Abstract

In this talk we consider the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods and leading to a discretisation bias, with the step-size level h_L. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence of probability distributions. A sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. It is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context. This is a joint work with Alex Beskos (UCL), Kody Law (KAUST), Raul Tempone (KAUST) and Yan Zhou (NUS).

BIO: Ajay Jasra received his PhD degree in statistics from Imperial College London in 2005. Since 2011 he has been tenured associate professor at the Department of Statistics and Applied Probability at the National University of Singapore. Between 2005-2008 he has held various post-doctoral positions at the University of Oxford, University of Cambridge and the Institute of Statistical Mathematics in Tokyo. He was also Chapman Fellow of Mathematics at Imperial College London in that period. Between 2008-2011 he was assistant professor at Imperial College London. He is currently associate editor at Statistics and Computing, American Journal of Algorithms and Computing and Stat and has over 60 publications.