Wacker PK, Blömker D, Schillings C (2018)
Publication Status: Published
Publication Type: Journal article
Publication year: 2018
Publisher: SIAM PUBLICATIONS
Book Volume: 56
Pages Range: 2537-2562
Journal Issue: 4
DOI: 10.1137/17M1132367
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 SDE. We focus here on a toy model with similar properties to the one arising in the context of an 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 nonglobally 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 cutoff or localization, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence.
APA:
Wacker, P.K., Blömker, D., & Schillings, C. (2018). A STRONGLY CONVERGENT NUMERICAL SCHEME FROM ENSEMBLE KALMAN INVERSION. SIAM Journal on Numerical Analysis, 56(4), 2537-2562. https://dx.doi.org/10.1137/17M1132367
MLA:
Wacker, Philipp Konstantin, Dirk Blömker, and Claudia Schillings. "A STRONGLY CONVERGENT NUMERICAL SCHEME FROM ENSEMBLE KALMAN INVERSION." SIAM Journal on Numerical Analysis 56.4 (2018): 2537-2562.
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