A strongly convergent numerical scheme from EnKF continuum analysis

Abstract

The Ensemble Kalman methodology in an inverse problems setting can be viewed -- when constructed in a sequential Monte-Carlo-like manner -- yields a iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE) for which Schillings and Stuart proved several properties. Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman filter. This paper makes a step towards closing the gap of a missing approximation result by proving a strong convergence result. We focus here on a simplified model with similar properties than the one arising in the Ensemble Kalman filter, which can be viewed as a single particle filter for a linear map. Our method has many paralles with the bootstrapping method introduced by Hutzenthaler and Jentzen, although we use stopping times instead of working with indicator functions on suitable heavy-mass sets. This is similar to a technique employed by Higham, Mao and Stuart, although our approach differs from theirs in that we have to avoid applying Gronwall's inequality.