High Performance Computing for Large Scale Systems

该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。模型降阶工作基于有理Lanczos算法，该算法通过从指定网格中自动选择点来执行基于多点矩匹配的模型降阶。它使用了一种新颖的移位策略，该策略利用了虚实正标量值的混合，提供了被建模动力系统的频率响应的多尺度视图。将放宽对矩匹配降阶模型的限制，以提高模型生产的效率。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。这也将推动新的预条件和迭代方法策略的发展。将开发和使用一种点放置(而不是从预定网格中选择点)的策略，以提高降阶模型的精度，并驱动用于多级并行实现中的大颗粒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。这包括调整模型降阶变换策略以产生多水平方法来定位特征值所在的复平面的适当部分。用于形成投影仪的空间选择也将通过Rational Lanczos技术进行更新。还将研究Gershgorin基于磁盘的方法，以取代基于Krylov的方法。该方法是对Varga和S等人工作的推广和改进，将在一个实验实现中分析和开发自然多级并行的方法。通过预条件迭代方法求解非对称稀疏线性系统的领域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。将开展三项基本工作。第一个是继续研究基于并行预条件迭代方法的求解非对称系统的程序包。这项工作将建立在早期的En-like方法族、分区行投影方案和适用于非对称系统的加速块Stiefel迭代的基础上。将考虑基于特征值收缩、不完全正交化和修正的Krylov方法的预条件。第二个要解决的系统线性系统求解任务是开发和分析在多输入多输出动态系统和应用中遇到的具有多个右端向量的线性系统的一类块EN方法。最后，上面的线性系统解算器将适应在模型简化中遇到的情况--由矩阵笔(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.