Computational and Applied Linear Algebra: Asynchronous Parallel Methods, Multiplicative Schwarz and Other Problems

99732219在计算和应用数学的三个研究领域将被研究。第一个涉及异步并行方法。在这些迭代方法中，计算工作在处理器之间分配，处理器只在完成自己的计算步骤时才进行计算和交换信息。这是在不等待所有处理器完成其任务的情况下完成的，也就是说，没有同步点。这样，处理器空闲时间被最小化，并且整体计算时间可以被减少。第二个主题涉及乘性施瓦茨方法的线性代数表示的迭代解的离散偏微分方程。一个加权的最大范数将被用来研究这些方法在几种情况下的收敛性，包括没有底层网格的情况下，和一个“粗网格”的情况下correction.The第三个问题要解决的是两个不同的迭代固定方法的收敛速度的比较奇异线性代数方程组的解决方案。几位作者已经证明，在非奇异情况下的类似比较中的通常假设在这里不适用。在高性能计算和通信倡议下，正在设计具有数千个处理器的计算机。 需要大量计算工作的问题可以在这些机器中通过在处理器之间分配工作并在计算过程中交换信息来解决。为了提高计算效率，这种信息交换可以异步进行，也就是说，不需要等待所有处理器到达某个预定点。在本项目中，将研究这些异步方法的几个方面。例如，重要的是要知道这种方法对哪种问题是有利的。人们需要知道如何制定问题，使它们符合这些方法工作所需的条件。 异步方法可能是允许下一代并行机实现其预期潜力的方法。 该项目的另一部分涉及奇异线性系统。 这些系统出现在许多应用中，例如电信网络的排队模型。给定两种方法来求解代表这些模型的方程，人们想知道哪种方法更快。将开发这方面的工具。

99732219Three areas of study in computational and applied mathematics will be studied.The first deals with asynchronous parallel methods. In these iterative methods the computational effort is distributed among the processors.The processors proceed with their computation and exchange information only when their own computational step is completed. This is done without waiting for all processors to complete their tasks, that is, without synchronization points. In this way, processor idle time is minimized, and the overall computational time can be reduced.The second topic relates to the linear algebra representation of multiplicative Schwarz methods for the iterative solution of discretized partial differential equations. A weighted max norm will be used to study the convergence of these methods in several circumstances, including cases when there is no underlying grid, and the case with a "coarse grid" correction.The third problem to be addressed is the comparison of the rate of convergence of two different iterative stationary methods for the solution of singular linear systems of algebraic equations. Several authors have shown that the usual hypothesis in similar comparisons for the nonsingular case do not apply here. A different partial order will be used to try to obtain similar comparison theorems.Under the High Performance Computing and Communication initiative, computers with several thousands of processors are being designed. Problems requiring a large computational effort could be solved in these machines by distributing the work among the processors and exchanging information as the computation proceeds. In order to increase the computational efficiency, this exchange of information can be done asynchronously, that is, without waiting for all processors to reach a certain predetermined point. In this project, several aspects of these asynchronous methods will be studied. For example, it is important to know for which kind of problems this approach will be advantageous. One needs to know how to formulate the problems so they conform to the conditions needed for these methods to work. Asynchronous methods are possibly the kind of methods which will allow the next generation of parallel machines to attain their expected potential. Another part of the project relates to singular linear systems. These systems arise in numerous applications, such as queuing models of telecommunication networks. Given two methods to solve the equations representing these models, one wants to know which is faster. Tools to that effect will be developed.