Anasazi software for the numerical solution of large-scale eigenvalue problems

Baker, Christopher G.; Hetmaniuk, Ulrich; Lehoucq, Richard B.; Thornquist, Heidi K.

doi:10.1145/1527286.1527287

Cited by 99 publications

(89 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…[67,68,69] However, regarding eigenvalue solvers, the emphasis in the PETSc environment is currently placed on iterative eigenvalue solvers, e.g., in the Scalable Library for Eigenvalue Problem Computations (SLEPc) [70] which is based on PETSc. Many more iterative solvers for large-scale sparse eigenvalue problems are available, e.g., via the Anasazi package [71] in the Trilinos software project [72]. However, the focus of the present paper is on direct methods.…”

Section: Recent Developments In Parallel Eigenvalue Solversmentioning

confidence: 99%

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

Marek

Blüm

Johanni

et al. 2014

J. Phys.: Condens. Matter

250

261

View full text Add to dashboard Cite

Abstract. Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N 3 ) with the size of the investigated problem, N (e.g., the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the ELPA library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the ScaLAPACK library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g., Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPIOpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem sizes arising in the electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a ELPA -Eigenvalue Solutions for Electronic Structure Theory 2 matrix of dimension 260,000, scalability up to 295,000 CPU cores has been shown on BlueGene/P.

show abstract

Section: Recent Developments In Parallel Eigenvalue Solversmentioning

confidence: 99%

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

Marek

Blüm

Johanni

et al. 2014

J. Phys.: Condens. Matter

250

261

View full text Add to dashboard Cite

show abstract

“…The main parallel libraries that provide eigensolvers similar to ours are summarized in Table 2.1: Anasazi [Baker et al, 2009], that includes a customizable block Generalized Davidson (that with a little coding can behave like a basic Jacobi-Davidson) and LOBPCG; BLOPEX [Knyazev et al, 2007], that provides a reference implementation of LOBPCG; and PRIMME [Stathopoulos and McCombs, 2010], that includes many methods that fit in the Generalized Davidson scheme, although currently without support for generalized problems. As Table 2.1 shows, none of the mentioned libraries support non-Hermitian problems.…”

Section: Implementation Of Davidson Methods In Slepcmentioning

confidence: 99%

Parallel Implementation

Trobec¹,

Kosec²

2015

Parallel Scientific Computing

View full text Add to dashboard Cite

ResumenEl problema de valores propios (también llamado de autovalores) está presente en multitud dé areas científicas a través de, por ejemplo, la resolución de ecuaciones en derivadas parciales, reducción de modelos y cálculo de funciones matriciales, entre otras muchas aplicaciones. Si los problemas son de dimensión moderada (menor a 10 6 ), pueden ser abordados mediante los llamados métodos directos, como el algoritmo iterativo QR o el método de divide y vencerás. Sin embargo, si el problema es de gran dimensión y sólo se requieren unas pocas soluciones, comparado con el tamaño del problema, los métodos iterativos pueden resultar más eficientes. Además, los métodos iterativos pueden ofrecer mejor rendimiento en arquitecturas de altas prestaciones, como las plataformas paralelas de memoria distribuida, en las que existe un cierto número de nodos computacionales con espacio de memoria propio y sólo pueden compartir información y sincronizarse mediante el paso de mensajes.Esta tesis aborda la implementación de métodos de tipo Davidson, destacando Generalized Davidson y Jacobi-Davidson, una clase de métodos iterativos que pueden ser competitivos en casos especialmente difíciles como calcular valores propios en el interior del espectro o cuando la factorización de matrices es prohibitiva o ineficiente, y sólo es posible una factorización aproximada. La implementación se desarrolla en SLEPc (Scalable Library for Eigenvalue Problem Computations), librería de software libre para la resolución de problemas de gran tamaño de valores propios, problemas cuadráticos de valores propios y problemas de valores singulares, entre otros.Como resultado de esta tesis, SLEPc incorpora una implementación de Generalized Davidson y Jacobi-Davidson para problemas estándares y generalizados, tanto hermitianos como no hermitianos, característica a destacar ya que los problemas no hermitianos no están soportados en otras librerías libres y paralelas con métodos de Davidson. Además de incorporar mejoras en la convergencia de los métodos, como la extracción de pares propios mediante el método de Rayleigh-Ritz armónico, se han realizado otras optimizaciones para mejorar el rendimiento, como evitar la aritmética compleja cuando las matrices son reales (aunque los pares propios puedan tener parte imaginaria) o agrupar las operaciones a bloques para aprovechar la memoria caché. Además se ha presentado un método nuevo para expandir el subespacio, llamado GD2, que ofrece mejores resultados en comparación con Generalized Davidson cuando el precondicionador está muy alejado del ideal.La estabilidad numérica y el rendimiento computacional de la implementación se han evaluado mediante una batería de problemas procedentes de aplicaciones reales, y comparado con otras librerías como PRIMME o Anasazi.Finalmente se presenta la integración de la implementación en dos aplicaciones científicas. Por un lado, Jacobi-Davidson ha mejorado las prestaciones de los cálculos de valores propios y los estudios multiparamétricos que aparecen en GENE, un código que re...

show abstract

“…We used SLEPc for our initial implementation, which we describe and for which we present benchmarking results in this paper. Other suitable eigensolver packages exist as well, such as Anasazi [9], and we may evaluate and benchmark some of these other packages for our application in the future.…”

Section: Methodsmentioning

confidence: 99%

Benchmarking parallel eigen decomposition for residuals analysis of very large graphs

Rutledge

Miller

Beard

2012

2012 IEEE Conference on High Performance Extreme Computing

View full text Add to dashboard Cite

Abstract-Graph analysis is used in many domains, from the social sciences to physics and engineering. The computational driver for one important class of graph analysis algorithms is the computation of leading eigenvectors of matrix representations of a graph. This paper explores the computational implications of performing an eigen decomposition of a directed graph's symmetrized modularity matrix using commodity cluster hardware and freely available eigensolver software, for graphs with 1 million to 1 billion vertices, and 8 million to 8 billion edges. Working with graphs of these sizes, parallel eigensolvers are of particular interest. Our results suggest that graph analysis approaches based on eigen space analysis of graph residuals are feasible even for graphs of these sizes.

show abstract

Anasazi software for the numerical solution of large-scale eigenvalue problems

Cited by 99 publications

References 24 publications

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science

Parallel Implementation

Benchmarking parallel eigen decomposition for residuals analysis of very large graphs

Contact Info

Product

Resources

About