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Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices
Author(s) -
Benner Peter,
Mach Thomas
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110360
Subject(s) - preconditioner , conjugate gradient method , mathematics , eigenvalues and eigenvectors , matrix (chemical analysis) , inverse , discretization , positive definite matrix , block (permutation group theory) , combinatorics , block matrix , operator (biology) , mathematical analysis , linear system , algorithm , geometry , physics , materials science , quantum mechanics , composite material , biochemistry , chemistry , repressor , transcription factor , gene
We 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( n log 2 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 M μ = ( M − μ I ) 2 . The matrix M μ is symmetric, positive definite, too. Numerical experiments illustrate the behavior of the suggested approach. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)