Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices

Abstract

AbstractWe present a method of almost linear complexity to approximate some (inner) eigenvalues of symmetric self‐adjoint integral or differential operators. Using ℋ‐arithmetic the discretisation of the operator leads to a large hierarchical (ℋ‐) matrix M. We assume that M is symmetric, positive definite. Then we compute the smallest eigenvalues by the locally optimal block preconditioned conjugate gradient method (LOBPCG), which has been extensively investigated by Knyazev and Neymeyr.Hierarchical matrices were introduced by W. Hackbusch in 1998. They are data‐sparse and require only O(nlog2 n) storage. There is an approximative inverse, besides other matrix operations, within the set of ℋ‐matrices, which can be computed in linear‐polylogarithmic complexity. We will use the approximative inverse as preconditioner in the LOBPCG method.Further we combine the LOBPCG method with the folded spectrum method to compute inner eigenvalues of M. This is equivalent to the application of LOBPCG to the matrix Mμ = (M − μI)2. The matrix Mμ is symmetric, positive definite, too.Numerical experiments illustrate the behavior of the suggested approach. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)