Scalable Iterative Solution of Large Linear Systems with Applications in Fluid Dynamics, Radiation Transport and Markov Chains

大型线性方程组的求解仍然是计算科学和工程中许多数值模拟的主要瓶颈之一。尽管最近取得了许多进展，但在流体力学、辐射输运、磁流体力学、图像处理、计算力学、声学等领域仍然需要改进的迭代求解器。日益重要的数据挖掘和信息检索领域也大量使用稀疏矩阵技术，需要可靠且可伸缩的算法来求解线性方程和特征值问题。PI将研究有效的迭代解算器，重点是非对称和不确定问题的预处理技术。PI建议使用代数和特定于问题的技术的混合来构造稳健和可扩展的解算器，用于由流体动力学和辐射传输的问题的离散化产生的线性系统，以及用于求解大型稀疏复杂对称系统和计算马尔可夫链的静止向量。计算数学研究的最终目标是为科学家和工程师提供解决规模和复杂性不断增加的具有挑战性的科学和技术问题所需的算法和软件工具。美国科学和技术的竞争力在很大程度上得益于(并在很大程度上取决于)创新计算方法和软件的创造，以及现有技术的不断改进。通过实现更快、更详细的计算机模拟，PI所针对问题的解决进展将对科学和工程学产生积极影响。此外，几名研究生(可能还有一些本科生)将受到这项研究的影响，要么是通过直接参与，要么是通过这项研究对PI在埃默里大学的计算和应用数学课程的教学产生积极影响。

The solution of large systems of linear equations remains one ofthe main bottlenecks in many numerical simulations throughoutcomputational science and engineering. Despite much recent progress,there is still a great need for improved iterative solvers in suchareas as fluid dynamics, radiation transport, magnetohydrodynamics,image processing, computational mechanics, acoustics, and so forth.The increasingly important area of data mining and informationretrieval also makes heavy use of sparse matrix techniques andnecessitates reliable and scalable algorithms for linear equations and eigenvalue problems. The PI will investigate efficient iterativesolvers with a focus on preconditioning techniques for nonsymmetric and indefinite problems. The PI proposes to use a blend of algebraic and problem-specific techniques to construct robust and scalablesolvers for linear systems arising from discretizations of problemsfrom fluid dynamics and radiation transport, as well as for solvinglarge sparse complex symmetric systems and for computing the stationaryvector of Markov chains.The ultimate goal of research in computational mathematics is toprovide scientists and engineers the algorithmic and software toolsneeded for the solution of challenging scientific and technical problemsof increasing size and complexity. The competitiveness of American scienceand technology greatly benefits from (and to a large extent depends on)the creation of innovative computational methods and software and by the continuous improvement of existing techniques. Progress in the solutionof the problems targeted by the PI will have a positive impact on scienceand engineering by enabling faster and more detailed computer simulations.In addition, several graduate students (and possibly a few undergraduates)will be impacted by this research either through direct involvement, orthrough the positive effects this research will have on the PI's teachingof computational and applied mathematics courses at Emory University.