TY - JOUR

T1 - A parallel additive schwarz preconditioned jacobi-davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

AU - Hwang, Feng Nan

AU - Wei, Zih Hao

AU - Hwang, Tsung-Min

AU - Wang, Weichung

PY - 2010/4/20

Y1 - 2010/4/20

N2 - We develop a parallel Jacobi-Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi-Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc's efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schrödinger's equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi-Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.

AB - We develop a parallel Jacobi-Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi-Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc's efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schrödinger's equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi-Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.

KW - Jacobi-Davidson methods

KW - Parallel computing

KW - Polynomial eigenvalue problems

KW - Quantum dot simulation

KW - Restricted additive Schwarz preconditioning

KW - Schrödinger's equation

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U2 - 10.1016/j.jcp.2009.12.024

DO - 10.1016/j.jcp.2009.12.024

M3 - Article

AN - SCOPUS:78649447470

VL - 229

SP - 2932

EP - 2947

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 8

ER -