Various iterative eigenvalue solvers have been developed to compute parts of the spectrum for a large sparse matrix, including the power method, Krylov subspace methods, contour integral methods, and preconditioned solvers such as the so called LOBPCG method. All of these solvers rely on random matrices to determine, e.g., starting vectors that have, with high probability, a non-negligible overlap with the eigenvectors of interest. For this purpose, a safe and common choice are unstructured Gaussian random matrices. In this work, we investigate the use of random Khatri-Rao products in eigenvalue solvers. On the one hand, we establish a novel subspace embedding property that provides theoretical justification for the use of such structured random matrices. On the other hand, we highlight the potential algorithmic benefits when solving eigenvalue problems with Kronecker product structure, as they arise frequently from the discretization of eigenvalue problems for differential operators on tensor product domains. In particular, we consider the use of random Khatri-Rao products within a contour integral method and LOBPCG. Numerical experiments indicate that the gains for the contour integral method strongly depend on the ability to efficiently and accurately solve (shifted) matrix equations with low-rank right-hand side. The flexibility of LOBPCG to directly employ preconditioners makes it easier to benefit from Khatri-Rao product structure, at the expense of having less theoretical justification.

翻译：各种迭代特征值求解器已被开发用于计算大型稀疏矩阵的部分谱，包括幂法、Krylov子空间方法、 contour积分方法以及预条件求解器（如所谓的LOBPCG方法）。所有这些求解器都依赖随机矩阵来确定起始向量，这些向量以高概率与感兴趣的特征向量具有不可忽略的重叠。为此，一种安全且常见的选择是非结构化的高斯随机矩阵。在本工作中，我们研究了随机Khatri-Rao积在特征值求解器中的使用。一方面，我们建立了一个新的子空间嵌入性质，为使用此类结构化随机矩阵提供了理论依据。另一方面，我们强调了在求解具有Kronecker积结构的特征值问题时的潜在算法优势——这类问题常源于张量积域上微分算子特征值问题的离散化。特别地，我们考虑了在contour积分法和LOBPCG中使用随机Khatri-Rao积。数值实验表明，contour积分法的增益高度依赖于能否高效精确地求解具有低秩右端的（移位）矩阵方程。而LOBPCG灵活地直接使用预条件器的特点使其更易于受益于Khatri-Rao积结构，但代价是缺乏充分的理论依据。