ITLApplied  Computational Mathematics Division
ACMD Seminar Series
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Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) software package

Andrew Knyazev
University of Colorado Denver

Thursday, March 6, 2008 11:30-12:30,
Building 101, Lecture Room D
Thursday, March 6, 2008 09:30-10:30,
Room 5000


Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) [2] is a public package, written in C, that at present includes only one eigenxolver, Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG). BLOPEX supports parallel computations through an abstract layer. BLOPEX is incorporated in the HYPRE package from LLNL and is available as an external block to the PETSc package from ANL as well as a stand-alone serial library. Hypre and PETSc packages provide high quality multigrid and domain decomposition preconditioning on parallel clusters with distributed or shared memory architecture.

The LOBPCG method, suggested and developed in [1] in the past decade, recently attracts an increasing attention as a potential alternative to the shift-and-invert Lanczos and preconditioned Davidson methods due to its simplicity, robustness and fast convergence. Several MATLAB, C, C++ and FORTRAN implementations of the LOBPCG are developed by different groups, e. g., for such applications areas as structured mechanics and electronic structure calculations [3].

Main LOBPCG features: a matrix-free iterative method for computing several extreme eigenpairs of symmetric positive generalized eigenproblems; a user-defined preconditioner; robustness with respect to random initial approximations, variable preconditioners, and ill-conditioning of the stiffness matrix; apparently optimal convergence speed. Numerical comparisons suggest that LOBPCG may be a genuine block analog for eigenproblems of the standard preconditioned conjugate gradient method for symmetric linear systems.

We present initial scalability results using BLOPEX with Hypre and PETSc on one BlueGene/L box solving eigenvalue problems of record sizes.

[1] A.V. Knyazev, Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method. SIAM Journal on Scientific Computing 23(2001): 517-541.

[2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov, Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc. SIAM Journal on Scientific Computing 25(2007): 2224-2239.

[3] F. Bottin, S. Leroux, A. Knyazev, G. Zerah, Large scale ab initio calculations based on three levels of parallelization. Computational Material Science (2007)

Speaker Bio: Andrew Knyazev (AK) graduated from the Applied Mathematics and Computer Sciences Department of the Moscow State University in 1981, and obtained his Ph.D. in Numerical Mathematics at the Institute of Numerical Mathematics Russian Academy of Sciences (INM RAS) in 1985. He worked as a software engineer at the Kurchatov's Institute of Atomic Energy 81-83, a researcher at the INM RAS 83-92, a visitor at the Courant Institute of Mathematical Sciences, NYU, 92-94, and the University of Colorado Denver (UCD) since 94. AK authored over 50 papers and received research funding from the NSF and DOE, currently being supported by two grants from the Division of Mathematical Sciences NSF. AK obtained research and teaching awards at the UCD. AK served as a guest editor and on editorial boards of several international journals on numerical mathematics and organized a number of conferences and workshops on computational aspects of eigenvalue problems. AK's research interests are ranging from numerical solution of partial differential equations to electronic structure calculations in nanosciences and spectral clustering for data mining, but he is mostly known as an expert on eigenvalue problems in the numerical linear algebra community. AK is also involved in public software development, currently supervising the BLOPEX package.

Presentation Slides: PDF

Contact: W. F. Mitchell

Note: Visitors from outside NIST must contact Robin Bickel; (301) 975-3668; at least 24 hours in advance.

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