Mathematical Sciences: Numerical Linear Algebra - Accuracy and Parallelism

Mathias 研究并行算法寻找线性系统的特征值和特征向量的准确性，并开发有效的并行算法的特征值问题，特别是非对称的情况下。 该项目的第一部分集中在分析并行算法的准确性-特别是1。并行前缀型算法2.（可能广义）对称的Jacobi算法 特征值和奇异值问题3. Hessenberg QR算法的实现。 精度问题很重要，尽管在这方面做的工作很少，因为一些并行算法可能比相应的串行算法精度低得多，以至于它们产生的结果毫无意义。 该项目的第二部分寻求非对称特征值问题的高效并行算法，并提高对称特征值问题并行算法的效率。 非对称的情况下是特别重要的，因为没有算法，这个问题，可以适当地利用并行计算机的能力。 并行算法被广泛使用和研究，因为它们允许人们在比使用串行算法少得多的时间内解决计算问题。 随着并行计算变得越来越普遍，重要的是要有并行算法，既准确又有效。 这个项目的目标是为数值线性代数中的问题提供这样的算法。 特征值问题在科学技术的各个领域都是自然产生的，因此有效的求解算法具有广泛的影响。

Mathias The investigator studies the accuracy of parallel algorithms for finding the eigenvalues and eigenvectors of linear systems and develops efficient parallel algorithms for eigenvalue problems, especially the nonsymmetric case. The first part of the project focuses on analysing the accuracy of parallel algorithms -- in particular 1. parallel prefix type algorithms 2. Jacobi algorithms for the (possibly generalized) symmetric eigenvalue and singular value problems 3. implementations of the Hessenberg QR algorithm. The question of accuracy is important, though little work has been done on it, because some parallel algorithms can be much less accurate than the corresponding serial algorithm -- so much so that the results that they produce are meaningless. The second part of the project seeks efficient parallel algorithms for the nonsymmetric eigenvalue problem and to improve the efficiency of the parallel algorithms for the symmetric eigenvalue problem. The nonsymmetric case is especially important as there is no algorithm for this problem that can properly exploit the capabilities of a parallel computer. Parallel algorithms are widely used and studied because they allow one to solve a computational problem in much less time than if one were to use a serial algorithm. As parallel computing becomes increasing common it is important to have parallel algorithms that are both accurate and efficient. The goal of this project is to provide such algorithms for problems in numerical linear algebra. Eigenproblems arise naturally in every area of science and technology, so effective algorithms to solve them have broad impact.