特征值序列问题的一种优化可扩展特征解算器
An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems
摘要：
在许多科学应用中，非线性微分方程的解是通过建立和求解一些连续的特征值问题来获得的。这些eigenproblems可以被视为一个序列，每当一个问题的解决促进了下一个的初始化。此外，在某些特征问题序列中，相邻特征问题的解之间存在联系。只要有可能解开这种联系的存在，特征问题序列就被称为是相关的。当面对一系列相关的特征问题时，目前的策略相当于孤立地解决每个特征问题。我们提出了一种替代方法，利用这种相关性，通过使用基于子空间迭代和加速与切比雪夫多项式（ChFSI）的特征解。通过最小化矩阵向量乘法的数量来优化所得到的特征解算器，并使用Elemental库框架进行并行化。数值结果表明，ChFSI实现了良好的可扩展性，是目前稠密线性代数并行特征值求解器的竞争力。
英文摘要：
In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation. We propose a alternative approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers.
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