Parallel Efficiency of the Lanczos Method for Eigenvalue Problems

LBNL-42828 Parallel Efficiency of the Lanczos Method for Eigenvalue Problems ^ Kesheng Wu+and Horst Simon* Abstract Two of the commonly used versions of the Lanczos method for eigenvalues problems are the shift-and-invert Lanczos method and the restarted Lanczos method. In this talk, we will address two questions, is the shift-and-invert Lanczos method a viable option on massively parallel machines and which one is more appropriate for a given eigenvalue problem? Introduction T h i s talk is o n how to compute eigenvalues a n d eigenvectors of large sparse s y m m e t r i c matrices o n massively parallel machines. One of the the most c o m m o n l y used algorithms for this task is the Lanczos m e t h o d w h i c h projects the large eigenvalue p r o b l e m onto a low dimensional K r y l o v subspace [7, 10]. I n most cases, the K r y l o v subspace basis is built t h r o u g h a series of matrix-vector m u l t i p l i c a t i o n s , the whole Lanczos m e t h o d c a n be implemented w i t h a matrix-vector m u l t i p l i c a t i o n routine a n d a few simple vector operations. T h i s a l g o r i t h m is highly efficient on parallel machines a n d it is effective for c o m p u t i n g extreme a n d well separated eigenvalues. To compute the interior or not well-separated eigenvalues, the shift-and-invert Lanczos m e t h o d is one of the most effective methods. Since the eigenvalues of a m a t r i x A are related to the eigenvalues of (A — c r l ) by a simple relation X(A) = a + 1/X((A — c r l ) ) a n d the corresponding eigenvectors of A a n d (A — c r l ) are identical, the shift-and-invert scheme computes the extreme eigenvalues of (A — c r l ) a n d deduce the corresponding eigenvalues of A. W i t h appropriate choice of a, the extreme eigenvalues of (A — al)^ are well separated and can be easily computed. T h e shift-and-invert Lanczos m e t h o d needs to b u i l d a K r y l o v subspace basis of (A — c r l ) w h i c h is done by solving a series of linear systems involving the coefficient m a t r i x (A — a I). T h e linear systems need to be solved accurately a n d the only reliable means to accomplish this is by using a direct m e t h o d [5]. Because it is difficult to implement direct methods o n d i s t r i b u t e d parallel computers, efficient parallel i m p l e m e n t a t i o n has not been widely available u n t i l recently [1, 8]. T h i s talk w i l l present our s t u d y of the parallel efficiency of the shift-and-invert Lanczos m e t h o d using these newly available direct solvers. T h e study w i l l show that the shift-and-invert Lanczos m e t h o d is ' T h i s w o r k was s u p p o r t e d b y t h e D i r e c t o r , Office o f E n e r g y R e s e a r c h , Office o f L a b o r a t o r y P o l i c y a n d Infrastructure M a n a g e m e n t , of the U . S . D e p a r t m e n t of E n e r g y under C o n t r a c t N o . D E - A C 0 3 - 7 6 S F 0 0 0 9 8 . T h i s r e s e a r c h u s e d r e s o u r c e s o f t h e N a t i o n a l E n e r g y R e s e a r c h S c i e n t i f i c C o m p u t i n g C e n t e r , w h i c h is s u p p o r t e d b y t h e Office o f E n e r g y R e s e a r c h o f t h e U . S . D e p a r t m e n t o f E n e r g y . 'Lawrence Berkeley National Laboratory/NERSC, Berkeley, CA Email: {ksni, hdsimon}81bl.gov.