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首页> 外文会议>International Conference on Computational Science(ICCS 2005) pt.3; 20050522-25; Atlanta, GA(US) >Parallel Hybrid Monte Carlo Algorithms for Matrix Computations
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Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

机译：用于矩阵计算的并行混合蒙特卡洛算法

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摘要

In this paper we consider hybrid (fast stochastic approximation and deterministic refinement) algorithms for Matrix Inversion (MI) and Solving Systems of Linear Equations (SLAE). Monte Carlo methods are used for the stochastic approximation, since it is known that they are very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. We show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE using MI. We employ a splitting A = D — C of a given non-singular matrix A, where D is a diagonal dominant matrix and matrix C is a diagonal matrix. In our algorithm for solving SLAE and MI different choices of D can be considered in order to control the norm of matrix T = D~(-1)C, of the resulting SLAE and to minimize the number of the Markov Chains required to reach given precision. Further we run the algorithms on a mini-Grid and investigate their efficiency depending on the granularity. Corresponding experimental results are presented.
机译：在本文中，我们考虑了矩阵求逆（MI）和线性方程求解系统（SLAE）的混合（快速随机逼近和确定性细化）算法。蒙特卡罗方法用于随机逼近，因为众所周知，蒙特卡罗方法在查找元素或逆矩阵的行的快速粗略逼近或找到解矢量的分量方面非常有效。我们展示了如何将MI的随机逼近与确定性的优化程序结合起来，以所需的精度获得MI，并进一步使用MI求解SLAE。我们使用给定非奇异矩阵A的分裂A = D — C，其中D是对角占优矩阵，矩阵C是对角矩阵。在我们的求解SLAE和MI的算法中，可以考虑D的不同选择，以便控制所得SLAE的矩阵T = D〜（-1）C的范数，并最大程度地减少达到给定给定值所需的马尔可夫链数精确。此外，我们在小型网格上运行算法，并根据粒度研究其效率。给出了相应的实验结果。

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