A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

Abstract

Abstract. We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The general aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference to develop a derivative-free stable method easy to implement in applications where the PDE (forward) model is only accessible as a black box (e.g. with commercial software). The proposed regularizing ensemble Kalman method can be derived as an approximation of the regularizing Levenberg-Marquardt (LM) scheme [14] in which the derivative of the forward operator and its adjoint are replaced with empirical covariances from an ensemble of elements from the admissible space of solutions. The resulting ensemble method consists of an update formula that is applied to each ensemble member and that has a regularization parameter selected in a similar fashion to the one in the LM scheme. Moreover, an early termination of the scheme is proposed according to a discrepancy principle-type of criterion. The proposed method can be also viewed as a regularizing version of standard Kalman approaches which are often unstable unless ad-hoc fixes, such as covariance localization, are implemented.The aim of this paper is to provide a detailed numerical investigation of the regularizing and convergence properties of the proposed regularizing ensemble Kalman scheme; the proof of these properties is an open problem. By means of numerical experiments, we investigate the conditions under which the proposed method inherits the regularizing properties of the LM scheme of [14] and is thus stable and suitable for its application in problems where the computation of the Fréchet derivative is not computationally feasible. More concretely, we study the effect of ensemble size, number of measurements, selection of initial ensemble and tunable parameters on the performance of the method. The numerical investigation is carried out with synthetic experiments on two model inverse problems: (i) identification of conductivity on a Darcy flow model and (ii) electrical impedance tomography with the complete electrode model. We further demonstrate the potential application of the method in solving shape identification problems that arises from the aforementioned forward models by means of a level-set approach for the parameterization of unknown geometries.

Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High‐Resolution Simulations

“…Nonetheless, the cost remains orders of magnitude higher than for optimization techniques. Ensemble Kalman methods are easily parallelizable, derivative‐free alternatives to the classical optimization and Bayesian approaches (Houtekamer & Zhang, ). Although theory for them is less well developed, empirical evidence demonstrates behavior similar to derivative‐based algorithms in complex inversion problems, with a comparable number of forward model integrations (Iglesias, ). Ensemble methods for joint state and parameter estimation have recently been systematically developed (Bocquet & Sakov, , ; Carrassi et al, ), and they are emerging as a promising way to solve inverse problems and to obtain qualitative estimates of uncertainty.…”

On the incorporation of box-constraints for ensemble Kalman inversion

“…It was first proposed by Iglesias et al [19,27], offering a cheaper approximation of the solution compared to traditional methods. Since its formulation a number of research directions have been considered such as applications, building theory and applying uncertainty quantification techniques [9,10,12,18,32]. However, the basic version of the EKI does not allow to incorporate additional constraints on the parameters, which often arise in many applications due to additional knowledge on the system.…”

Filter based methods for statistical linear inverse problems

“…In this basic form of the algorithm regularization is present due to dynamical preservation of a subspace spanned by the ensemble during the iteration. The paper [10] gives further insight into the development of regularization for these ensemble Kalman inversion methods, drawing on links with the Levenberg-Marquardt scheme [7]. In this paper our aim is to further the study of filters for the solution of inverse problems, going beyond the ensemble Kalman filter to encompass the study of other filters such as 3DVAR and the Kalman filter itself -see [13] for an overview of these filtering methods.…”