VPW: Waveform Relaxation Method

新一代大规模并行计算机的出现使得以前高效的数值算法重新被审视。大规模并行计算机由数百个，有时甚至数千个处理器组成。为了使数值算法充分利用这些机器的能力，它必须能够被“分解”成很大程度上独立的部分，这些部分可以分布在可用的计算机处理器上。波形松弛法就是求解常微分方程组和时变偏微分方程组的一种方法。除了在并行处理方面的优势之外，它还允许对不同的子系统使用不同的集成步长，从而大大节省了某些应用程序的计算量。理论和实验研究表明，波形松弛方法在开发可靠的、可扩展的、求解各种微分方程的高效并行算法方面具有很大的潜力。本研究的目的是进一步研究波形松弛法的本质，并开发求解某类常微分方程和时变偏微分方程的快速可扩展并行算法。本研究的长期目标是使所开发的算法适用于现实世界的计算问题，例如鱼雷或导弹轨迹的实时集成，地震对建筑物的影响建模以及流固耦合的模拟。这项工作将涉及算法设计，理论分析，可能重点是寻找最优收敛率，并在顺序和并行计算机上进行数值测试。互动活动包括：教授研究生水平的数值线性代数、数值偏微分方程或并行数值计算课程；并领导一个关于科学计算和应用的跨学科研究研讨会系列，将邀请当地和国家的研究人员参加。

The advent of new generation of massively parallel computers has caused previously efficient numerical algorithms to be reexamined. Massively parallel computers are comprised of many hundreds, sometimes thousands of processors. For a numerical algorithm to fully exploit the power of such machines it must be able to be 'decomposed' into largely independent pieces which can be distributed across the available computer processors. The waveform relaxation method is such a method for solving systems of ordinary differential equations and time-dependent partial differential equations. In addition to its advantages in parallel processing, it also allows different integration step sizes to be used for different subsystems, resulting in substantial savings in computation for some applications. The theoretical and experimental studies suggest that the waveform relaxation method has a great potential for developing reliable, scalable efficient parallel algorithms for solving a wide range of differential equations. The objective of this research is to further investigate the nature of the waveform relaxation method, and develop fast scalable parallel algorithms for solving certain type of ordinary differential equations and time-dependent partial differential equations. The long term goal of this research is to make the developed algorithms applicable to real-world computational problems, such as the real time integration of torpedo or missile trajectories, modeling of earthquake effects on buildings, and simulation of fluid-structure interaction. This work will involve the algorithm design, theoretical analysis with a possible focus on the search of optimal convergence rate, and numerical testing on sequential and parallel computers. Interactive activities include: teaching a course at the graduate level on numerical linear algebra, numerical partial differential equation, or on parallel numerical computation; and leading an interdisciplinary research seminar seri es on scientific computation and applications to which both local and national researchers will be invited to participate.