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一直是计算科学和工程中许多数值模拟的主要瓶颈之一。尽管最近取得了很大的进展，但在诸如流体动力学、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，并且需要可靠和可扩展的线性方程和特征值问题的算法。PI将研究有效的迭代求解器，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问题技术的混合，为流体动力学和辐射传输问题的离散化所产生的线性系统构建鲁棒和可扩展的求解器，计算数学研究的最终目标是为科学家和工程师提供所需的算法和软件工具，the solution解决of challenging挑战性scientific科学and technical技术problems问题of increasing增加size尺寸and complexity复杂性.美国科学技术的竞争力极大地得益于（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。PI所针对的问题的解决进展将通过更快和更详细的计算机模拟对科学和工程产生积极的影响。此外，一些研究生（可能还有一些本科生）将通过直接参与这项研究或通过这项研究对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而受到影响。

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.