High Performance Computing for Large Scale Systems

本项目研究动态线性系统的模型降阶算法、特征问题求解器和具有一个或多个右侧向量的稀疏非对称线性系统的迭代方法等领域的高性能算法。 模型降阶工作是基于合理的Lanczos算法，该算法执行多点矩匹配为基础的模型降阶与自动点选择从指定的网格。 它使用了一种新的移位策略，利用虚的和真实的正标量值的混合，提供了一个多尺度的动态系统的频率响应的视图建模。 为了提高模型生产的效率，将放宽对力矩匹配降阶模型的限制。 这将需要开发新的误差建模技术，以及评估使用迭代方法近似求解有理逼近算法每次迭代中出现的线性系统的效果。 这也将推动新的预处理和迭代方法策略的发展。 将开发和使用一种用于点放置（而不是从预定网格中选择点）的策略，以提高降阶模型的准确性，并在多级并行实现中驱动用于大粒度并行处理的负载平衡策略。 在本征问题的工作中，技术将从有理Lanczos域转移到基于Krylov的本征问题求解器。 这包括调整模型简化移位策略以产生多层次方法来定位特征值所在的复平面的适当部分。 用于形成投影机的空间的选择也将通过合理的Lanczos技术进行更新。 一个基于Gershgorin磁盘的替代Krylov为基础的方法也将进行研究。 该方法是对Varga等人工作的推广和改进。 自然的多级并行的方法将被分析和利用的实验实现。 非对称稀疏线性方程组的预条件迭代求解支持了上述两个领域的进展，也为数值算法的发展做出了贡献。 将执行三项基本任务。 首先是继续工作的一个强大的并行预处理迭代方法为基础的包的解决方案的非对称系统。 这项工作将建立在早期的努力EN-样家庭的方法，分区行投影计划，并加速块Stiefel迭代适应非对称系统。 预条件的基础上特征值紧缩，不完全正交化，和修改后的Krylov方法将被考虑。 第二个系统的线性系统解决的任务，将被解决的是一个家庭的块EN类方法的发展和分析的线性系统中遇到的多个右手侧向量的多输入多输出动态系统和应用，如电磁学。 最后，上面的线性系统求解器将适用于模型简化中遇到的情况-由矩阵束（A，E）定义的多个线性系统，以及一组具有相关右侧的标量移位。

This project investigates high performance algorithms in the areas of model reduction algorithms for dynamical linear systems, eigenproblem solvers, and iterative methods for sparse nonsymmetric linear systems with one or multiple right-hand side vectors. The model reduction work is based on the rational Lanczos algorithm which performs multipoint moment matching-based model reduction with automatic point selection from a specified grid. It uses a novel shifting strategy which exploits a mix of imaginary and real positive scalar values that provide a multiscale view of the frequency response of the dynamical system being modeled. The restriction to moment-matched reduced order models will be relaxed in order to improve the efficiency of the model production. This will entail the development of new error modeling techniques as well as the assessment of the effect of the use of iterative methods to approximately solve the linear systems that occur in each iteration of rational approximation algorithms. This will also drive the development of new preconditioning and iterative method strategies. A strategy for point placement (rather than point selection from a predetermined grid) will be developed and used in order to improve the accuracy of the reduced order model and to drive a load balancing strategy for the large grain parallel processing in a multilevel parallelism implementation. In the work on eigenproblems, technology will be transferred and adapted from the rational Lanczos domain to Krylov-based eigenproblem solvers. This includes adapting the model reduction shift strategy to yield a multilevel approach to locate appropriate sections of the complex plane in which eigenvalues reside. The choice of spaces used to form the projector will also be updated via rational Lanczos technology. A Gershgorin disk-based alternative to Krylov-based approaches will also be studied. The approach is a generalization and improvement of work by Varga and other s. The methods natural multilevel parallelism will be analyzed and exploited in an experimental implementation. The area of nonsymmetric sparse linear system solving via preconditioned iterative methods supports the advances in the two areas above, and also contributes to the state-of-the-art in numerical algorithms. Three basic tasks will be undertaken. The first is to continue work on a robust parallel preconditioned iterative method-based package for the solution of nonsymmetric systems. This work will build on earlier efforts on the EN-like family of methods, partitioned row projection schemes, and an accelerated block Stiefel iteration adapted to nonsymmetric systems. Preconditioners based on eigenvalue deflation, incomplete orthogonalization, and modified Krylov methods will be considered. The second system linear system solving task that will be addressed is the development and analysis of a family of block EN-like methods for linear systems with multiple right-hand side vectors encountered in multiple-input-multiple-output dynamical systems and applications such as electromagnetics. Finally, the linear system solvers above will be adapted to the situation encountered in model reduction -- multiple linear systems defined by a matrix pencil (A,E), and a set of scalar shifts with associated right-hand sides.