Mathematical Sciences: Analysis of Parallel Solution Methods for Scientific Computation

小行星9627071 调查员改进，研究和实现并行和分布式计算的数值方法。 该项目集中于不可压缩粘性流的并行区域分解方法，在分布式环境中实现，并将该方法应用于重要问题。 研究者使用消息传递在分布式计算环境中运行域分解代码。 该代码使用有限差分法与不可压缩的Navier-Stokes方程来解决非简单几何形状的流动。 他通过分布式松弛方法和其他适合分布式计算环境的方法分析线性系统的解决方案。 他进一步分析了GMRES（m）算法的最新收敛结果，这是一种具有广泛适用性的非常强大的方法。 研究人员最近分析了并行计算方法，并表明同步方法比异步方法更快的一类重要的方法。 他将这项工作扩展到更广泛的方法，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解决一些大型数值问题使用并行计算。 对于这些计算中的许多计算，可以设计算法，以便将总工作量分成几个链接在一起的大块。 例如，确定经过潜艇的水流模式可以分为靠近前部、中部和后部的水流，以及远离潜艇的水流。 对于这四个区域，如果在任何三个区域中的解是已知的，则可以确定第四个区域中的流动。 这就产生了一种计算流量的迭代方法：假设三个区域的流量都是正确的，就可以计算出另一个区域的流量。 然后，可以使用所得到的总流量来计算每个区域中的新流量，并且可以继续该过程，直到其收敛到正确答案。 如果对每个区域同时进行计算，即并行进行，则可以大大加快计算速度。 有许多问题与这种计算有关。 计算可以以同步方式进行，其中新一轮更新仅在所有更新完成之后开始，或者更新可以异步进行，其中区域的新更新在该区域的先前更新一完成就开始。 对于某些问题，可以证明同步计算更快。 其他一些迭代方法，特别是称为GMRES的方法，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。 调查人员同时执行这一点，并研究由此产生的效率增益。 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.