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

Feng Nan Hwang, Zih Hao Wei, Tsung-Min Hwang, Weichung Wang

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21 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)2932-2947
Number of pages16
JournalJournal of Computational Physics
Issue number8
Publication statusPublished - 2010 Apr 20



  • Jacobi-Davidson methods
  • Parallel computing
  • Polynomial eigenvalue problems
  • Quantum dot simulation
  • Restricted additive Schwarz preconditioning
  • Schrödinger's equation

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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