A Strongly Convergent Numerical Scheme from Ensemble Kalman Inversion

Abstract

The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion.The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar stochastic differential equation. We focus here on a toy model with similar properties than the one arising in the context of Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with non-globally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler-Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming.We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cut-off or localisation, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence. 1 stay near the origin and have p-moments at least up to p < 3. Then, any moment of the difference between u andũ n will explode for h → 0.There has been significant progress in the field of strongly convergent numerical schemes for SDEs with non-global Lipschitz-continuous nonlinearities. Standard references for numerical methods for SDEs, like [17] and [25,22], show strong convergence of the Euler-Maruyama method only for globally Lipschitz-continuous drift and diffusion terms. Higham, Mao and Stuart [10] proved a conditional result about strong convergence of the Euler-Maruyama discretization for non-globally Lipschitz SDEs given that moments of both solutions and the discretization stay bounded. This means that the question of strong convergence was replaced by the question whether moments of the numerical scheme stay bounded, but Hutzenthaler, Jentzen and Kloeden [12] answered the latter to the negative, even proving that moments of the Euler-Maruyama scheme always explode in finite time if either drift or diffusion term are not globally Lipschitz. Instead, they proposed a slight modification of the EM method, the so-called "tamed" EM method (and implicit equivalents) in [13,11].The numerical scheme (2) arising in the EnKF analysis bears resemblance to the "tamed" methods used throughout the literature. In case only the drift is non-globally Lipschitz a similar idea to (2) was used already in [13], but there the drift-tamed nonlinearity is strictly bounded by one, while in EnKF it is still allowed to grow linearly. In increment-ta...

Parameterizations for ensemble Kalman inversion

“…There is very little analysis of EKI, especially in the fixed, small, ensemble size setting where it is most powerful. Existing work in this direction may be found in [7,48]; it would be of interest to extend these analyses to the parameterizations introduced here. Furthermore, from a practical perspective, it would be interesting to extend the deployment of the methods introduced here to the study of further applications.…”

Kinetic methods for inverse problems

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