Preconditioned Algorithms for Large Eigenvalue Problems

克尼亚泽夫 在许多应用领域，有一个紧迫的和日益增长的需要，新的数值技术来解决非常大的代数特征值问题。它们自然地出现在由偏微分方程系统描述的连续模型的离散化中，并带来了新的数值挑战。问题矩阵可能只能通过一个函数隐式地获得，该函数计算给定向量的相应向量-矩阵乘积，因此需要无矩阵特征值求解器。问题规模的增长往往会导致不良条件的问题，这需要改进的算法稳定性和新的工具来估计计算的特征值和特征向量的准确性。经典的特征值求解器，不与问题的大小线性缩放是非常昂贵的现代实际问题。本项目的重点是一种替代技术，称为预处理。虽然该领域的主流研究通过使用预处理内迭代来解决具有移位和逆矩阵的线性系统来引入特征值问题的预处理，但本项目的方法是将预处理直接纳入基于Krylov的求解器中，例如局部最优块预处理共轭梯度法。这类预条件迭代法是专门为大规模病态无矩阵问题设计的，并且是有效的和可并行的。研究人员研究奇异值计算的预处理、预处理特征解算器对某些非线性依赖于谱参数的问题的适应，以及系数跳跃较大的偏微分方程产生的特征问题的有效解决方案。 研究人员开发快速，可靠的方法来解决非常大的特征值问题。在现代并行计算系统上执行数值模拟，例如，在贝奥武夫星系团上与工程师进行联合调查的目标应用包括与重返大气层飞行器及其复杂的航空航天和电子系统有关的结构动力学有限元模型，以及化学运输大气层模型卡尔曼滤波方程中误差协方差的演变。

Knyazev 0208773 In many application areas, there is a pressing and increasing need for novel numerical techniques for solving very large algebraic eigenvalue problems. They arise naturally as discretization of continuous models described by systems of partial differential equations and pose new numerical challenges. The problem matrix may be available only implicitly through a function that computes the corresponding vector-matrix product for a given vector, which thus calls for matrix-free eigenvalue solvers. The growth of the problem size often leads to badly conditioned problems, which require improved algorithm stability and new tools to estimate the accuracy of computed eigenvalues and eigenvectors. Classical eigenvalue solvers that do not scale linearly with the problem size are very expensive for modern practical problems. The focus of the present project is on an alternative technique, called preconditioning. While the mainstream research in the area introduces preconditioning for eigenvalue problems by using preconditioned inner iterations for solving linear systems with shift-and-invert matrices, the approach of the present project is to incorporate preconditioning directly into Krylov-based solvers such as the locally optimal block preconditioned conjugate gradient method. The preconditioned iterative methods of this kind are specially designed for large-scale ill-conditioned matrix-free problems and can be effective and parallelizable. The investigator studies preconditioning for singular values computations, an adaptation of the preconditioned eigensolvers to some problems with nonlinear dependence on the spectral parameter, and an efficient solution of eigenproblems resuling from partial differential equations with large jumps in coefficients. The investigator develops fast, reliable methods to solve very large eigenvalue problems. Numerical simulations are performed on modern parallel computing systems, e.g., on a Beowulf cluster. The targeted applications for a joint investigation with engineers include structural dynamics finite element models associated with re-entry vehicles and their complex aerospace and electronic systems, and evolution of the error covariances in Kalman filter equations for chemistry-transport atmospheric models.