Cluster robustness of preconditioned gradient subspace iteration eigensolvers (Q2368744)

Consider the solution of the generalized eigenvalue problem \(Lx = \lambda Mx\), where both \(L\) and \(M\) are symmetric positive definite matrices. The smallest eigenvalue \(\lambda\) minimizes the generalized Rayleigh quotient \((Lx,x)/(Mx,x)\), a relation from which a gradient method for computing \(\lambda\) can be derived. Preconditioning in the sense of this paper amounts to replacing the standard scalar product \((\cdot,\cdot)\) by a weighted scalar product \((\cdot,\cdot)_{K^{-1}} = (K^{-1}\cdot,\cdot)\), where \(K\) is a suitably chosen symmetric positive definite matrix. This paper is concerned with subspace variants of the preconditioned gradient method, as discussed, by \textit{A. V. Knyazev} and \textit{K. Neymeyr} [Linear Algebra Appl. 358, No. 1--3, 95--114 (2003; Zbl 1037.65039)]. An important advantage of such subspace variants is that their convergence tends to be much less affected by clusters of eigenvalues, provided that the cluster as a whole is either captured or omitted by the subspace. The paper provides a theoretical foundation of this phenomenon. For this purpose, it turns out to be difficult to work with individual distances between the Ritz values and the desired eigenvalues. Instead, convergence bounds for the sum of these distances are derived. The main results of this paper present bounds which are cluster robust in the sense that they are essentially independent of inner distances, i.e., distances between the desired eigenvalues. Apart from establishing new results, the paper provides an excellent overview of some recent developments in the area of preconditioned eigenvalue solvers.