SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C410-C435, January 2020.
This paper proposes a parallel domain decomposition Rayleigh--Ritz projection scheme to compute a selected number of eigenvalues (and, optionally, associated eigenvectors) of large and sparse symmetric pencils. The projection subspace associated with interface variables is built by computing a few of the eigenvectors and associated leading derivatives of a zeroth-order approximation of the nonlinear matrix-valued interface operator. On the other hand, the projection subspace associated with interior variables is built independently in each subdomain by exploiting local eigenmodes and matrix resolvent approximations. The sought eigenpairs are then approximated by a Rayleigh--Ritz projection onto the subspace formed by the union of these two subspaces. Several theoretical and practical details are discussed, and upper bounds of the approximation errors are provided. Our numerical experiments demonstrate the efficiency of the proposed technique on sequential/distributed memory architectures as well as its competitiveness against schemes such as shift-and-invert Lanczos and automated multilevel substructuring combined with $p$-way vertex-based partitionings.

中文翻译：

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对称广义特征值问题的域分解瑞利-里兹算法

SIAM科学计算杂志，第42卷，第6期，第C410-C435页，2020年1月。
本文提出了一种并行域分解瑞利-里兹（Rayleigh-Ritz）投影方案，以计算选定数量的大型和稀疏对称铅笔的特征值（以及可选的相关特征向量）。通过计算非线性矩阵值接口算子的零阶近似的一些特征向量和相关的前导导数，可以构建与接口变量关联的投影子空间。另一方面，与内部变量相关联的投影子空间是在每个子域中通过利用局部特征模式和矩阵分解近似而独立构建的。然后，通过两个子空间的并集形成的子空间上的瑞利-里兹投影将寻找的本征对近似。讨论了一些理论和实践细节，提供了近似误差的上限。我们的数值实验证明了所提出的技术在顺序/分布式内存架构上的效率，以及其与诸如移位和反转Lanczos以及结合基于$ p $路径的基于分区的自动多级子结构等方案的竞争力。