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Register a new userUnscented Kalman Inversion: Efficient Gaussian Approximation to the Posterior Distribution
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Added by
Daniel Z. Huang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 approximate the mean and covariance of the posterior distribution for well-posed inverse problems with large observation data. Theoretical guarantees for both linear and nonlinear inverse problems are presented. Numerical results, including learning of permeability parameters in subsurface flow and of the Navier-Stokes initial condition from solution data at positive times are presented. The results obtained by the UKI require only $O(10)$ iterations, and match well with the expected results obtained by the Markov Chain Monte Carlo method.
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A useful approach to solve inverse problems is to pair the parameter-to-data map with a stochastic dynamical system for the parameter, and then employ techniques from filtering to estimate the parameter given the data. Three classical approaches to filtering of nonlinear systems are the extended, ensemble and unscented Kalman filters. The extended Kalman inversion (ExKI) is impractical when the forward map is not readily differentiable and given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Ensemble Kalman inversion (EKI) has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we demonstrate that unscented Kalman methods also provide an effective tool for derivative-free inversion in the setting of black-box forward models, introducing unscented Kalman inversion (UKI). Theoretical analysis is provided for linear inverse problems, and a smoothing property of the data mis-fit under the unscented transform is explained. We provide numerical experiments, including various applications: learning subsurface flow permeability parameters; learning the structure damage field; learning the Navier-Stokes initial condition; and learning subgrid-scale parameters in a general circulation model. The theory and experiments show that the UKI outperforms the EKI on parameter learning problems with moderate numbers of parameters and outperforms the ExKI on problems where the forward model is not readily differentiable, or where the derivative is very sensitive. In particular, UKI based methods are of particular value for parameter estimation problems in which the number of parameters is moderate but the forward model is expensive and provided as a black box which is impractical to differentiate.
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 advantages of the low-rank covariance structure to reduce the number of forward problem evaluations and the memory cost, related to the need to propagate large covariance matrices. And we leverage reduced-order model techniques to further speed up these forward evaluations. The effectiveness of the enhanced UKI is demonstrated on a barotropic model inverse problem with O($10^5$) unknown parameters and a 3D generalized circulation model (GCM) inverse problem, where each iteration is as efficient as that of gradient-based optimization methods.
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 weight
In this work a general approach to compute a compressed representation of the exponential $exp(h)$ of a high-dimensional function $h$ is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.
In this work we marry multi-index Monte Carlo with ensemble Kalman filtering (EnKF) to produce the multi-index EnKF method (MIEnKF). The MIEnKF method is based on independent samples of four-coupled EnKF estimators on a multi-index hierarchy of resolution levels, and it may be viewed as an extension of the multilevel EnKF (MLEnKF) method developed by the same authors in 2020. Multi-index here refers to a two-index method, consisting of a hierarchy of EnKF estimators that are coupled in two degrees of freedom: time discretization and ensemble size. Under certain assumptions, the MIEnKF method is proven to be more tractable than EnKF and MLEnKF, and this is also verified in numerical examples.
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