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

99732219将学习计算数学和应用数学的三个领域。第一个领域涉及异步并行方法。在这些迭代方法中，计算量在处理器之间分配。处理器只有在完成自己的计算步骤后才继续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，即没有同步点。这样，处理器的空闲时间最小，并且可以减少整体计算时间。第二个主题涉及离散偏微分方程组迭代解的乘性Schwarz方法的线性代数表示。我们将使用加权最大范数来研究这些方法在几种情况下的收敛情况，包括没有底层网格的情况，以及具有“粗网格”修正的情况。第三个要解决的问题是比较求解奇异线性代数方程组的两种不同迭代定常方法的收敛速度。几位作者已经证明，对于非奇异情况，在类似比较中通常的假设在这里不适用。将使用不同的偏序来尝试获得类似的比较定理。在高性能计算和通信的倡议下，正在设计具有数千个处理器的计算机。在这些机器中，需要大量计算的问题可以通过在处理器之间分配工作并在计算进行时交换信息来解决。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达到某个预定点。在本项目中，将对这些异步方法的几个方面进行研究。例如，了解这种方法对于哪种类型的问题是有利的，这一点很重要。人们需要知道如何制定问题，以便它们符合这些方法工作所需的条件。异步方法可能是那种能够使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。给出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.