A parallel additive Schwarz preconditioned Jacobi–Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

Hwang, Feng Nan; Wei, Zih-Hao; Huang, Tsung-Ming; Wang, Weichung

doi:10.1016/j.jcp.2009.12.024

Cited by 27 publications

(23 citation statements)

References 44 publications

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“…Results are summarized in Table 1. The first problems arise in the computation of the electronic structure of quantum dots via discretization of the Schrödinger equation [19]. The rest belong to the NLEVP collection [8], all of them polynomial eigenproblems except the last one (loaded string) which is a rational eigenproblem that we have used to illustrate how general nonlinear eigenproblems can be solved via polynomial interpolation.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Projection methods for large-scale polynomial eigenvalue problems may opt to project the problem directly by imposing a Galerkin-type condition on the residual associated with the polynomial eigenproblem, see, e.g., [18,19,5]. An alternative approach is to apply a standard projection method such as Arnoldi to the linearization of the matrix polynomial.…”

mentioning

confidence: 99%

“…The computational experiments of §5 were carried out on the supercomputer Tirant at Universitat de València. The matrices from the quantum dot simulation used in §5 were kindly provided by the authors of [19].…”

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confidence: 99%

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Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

Campos¹,

Román²

2016

SIAM J. Sci. Comput.

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Abstract. Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial, but exploit the block structure with the aim of being memory-efficient in the representation of the Krylov subspace basis. The problem may appear in the form of a low-degree polynomial (quartic or quintic, say) expressed in the monomial basis, or a high-degree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a non-monomial basis. We have implemented a parallel solver in SLEPc covering both cases, that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart, and illustrate the robustness and performance of the solver with some numerical experiments.

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Section: Numerical Resultsmentioning

confidence: 99%

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confidence: 99%

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Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

Campos¹,

Román²

2016

SIAM J. Sci. Comput.

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“…Table 1 summarizes the test cases, providing information about the degree of the polynomial, the matrix size, the number of requested eigenpairs and the target value around which eigenvalues are sought. The first problems arise in the computation of the electronic structure of quantum dots via discretization of the Schrödinger equation [13]. The rest belong to the NLEVP collection [5].…”

Section: Computational Resultsmentioning

confidence: 99%

Parallel iterative refinement in polynomial eigenvalue problems

Campos

Román

2016

Numerical Linear Algebra App

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Methods for the polynomial eigenvalue problem sometimes need to be followed by an iterative refinement process to improve the accuracy of the computed solutions. This can be accomplished by means of a Newton iteration tailored to matrix polynomials. The computational cost of this step is usually higher than the cost of computing the initial approximations, due to the need of solving multiple linear systems of equations with a bordered coefficient matrix. An effective parallelization is thus important, and we propose different approaches for the message-passing scenario. Some schemes use a subcommunicator strategy in order to improve the scalability whenever direct linear solvers are used. We show performance results for the various alternatives implemented in the context of SLEPc, the Scalable Library for Eigenvalue Problem Computations.

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“…Considerando un vector w ortogonal al residuo, w * P (µ)u = 0, y multiplicando la ecuación anterior por w * , se obtiene una expresión para ∆λ, que al ser sustituida en (1.21), y dado que t = (I − uu * )t, produce la ecuación de corrección del método de Jacobi-Davidson (1.20). Una de las opciones que con más frecuencia se escogen para el subespacio de test [16,91,64,48,46] es la de hacerlo coincidir con el de búsqueda (proyección ortogonal). Otras opciones para W dan distintas propiedades de convergencia [27,83,81].…”

Section: Jacobi-davidsonunclassified

Implementación paralela de métodos iterativos para la resolución de problemas polinómicos de valores propios

González¹,

Carmén²

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Quiero expresar mi más profundo agradecimiento a José Román, por su excelente trabajo como director y por su dedicación y valiosa aportación a esta tesis. Su gran entrega como investigador ha sido para mí un ejemplo y un estímulo constantes. En segundo lugar, me gustaría mencionar al profesor Daniel Kressner ya que algunos de sus trabajos han sido fuente de inspiración en mayor o menor medida de ideas que considero relevantes para esta tesis. También quiero manifestar mi agradecimiento a Andrés Tomás y Eloy Romero por sus contribuciones a SLEPc, algunas de las cuales han servido de referencia para los desarrollos software realizados en esta tesis. Mi reconocimiento al Ministerio de Educación Cultura y Deporte por la beca otorgada para la realización de esta tesis, y a la Red Española de Supercomputación por los recursos computacionales proporcionados para este trabajo. En elámbito personal, me gustaría agradecer, de nuevo a José Román, el permitirme entrar a formar parte del proyecto SLEPc; a mi compañero de equipo Alejandro Lamas, sus constantes muestras deánimo y su sentido del humor; a los compañeros de Departamento, Fernando Alvarruiz y David Guerrero, la buena disposición que siempre han mostrado para escucharme y ayudarme; finalmente a mi familia, su apoyo continuado e incondicional.

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A parallel additive Schwarz preconditioned Jacobi–Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

Cited by 27 publications

References 44 publications

Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

Parallel iterative refinement in polynomial eigenvalue problems

Implementación paralela de métodos iterativos para la resolución de problemas polinómicos de valores propios

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