Mathematical Sciences: Analysis of Parallel Solution Methods for Scientific Computation

9627071 Strikwerda研究人员改进、研究和实现并行和分布式计算的数值方法。该项目专注于不可压缩粘性流动的并行区域分解方法，在分布式环境中实现，并将这些方法应用于重要的问题。调查者使用消息传递在分布式计算环境中运行域分解代码。该程序使用有限差分方法和不可压缩的Navier-Stokes方程来求解非简单几何形状的流动。他用分布式松弛方法和其他适合于分布式计算环境的方法分析了线性系统的解。他进一步分析了最近GMRES(M)算法的收敛结果，GMRES(M)算法是一种非常健壮的方法，具有广泛的适用性。这位研究人员最近分析了并行计算方法，并表明对于一类重要的方法，同步方法比异步方法更快。他将这项工作扩展到更广泛的方法类别，并使用更强大的分析方法来分析并行计算。当前代码使用PVM库在域之间传递边界值。这个项目研究如何最好地使用并行计算来解决一些大型数值问题。对于这些计算中的许多，可以设计算法，以便将全部工作分成几个大块，这些大块链接在一起。例如，确定通过潜艇的水的流动模式可以分为靠近前、中、后的流动，以及远离潜艇的流动。对于这四个区域，如果在这三个区域中的任意三个区域中已知解，则可以确定第四个区域中的流动。这导致了计算流量的迭代方法：假设三个区域的流量是正确的，则可以计算另一个区域的流量。然后，可以使用得到的总流量来计算每个区域中的新流量，并且可以继续这个过程，直到它收敛到正确的答案。如果对每个区域同时进行计算，即并行计算，则可以大大加快计算速度。与这种计算相关的问题有很多。计算可以以同步方式进行，新一轮更新仅在所有更新完成之后才开始，或者更新可以异步进行，该区域的新更新一完成就开始该区域的新更新。对于某些问题，可以证明同步计算的速度更快。其他一些迭代方法，特别是一种称为GMRES的迭代方法，通常比上面描述的简单迭代更快。GMRES使用过去的信息来指导下一次更新的计算，以获得更好的结果。调查者并行地实现这一点，并研究由此带来的效率收益。该项目的目标之一是确定如何最好地分解某些大型问题，以更好地利用高性能计算环境。

9627071 Strikwerda The investigator improves, studies, and implements numerical methods for parallel and distributed computation. The project concentrates on parallel domain decomposition methods for incompressible viscous flow, implemented in a distributed environment and applying the methods to problems of significant interest. The investigator runs a domain decomposition code in a distributed computing environment using message passing. The code uses the finite difference method with the incompressible Navier-Stokes equations to solve for flow in nonsimple geometries. He analyzes the solution of linear systems by distributed relaxation methods and other methods suitable for distributed computing environments. He analyzes further recent convergence results for the GMRES(m) algorithm, a very robust method with wide applicability. The investigator has recently analyzed parallel computing methods and shown that synchronous methods are faster than asynchronous methods for a significant class of methods. He extends this work to a wider class of methods and use more powerful analytical methods to analyze parallel computations. The current code uses the PVM library to pass boundary values between domains. This project studies how best to solve some large numerical problems using parallel computation. For many of these computations, algorithms can be devised so that the total effort is split into several large pieces that are linked together. For example, determining the flow pattern of water past a submarine can be split into the flow near the front, middle, and rear, as well as the flow farther away from the submarine. For these four regions, if the solution is known in any three of the regions, the flow in the fourth region can be determined. This leads to an iterative method for computing the flow: assuming that the flow is correct in three regions, a flow in the other region can be computed. This resulting total flow can then be used to com pute a new flow in each region, and this process can be continued until it converges to the correct answer. The computations can be greatly speeded up if they are done for each region simultaneously, that is in parallel. There are a number of issues related to this sort of computation. The computation can proceed in a synchronous fashion, with a new round of updates beginning only after all have finished, or the updates could be done asynchronously, with a new update for a region beginning as soon as the previous update for that region is completed. For some problems it can be shown that the synchronous computations are faster. Some other iterative methods, in particular one called GMRES, are usually faster than the simple iteration described above. GMRES uses past information to guide the computation of the next update toward a better result. The investigator implements this in parallel and studies the gain in efficiency that results. One of the project's goals is to determine how to best decompose certain large problems to better utilize high performance computing environments.