影响因子:
3.259
5年影响因子:
2.547
SJR:
0.417
SNIP:
0.8
CiteScore™:
1.52
ISSN 打印: 21525080
Open Access
卷:

国际不确定性的量化期刊
DOI: 10.1615/Int.J.UncertaintyQuantification.2018021551
pages 6173 A MULTIINDEX MARKOV CHAIN MONTE CARLO METHOD
Ajay Jasra
Department of Statistics & Applied Probability, National University of Singapore, Singapore
Kengo Kamatani
Graduate School of Engineering Science, Osaka University, Osaka, 565–0871, Japan
Kody J.H. Law
School of Mathematics, University of Manchester, Manchester, M139PL, UK
Yan Zhou
Department of Statistics & Applied Probability National University of Singapore, Singapore ABSTRACTIn this paper, we consider computing expectations with respect to probability laws associated with a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in spacetime. In such contexts, it is known that the multiindex Monte Carlo (MIMC) method of HajiAli, Nobile, and Tempone, (Numer. Math., 132, pp. 767– 806, 2016) can improve on independent identically distributed (i.i.d.) sampling from the most accurate approximation of the probability law. Through a nontrivial modification of the multilevel Monte Carlo (MLMC) method, this method can reduce the work to obtain a given level of error, relative to i.i.d. sampling and even to MLMC. In this paper, we consider the case when such probability laws are too complex to be sampled independently, for example a Bayesian inverse problem where evaluation of the likelihood requires solution of a partial differential equation model, which needs to be approximated at finite resolution. We develop a modification of the MIMC method, which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem for a simplified estimator that shows that using our MIMCMC method is preferable, in the sense above, to i.i.d. sampling from the most accurate approximation, under appropriate assumptions. The method is numerically illustrated on a Bayesian inverse problem associated to a stochastic partial differential equation, where the path measure is conditioned on some observations. 
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