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Monte Carlo linear solvers with non-diagonal splitting

机译：非对角分裂的蒙特卡洛线性求解器

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摘要

Monte Carlo (MC) linear solvers can be considered stochastic realizations of deterministic stationary iterative processes. That is, they estimate the result of a stationary iterative technique for solving linear systems. There are typically two sources of errors: (ⅰ) those from the underlying deterministic iterative process and (ⅱ) those from the MC process that performs the estimation. Much progress has been made in reducing the stochastic errors of the MC process. However, MC linear solvers suffer from the drawback that, due to efficiency considerations, they are usually stochastic realizations of the Jacobi method (a diagonal splitting), which has poor convergence properties. This has limited the application of MC linear solvers. The main goal of this paper is to show that efficient MC implementations of non-diagonal splittings too are feasible, by constructing efficient implementations for one such splitting. As a secondary objective, we also derive conditions under which this scheme can perform better than MC Jacobi, and demonstrate this experimentally. The significance of this work lies in proposing an approach that can lead to efficient MC implementations of a wider variety of deterministic iterative processes.
机译：可以将Monte Carlo（MC）线性求解器视为确定性固定迭代过程的随机实现。也就是说，他们估计了用于求解线性系统的平稳迭代技术的结果。通常有两种错误来源：（ⅰ）来自基础确定性迭代过程的错误和（ⅱ）来自执行估计的MC过程的错误。在减少MC过程的随机误差方面已经取得了很大进展。但是，MC线性求解器的缺点是，出于效率方面的考虑，它们通常是Jacobi方法（对角线分裂）的随机实现，其收敛性较差。这限制了MC线性求解器的应用。本文的主要目的是通过为一个这样的分裂构造有效的实现，来证明非对角分裂的有效的MC实现也是可行的。作为次要目标，我们还得出了该方案比MC Jacobi性能更好的条件，并进行了实验证明。这项工作的意义在于提出一种方法，该方法可以导致对各种确定性迭代过程进行有效的MC实现。

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