Abstract
In the numerical solution of large-scale eigenvalue problems, Davidson-type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson-type methods, the factorization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi-Davidson is one of the most effective variants. However, parallel implementations of this method are not widely available, particularly for non-symmetric problems. We present a parallel implementation to be released in SLEPc, the Scalable Library for Eigenvalue Problem Computations, and test it in the context of a highly scalable plasma turbulence simulation code. We analyze its parallel efficiency and compare it with Krylov-type eigensolvers.
This work was partially supported by the Spanish Ministerio de Ciencia e Innovación under project TIN2009-07519.
Chapter PDF
Similar content being viewed by others
References
Arbenz, P., Becka, M., Geus, R., Hetmaniuk, U., Mengotti, T.: On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Computing 32(2), 157–165 (2006)
Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.): Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Society for Industrial and Applied Mathematics, Philadelphia (2000)
Baker, C.G., Hetmaniuk, U.L., Lehoucq, R.B., Thornquist, H.K.: Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Transactions on Mathematical Software 36(3) 13:1–13:23 (2009)
Balay, S., Buschelman, K., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.0.0, Argonne National Laboratory (2008)
Bollhöfer, M., Notay, Y.: JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices. Computer Physics Communications 177(12), 951–964 (2007)
Dannert, T., Jenko, F.: Gyrokinetic simulation of collisionless trapped-electron mode turbulence. Physics of Plasmas 12(7), 072309 (2005)
Fokkema, D.R., Sleijpen, G.L.G., van der Vorst, H.A.: Jacobi–Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM Journal on Scientific Computing 20(1), 94–125 (1999)
Hernandez, V., Roman, J.E., Tomas, A.: Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Computing 33(7-8), 521–540 (2007)
Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Transactions on Mathematical Software 31(3), 351–362 (2005)
Jacobi, C.G.J.: Über ein leichtes Verfahren die in der Theorie der Säculärstörungen vorkommenden Gleichungen numerisch aufzulösen. Crelle’s J. 30, 51–94 (1846)
Morgan, R.B.: Computing interior eigenvalues of large matrices. Linear Algebra and its Applications, 154–156, 289–309 (1991)
Paige, C.C., Parlett, B.N., van der Vorst, H.A.: Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications 2(2), 115–133 (1995)
Roman, J.E., Kammerer, M., Merz, F., Jenko, F.: Fast eigenvalue calculations in a massively parallel plasma turbulence code. In: Parallel Computing (to appear, 2010), doi:10.1016/j.parco.2009.12.001
Romero, E., Cruz, M.B., Roman, J.E., Vasconcelos, P.B.: A Jacobi-Davidson parallel implementation for unsymmetric eigenproblems. In: Vector and Parallel Processing – VECPAR 2010 (to appear, 2010)
Sleijpen, G.L.G., Booten, A.G.L., Fokkema, D.R., van der Vorst, H.A.: Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numerical Mathematics 36(3), 595–633 (1996)
Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM Journal on Matrix Analysis and Applications 17(2), 401–425 (1996)
Sleijpen, G.L.G., van der Vorst, H.A., Meijerink, E.: Efficient expansion of subspaces in the Jacobi–Davidson method for standard and generalized eigenproblems. Electronic Transactions on Numerical Analysis 7, 75–89 (1998)
Sleijpen, G., Fokkema, D.: BiCGstab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum. Electronic Transactions on Numerical Analysis 1, 11–32 (1993)
Stathopoulos, A., McCombs, J.R.: PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description. Tech. Rep. WM-CS-2006-08, College of William & Mary (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Romero, E., Roman, J.E. (2010). A Parallel Implementation of the Jacobi-Davidson Eigensolver and Its Application in a Plasma Turbulence Code. In: D’Ambra, P., Guarracino, M., Talia, D. (eds) Euro-Par 2010 - Parallel Processing. Euro-Par 2010. Lecture Notes in Computer Science, vol 6272. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15291-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-15291-7_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15290-0
Online ISBN: 978-3-642-15291-7
eBook Packages: Computer ScienceComputer Science (R0)