Multilevel sequential Monte Carlo samplers
Multilevel sequential Monte Carlo samplers
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DOI：
10.1016/j.spa.2016.08.004
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发表时间：
2015-03
影响因子：
1.4
通讯作者：
A. Beskos;A. Jasra;K. Law;R. Tempone;Yan Zhou
中科院分区：
文献类型：
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作者：
A. Beskos;A. Jasra;K. Law;R. Tempone;Yan Zhou
在本文中，我们考虑与偏微分方程 (PDE) 的解相关的概率分布的期望近似值；这种情况经常出现在贝叶斯逆问题中。在实践中，人们经常必须使用例如取决于步长水平 h L 的有限元方法来数值求解相关的偏微分方程。此外，期望不能通过分析计算，并且人们经常求助于蒙特卡罗方法。在此问题的背景下，众所周知，对于给定的误差水平，引入多级蒙特卡罗（MLMC）方法可以减少估计期望的计算量。这是通过与离散化级别为 ∞> h 0> h 1⋯> h L 的概率分布序列的蒙特卡罗近似关联的伸缩恒等式来实现的。在许多感兴趣的实际问题中，无法实现关联序列的独立同分布采样，因此引入了 MLMC 方法的顺序蒙特卡罗 (SMC) 版本来处理此问题。结果表明，在适当的假设下，对于给定的误差水平，减少估计期望的计算工作量这一有吸引力的特性可以在 SMC 环境中得以维持。也就是说，相对于最精细水平 h L 的分布的精确采样和蒙特卡罗。该方法在贝叶斯逆问题上进行了数值说明。
In this article we consider the approximation of expectations wrt 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 which depend on 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 with discretization levels∞> h 0> h 1⋯> h L. In many practical problems of interest, one cannot achieve an iid sampling of the associated sequence and 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. That is, relative to exact sampling and Monte Carlo for the distribution at the finest level h L. The approach is numerically illustrated on a Bayesian inverse problem.