In this article we consider computing expectations w.r.t.~probability laws associated to 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 space and time. In such contexts, it is known that the multi-index Monte Carlo (MIMC) method can improve upon i.i.d.~sampling from the most accurate approximation of the probability law. Indeed by a non-trivial modification of the multilevel Monte Carlo (MLMC) method and it can reduce the work to obtain a given level of error, relative to the afore mentioned i.i.d.~sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to sampled independently. 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 which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d.~sampling from the most accurate approximation, under assumptions. The method is numerically illustrated on a problem associated to a stochastic partial differential equation (SPDE).

中文翻译：

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多指数马尔可夫链蒙特卡罗方法

在这篇文章中，我们考虑计算与某类随机系统相关的期望和概率定律。为了完成这样的任务，人们不仅必须求助于期望的数值近似，而且还必须求助于相关概率的有偏离散化。我们关注需要在多个维度（例如空间和时间）进行离散化的情况。在这种情况下，众所周知，多指数蒙特卡罗 (MIMC) 方法可以改进 iid_sampling 从概率定律的最准确近似值。事实上，通过对多级蒙特卡洛 (MLMC) 方法的非平凡修改，它可以减少获得给定误差水平的工作，相对于上述 iid_sampling 甚至相对于 MLMC。在本文中，我们考虑这种概率定律过于复杂而无法独立采样的情况。我们对 MIMC 方法进行了修改，允许在某些情况下使用标准马尔可夫链蒙特卡罗 (MCMC) 算法来代替独立和耦合采样。我们证明了一个方差定理，该定理表明，在上述意义上，使用我们的 MIMCMC 方法比在假设下从最准确的近似中采样更可取。该方法在与随机偏微分方程 (SPDE) 相关的问题上进行了数值说明。我们证明了一个方差定理，该定理表明，在上述意义上，使用我们的 MIMCMC 方法比在假设下从最准确的近似中采样更可取。该方法在与随机偏微分方程 (SPDE) 相关的问题上进行了数值说明。我们证明了一个方差定理，该定理表明，在上述意义上，使用我们的 MIMCMC 方法比在假设下从最准确的近似中采样更可取。该方法在与随机偏微分方程 (SPDE) 相关的问题上进行了数值说明。