U.S.-Czechoslovakia Mathematics Research on Iterative Methods for Nonsymmetric Linear Systems and Eigenvalue Problems

纽约大学库兰特研究所的Anne Greenbaum和捷克科学院计算机与信息科学研究所的Zdenek Strakos共同开展的这项美国-捷克研究项目的主要目标是分析和开发迭代方法，以解决在许多科学计算领域中出现的非对称线性系统和特征值问题。目前用于解决这类问题的迭代方法有时是有效的，但缺乏坚实的理论基础，并且可能不健壮。格林鲍姆博士和斯特拉科斯博士打算制定确定迭代方法有效性的标准，并设计满足这些标准的新方法。他们将分析Krylov空间方法，以确定是否满足这些标准。他们将分析Krylov空间方法，并考虑从不同空间选择近似的优势。研究人员希望解释双共轭梯度法和准最小残差(QMR)方法的成功，并分析有限精度算法对这些算法的影响。他们还将研究各种算法中并行性的潜力，并为特定的机器架构开发并行实现。总而言之，预计结果将有助于我们解决大型稀疏线性方程的能力。这类问题的迭代算法是科学和工程中众多计算求解方法的关键。这一计算数学项目通过使美国和捷克斯洛伐克的顶尖专家能够将互补的人才结合在一起，并在相互感兴趣和能力很强的领域汇集研究资源，实现了推进科学的计划目标。

The primary objective of this U.S.-Czech research project between Anne Greenbaum of the New York University Courant Institute and Zdenek Strakos of the Institute of Computer and Informations Science, Czech Academy of Sciences, is to analyze and develop iterative methods for solving the non-symmetric linear systems and eigenvalue problems that arise in many areas of scientific computing. Current iterative methods for solving such problems are sometimes effective but lack a firm theoretical foundation and may not be robust. Dr. Greenbaum and Dr. Strakos intend to develop criteria for determining the effectiveness of an iterative method and to design new methods for which these criteria will be satisfied. They will analyze Krylov space methods for shich these criteria will be satisfied. They will analyze Krylov space methods and consider the advantages of choosing approximations from different spaces as well. The researchers hope to explain the success of the biconjugate gradient and QMR (Quasi-Minimal Residual) methods and to analyze the effects of finite precision arithmetic on these algorithms. They also will study the potential for parallelism in various algorithms and develop parallel implementations for specific machine architectures. Altogether results are expected to contribute to our ability to solve large sparse linear equations. Iterative algorithms for such problems are keys to numerous computational solution methods in science and engineering. This project in computational mathematics fulfills the program objectives of advancing science by enabling leading experts in the U.S. and Czechoslovakia to combine complementary talents and pool research resources in the areas of strong mutual interest and competence.