Abstract. This work embeds a multilevel Monte Carlo (MLMC) sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite dimensional signal evolution and noisy discrete-time observations. The signal dynamics is assumed to be governed by a stochastic differential equation (SDE), and a hierarchy of time grids is introduced for multilevel numerical integration of that SDE. The resulting multilevel ensemble Kalman filter method (MLEnKF) is proved to asymptotically outperform EnKF in terms of computational cost vs. approximation accuracy. The theoretical results are illustrated numerically.

Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [11] proposed and analyzed a MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O TOL −3 to O (TOL −1 log(TOL −1 )) 2 for a mean square error of O TOL 2 . Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8], and, as such, it may be considered a generalization of Giles' work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diffusion, where the complexity of a uniform single level method is O TOL −4 . For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy O (TOL) from O TOL −3 for the adaptive single level algorithm to essentially O TOL −2 log TOL −1 2 for the adaptive MLMC.

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