methodology converges to the true posterior in the Wasserstein-1 distance and that sampling from the latent space is equivalent to sampling in the high-dimensional space in a weak sense. The method is showcased on three test cases where we perform both state and parameter estimation simultaneously. The approach is shown to be up to two orders of magnitude more accurate than alternative approaches while also being up to an order of magnitude computationally faster, in several test cases, including the important engineering setting of detecting leaks in pipelines.

show abstract

“…Earlier works are found in [34,16,29]. See also the numerical analysis and other follow up works in [7,8,22,41].…”

mentioning

confidence: 92%

Constrained Ensemble Langevin Monte Carlo

Ding¹,

Li²

2022

FoDS

Self Cite

View full text Add to dashboard Cite

<p style='text-indent:20px;'>The classical Langevin Monte Carlo method looks for samples from a target distribution by descending the samples along the gradient of the target distribution. The method enjoys a fast convergence rate. However, the numerical cost is sometimes high because each iteration requires the computation of a gradient. One approach to eliminate the gradient computation is to employ the concept of "ensemble." A large number of particles are evolved together so the neighboring particles provide gradient information to each other. In this article, we discuss two algorithms that integrate the ensemble feature into LMC, and the associated properties.</p><p style='text-indent:20px;'>In particular, we find that if one directly surrogates the gradient using the ensemble approximation, the algorithm, termed Ensemble Langevin Monte Carlo, is unstable due to a high variance term. If the gradients are replaced by the ensemble approximations only in a constrained manner, to protect from the unstable points, the algorithm, termed Constrained Ensemble Langevin Monte Carlo, resembles the classical LMC up to an ensemble error but removes most of the gradient computation.</p>

show abstract

Ensemble Kalman inversion: mean-field limit and convergence analysis

Cited by 38 publications

References 28 publications

Localization in Ensemble Kalman inversion

Localization in Ensemble Kalman inversion

Markov Chain Generative Adversarial Neural Networks for Solving Bayesian Inverse Problems in Physics Applications

Constrained Ensemble Langevin Monte Carlo

Contact Info

Product

Resources

About