Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble Kalman Filter. Despite their popularity, they are, however, not unbiased when the forward map is nonlinear. Important Sampling (IS), on the other hand, obtains the unbiased sampling at the expense of large variance of weights, leading to slow convergence of high moments.
We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this paper. It follows the same gradient flow as that of EnKI/EnSRF with weight corrections. Compared to the classical methods, the new methods are unbiased, and compared with IS, the method has bounded weight variance. Both properties will be proved rigorously in this paper. We further discuss the stability of the underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally in nonlinear settings. Numerical evidence will be demonstrated at the end.
Ensemble Kalman inversion is a parallelizable derivative-free method to solve inverse problems. The method uses an ensemble that follows the Kalman update formula iteratively to solve an optimization problem. The ensemble size is crucial to capture the correct statistical information in estimating the unknown variable of interest. Still, the ensemble is limited to a size smaller than the unknown variable's dimension for computational efficiency. This study proposes a strategy to correct the sampling error due to a small ensemble size, which improves the performance of the ensemble Kalman inver…
We study the use of novel techniques arising in machine learning for inverse problems. Our approach replaces the complex forward model by a neural network, which is trained simultaneously in a one-shot sense when estimating the unknown parameters from data, i.e. the neural network is trained only for the unknown parameter. By establishing a link to the Bayesian approach to inverse problems, an algorithmic framework is developed which ensures the feasibility of the parameter estimate w.r. to the forward model. We propose an efficient, derivative-free optimization method based on variants of the…
The Ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to pat…
Convergence Analysis of Ensemble Kalman Inversion: The Linear, Noisy Case
Institute for Mathematics, University of Mannheim（曼海姆大学数学研究所）；Department of Computing and Mathematical Sciences, California Institute of Technology（加州理工学院计算与数学科学系）
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 …
The Bayesian approach to inverse problems is widely used in practice to infer unknown parameters from noisy observations. In this framework, the ensemble Kalman inversion has been successfully applied for the quantification of uncertainties in various areas of applications. In recent years, a complete analysis of the method has been developed for linear inverse problems adopting an optimization viewpoint. However, many applications require the incorporation of additional constraints on the parameters, e.g. arising due to physical constraints. We propose a new variant of the ensemble Kalman inv…
The unscented Kalman inversion (UKI) method presented in [1] is a general derivative-free approach for the inverse problem. UKI is particularly suitable for inverse problems where the forward model is given as a black box and may not be differentiable. The regularization strategies, convergence property, and speed-up strategies [1,2] of the UKI are thoroughly studied, and the method is capable of handling noisy observation data and solving chaotic inverse problems. In this paper, we study the uncertainty quantification capability of the UKI. We propose a modified UKI, which allows to well appr…
The interaction between the foundation structures and the soil has been developed for many engineering applications. For the determination of the stress in foundation structure it is needed to determine the influence of the stiffness of soil with respect to the displacement w of the deformable plate (direct problem), and viceversa, how the stiffness of the foundation structure affects the resulting subsidence (inverse problem). In this paper, we deal with the Winkler mathematical model and propose to use an efficient Ensemble Kalman Inversion scheme (EKI) that regularizes iteratively the ill-p…
Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in th…
Many data-science problems can be formulated as an inverse problem, where the parameters are estimated by minimizing a proper loss function. When complicated black-box models are involved, derivative-free optimization tools are often needed. The ensemble Kalman filter (EnKF) is a particle-based derivative-free Bayesian algorithm originally designed for data assimilation. Recently, it has been applied to inverse problems for computational efficiency. The resulting algorithm, known as ensemble Kalman inversion (EKI), involves running an ensemble of particles with EnKF update rules so they can co…
The Ensemble Kalman Filter (EnKF) belongs to the class of iterative particle filtering methods and can be used for solving control--to--observable inverse problems. In this context, the EnKF is known as Ensemble Kalman Inversion (EKI). In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one--dimensional linear stability analysis reveals possible drawbacks in the phase space of moments provided by the continuous limits of the EKI, but observed also in the multi--dimensional setting. I…
Mathematical modeling and simulation of complex physical systems based on partial differential equations (PDEs) have been widely used in engineering and industrial applications. To enable reliable predictions, it is crucial yet challenging to calibrate the model by inferring unknown parameters/fields (e.g., boundary conditions, mechanical properties, and operating parameters) from sparse and noisy measurements, which is known as a PDE-constrained inverse problem. In this work, we develop a novel bi-fidelity (BF) ensemble Kalman inversion method to tackle this challenge, leveraging the accuracy…
The unscented Kalman inversion (UKI) presented in [1] is a general derivative-free approach to solving the inverse problem. UKI is particularly suitable for inverse problems where the forward model is given as a black box and may not be differentiable. The regularization strategy and convergence property of the UKI are thoroughly studied, and the method is demonstrated effectively handling noisy observation data and solving chaotic inverse problems. In this paper, we aim to make the UKI more efficient in terms of computational and memory costs for large scale inverse problems. We take advantag…
The spatial dependent unknown acoustic source is reconstructed according noisy multiple frequency data on a remote closed surface. Assume that the unknown function is supported on a bounded domain. To determine the support, we present a statistical inversion algorithm, which combines the ensemble Kalman filter approach with level set technique. Several numerical examples show that the proposed method give good numerical reconstruction.
Ensemble Kalman Inversion as an Inertial Interacting Particle System
Institute for Geometry and Applied Mathematics, RWTH Aachen University（亚琛工业大学几何与应用数学研究所）；Department of Mathematics and Applied Mathematics, University of Pretoria（比勒陀利亚大学数学与应用数学系）；Department of Mathematics, Sapienza University of Rome（罗马第一大学数学系）
We propose a new regularisation strategy for the classical ensemble Kalman inversion (EKI) framework. The strategy consists of: (i) an adaptive choice for the regularisation parameter in the update formula in EKI, and (ii) criteria for the early stopping of the scheme. In contrast to existing approaches, our parameter choice does not rely on additional tuning parameters which often have severe effects on the efficiency of EKI. We motivate our approach using the interpretation of EKI as a Gaussian approximation in the Bayesian tempering setting for inverse problems. We show that our parameter c…
The ensemble Kalman inversion (EKI) is a particle based method which has been introduced as the application of the ensemble Kalman filter to inverse problems. In practice it has been widely used as derivative-free optimization method in order to estimate unknown parameters from noisy measurement data. For linear forward models the EKI can be viewed as gradient flow preconditioned by a certain sample covariance matrix. Through the preconditioning the resulting scheme remains in a finite dimensional subspace of the original high-dimensional (or even infinite dimensional) parameter space and can …
This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because o…
This paper uses a probabilistic approach to analyze the converge of an ensemble Kalman filter solution to an exact Kalman filter solution in the simplest possible setting, the scalar case, as it allows us to build upon a rich literature of scalar probability distributions and non-elementary functions. To this end we introduce the bare-bones Scalar Pedagogical Ensemble Kalman Filter (SPEnKF). We show that in the asymptotic case of ensemble size, the expected value of both the analysis mean and variance estimate of the SPEnKF converges to that of the true Kalman filter, and that the variances of…
The ensemble Kalman inversion (EKI) for the solution of Bayesian inverse problems of type $y = A u +\varepsilon$, with $u$ being an unknown parameter, $y$ a given datum, and $\varepsilon$ measurement noise, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose initial empirical measure is sampled from the prior, evolving over an artificial time $t$ towards an approximate solution of the inverse problem, with $t=1$ emulating the posterior, and $t\to\infty$ corresponding to the under-regul…
We design and analyse the performance of a multilevel ensemble Kalman filter method (MLEnKF) for filtering settings where the underlying state-space model is an infinite-dimensional spatio-temporal process. We consider underlying models that needs to be simulated by numerical methods, with discretization in both space and time. The multilevel Monte Carlo (MLMC) sampling strategy, achieving variance reduction through pairwise coupling of ensemble particles on neighboring resolutions, is used in the sample-moment step of MLEnKF to produce an efficient hierarchical filtering method for spatio-tem…
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 …
In this article we consider the development of an unbiased estimator for the ensemble Kalman--Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology which can be viewed as a continuous-time analogue of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work [Rhee \& Glynn 2010, [31]] which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization, and through the number of samples at each level. Our estimator will be specific to …
Recent researches in data assimilation lead to the introduction of the parametric Kalman filter (PKF): an implementation of the Kalman filter, where the covariance matrices are approximated by a parameterized covariance model. In the PKF, the dynamics of the covariance during the forecast step relies on the prediction of the covariance parameters. Hence, the design of the parameter dynamics is crucial while it can be tedious to do this by hand. This contribution introduces a python package, SymPKF, able to compute PKF dynamics for univariate statistics and when the covariance model is paramete…
Data assimilation provides algorithms for widespread applications in various fields. It is of practical use to deal with a large amount of information in the complex system that is hard to estimate. Weather forecasting is one of the applications, where the prediction of meteorological data are corrected given the observations. Numerous approaches are contained in data assimilation. One specific sequential method is the Kalman Filter. The core is to estimate unknown information with the new data that is measured and the prior data that is predicted. As a matter of fact, there are different impr…
RWTH Aachen University（亚琛工业大学）；Technische Universität Braunschweig（布伦瑞克工业大学）；KAUST（阿卜杜拉国王科技大学）
This paper presents the machine learning-based ensemble conditional mean filter (ML-EnCMF) -- a filtering method based on the conditional mean filter (CMF) previously introduced in the literature. The updated mean of the CMF matches that of the posterior, obtained by applying Bayes' rule on the filter's forecast distribution. Moreover, we show that the CMF's updated covariance coincides with the expected conditional covariance. Implementing the EnCMF requires computing the conditional mean (CM). A likelihood-based estimator is prone to significant errors for small ensemble sizes, causing the f…
