A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, PA.C. G. Baker, U. L. Hetmaniuk, R. B. Lehoucq, and H. K. Thornquist. 2009. Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Trans. Math. Softw. 36, 3, 13:1--13:23.S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, L. C. McInnes, B. Smith, and H. Zhang. 2011. PETSc users manual. Tech. Rep. ANL-95/11-Revision 3.2, Argonne National Laboratory.S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith. 1997. Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, Eds., Birkhaüser, 163--202.M. A. Brebner and J. Grad. 1982. Eigenvalues of Ax =λ Bx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process. Linear Algebra Appl. 43, 99--118.C. Campos, J. E. Roman, E. Romero, and A. Tomas. 2011. SLEPc users manual. Tech. Rep. DSICII/24/02 - Revision 3.2, D. Sistemes Informàtics i Computació, Universitat Politècnica de València. http://www.grycap.upv.es/slepc.T. Dannert and F. Jenko. 2005. Gyrokinetic simulation of collisionless trapped-electronmode turbulence. Phys. Plasmas 12, 7, 072309.E. R. Davidson. 1975. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17, 1, 87--94.T. A. Davis and Y. Hu. 2011. The University of Florida Sparse Matrix Collection. ACM Trans. Math. Softw. 38, 1, 1:1--1:25.H. C. Elman, A. Ramage, and D. J. Silvester. 2007. Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33, 2. Article 14.T. Ericsson and A. Ruhe. 1980. The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comp. 35, 152, 1251--1268.M. Ferronato, C. Janna, and G. Pini. 2012. Efficient parallel solution to large-size sparse eigenproblems with block FSAI preconditioning. Numer. Linear Algebra Appl. 19, 5, 797--815.D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst. 1998. Jacobi--Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20, 1, 94--125.M. A. Freitag and A. Spence. 2007. Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem. Electron. Trans. Numer. Anal. 28, 40--64.M. Genseberger. 2010. Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi-Davidson method for large scale eigenvalue problems. App. Numer. Math. 60, 11, 1083--1099.V. Hernandez, J. E. Roman, and A. Tomas. 2007. Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33, 7--8, 521--540.V. Hernandez, J. E. Roman, and V. Vidal. 2005. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31, 3, 351--362.V. Heuveline, B. Philippe, and M. Sadkane. 1997. Parallel computation of spectral portrait of large matrices by Davidson type methods. Numer. Algor. 16, 1, 55--75.M. E. Hochstenbach. 2005a. Generalizations of harmonic and refined Rayleigh-Ritz. Electron. Trans. Numer. Anal. 20, 235--252.M. E. Hochstenbach. 2005b. Variations on harmonic Rayleigh--Ritz for standard and generalized eigenproblems. Preprint, Department of Mathematics, Case Western Reserve University.M. E. Hochstenbach and Y. Notay. 2006. The Jacobi--Davidson method. GAMM Mitt. 29, 2, 368--382.F.-N. Hwang, Z.-H. Wei, T.-M. Huang, and W. Wang. 2010. A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation. J. Comput. Phys. 229, 8, 2932--2947.A. V. Knyazev. 2001. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 2, 517--541.A. V. Knyazev, M. E. Argentati, I. Lashuk, and E. E. Ovtchinnikov. 2007. Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in HYPRE and PETSc. SIAM J. Sci. Comput. 29, 5, 2224--2239.J. Kopal, M. Rozložník, M. Tuma, and A. Smoktunowicz. 2012. Rounding error analysis of orthogonalization with a non-standard inner product. Numer. Math. 52, 4, 1035--1058.D. Kressner. 2006. Block algorithms for reordering standard and generalized Schur forms. ACM Trans. Math. Softw. 32, 4, 521--532.R. B. Lehoucq, D. C. Sorensen, and C. Yang. 1998. ARPACK Users' Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA.Z. Li, Y. Saad, and M. Sosonkina. 2003. pARMS: a parallel version of the algebraic recursive multilevel solver. Numer. Linear Algebra Appl. 10, 5--6, 485--509.J. R. McCombs and A. Stathopoulos. 2006. Iterative validation of eigensolvers: a scheme for improving the reliability of Hermitian eigenvalue solvers. SIAM J. Sci. Comput. 28, 6, 2337--2358.F. Merz, C. Kowitz, E. Romero, J. E. Roman, and F. Jenko. 2012. Multi-dimensional gyrokinetic parameter studies based on eigenvalues computations. Comput. Phys. Commun. 183, 4, 922--930.R. B. Morgan. 1990. Davidson's method and preconditioning for generalized eigenvalue problems. J. Comput. Phys. 89, 241--245.R. B. Morgan. 1991. Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154--156, 289--309.R. B. Morgan and D. S. Scott. 1986. Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Statist. Comput. 7, 3, 817--825.R. Natarajan and D. Vanderbilt. 1989. A new iterative scheme for obtaining eigenvectors of large, real-symmetric matrices. J. Comput. Phys. 82, 1, 218--228.M. Nool and A. van der Ploeg. 2000. A parallel Jacobi--Davidson-type method for solving large generalized eigenvalue problems in magnetohydrodynamics. SIAM J. Sci. Comput. 22, 1, 95--112.J. Olsen, P. Jørgensen, and J. Simons. 1990. Passing the one-billion limit in full configuration-interaction (FCI) calculations. Chem. Phys. Lett. 169, 6, 463--472.C. C. Paige, B. N. Parlett, and H. A. van der Vorst. 1995. Approximate solutions and eigenvalue bounds from Krylov subspaces. Numer. Linear Algebra Appl. 2, 2, 115--133.E. Romero and J. E. Roman. 2011. Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi--Davidson eigensolver. Concur. Comput.: Pract. Exp. 23, 17, 2179--2191.Y. Saad. 1993. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 2, 461--469.G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. 1996. Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36, 3, 595--633.G. L. G. Sleijpen and H. A. van der Vorst. 1996. A Jacobi--Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 2, 401--425.G. L. G. Sleijpen and H. A. van der Vorst. 2000. A Jacobi--Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42, 2, 267--293.G. L. G. Sleijpen, H. A. van der Vorst, and E. Meijerink. 1998. Efficient expansion of subspaces in the Jacobi--Davidson method for standard and generalized eigenproblems. Electron. Trans. Numer. Anal. 7, 75--89.A. Stathopoulos. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part I: Seeking one eigenvalue. SIAM J. Sci. Comput. 29, 2, 481--514.A. Stathopoulos and J. R. McCombs. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part II: Seeking many eigenvalues. SIAM J. Sci. Comput. 29, 5, 2162--2188.A. Stathopoulos and J. R. McCombs. 2010. PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description. ACM Trans. Math. Softw. 37, 2, 21:1--21:30.A. Stathopoulos and Y. Saad. 1998. Restarting techniques for the (Jacobi-)Davidson symmetric eigenvalue methods. Electron. Trans. Numer. Anal. 7, 163--181.A. Stathopoulos, Y. Saad, and C. F. Fischer. 1995. Robust preconditioning of large, sparse, symmetric eigenvalue problems. J. Comput. Appl. Math. 64, 3, 197--215.A. Stathopoulos, Y. Saad, and K. Wu. 1998. Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods. SIAM J. Sci. Comput. 19, 1, 227--245.G. W. Stewart. 2001. Matrix Algorithms. Volume II: Eigensystems. SIAM, Philadelphia, PA.H. A. van der Vorst. 2002. Computational methods for large eigenvalue problems. In Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., Vol. VIII, Elsevier, 3--179.H. A. van der Vorst. 2004. Modern methods for the iterative computation of eigenpairs of matrices of high dimension. Z. Angew. Math. Mech. 84, 7, 444--451.T. van Noorden and J. Rommes 2007. Computing a partial generalized real Schur form using the Jacobi--Davidson method. Numer. Linear Algebra Appl. 14, 3, 197--215.T. D. Young, E. Romero, and J. E. Roman. 2013. Parallel finite element density functional computations exploiting grid refinement and subspace recycling. Comput. Phys. Commun. 184, 1, 66--72

Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver

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 that has been included 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 a Krylov-Schur eigensolver. © 2011 John Wiley and Sons, Ltd..The authors are indebted to Florian Merz for providing us with the test cases and for his useful suggestions. The authors acknowledge the computer resources provided by the Barcelona Supercomputing Center (BSC). This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519.Romero Alcalde, E.; Román Moltó, JE. (2011). Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver. Concurrency and Computation: Practice and Experience. 23:2179-2191. https://doi.org/10.1002/cpe.1740S2179219123Hochstenbach, M. E., & Notay, Y. (2009). Controlling Inner Iterations in the Jacobi–Davidson Method. SIAM Journal on Matrix Analysis and Applications, 31(2), 460-477. doi:10.1137/080732110Heuveline, V., Philippe, B., & Sadkane, M. (1997). Numerical Algorithms, 16(1), 55-75. doi:10.1023/a:1019126827697Arbenz, P., Bečka, M., Geus, R., Hetmaniuk, U., & Mengotti, T. (2006). On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Computing, 32(2), 157-165. doi:10.1016/j.parco.2005.06.005Genseberger, M. (2010). Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi–Davidson method for large scale eigenvalue problems. Applied Numerical Mathematics, 60(11), 1083-1099. doi:10.1016/j.apnum.2009.07.004Stathopoulos, A., & McCombs, J. R. (2010). PRIMME. ACM Transactions on Mathematical Software, 37(2), 1-30. doi:10.1145/1731022.1731031Baker, C. G., Hetmaniuk, U. L., Lehoucq, R. B., & Thornquist, H. K. (2009). Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Transactions on Mathematical Software, 36(3), 1-23. doi:10.1145/1527286.1527287Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Romero, E., Cruz, M. B., Roman, J. E., & Vasconcelos, P. B. (2011). A Parallel Implementation of the Jacobi-Davidson Eigensolver for Unsymmetric Matrices. High Performance Computing for Computational Science – VECPAR 2010, 380-393. doi:10.1007/978-3-642-19328-6_35Romero, E., & Roman, J. E. (2010). A Parallel Implementation of the Jacobi-Davidson Eigensolver and Its Application in a Plasma Turbulence Code. Lecture Notes in Computer Science, 101-112. doi:10.1007/978-3-642-15291-7_11Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen*). (1846). Journal für die reine und angewandte Mathematik (Crelles Journal), 1846(30), 51-94. doi:10.1515/crll.1846.30.51G. Sleijpen, G. L., & Van der Vorst, H. A. (1996). A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 17(2), 401-425. doi:10.1137/s0895479894270427Fokkema, D. R., Sleijpen, G. L. G., & Van der Vorst, H. A. (1998). Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils. SIAM Journal on Scientific Computing, 20(1), 94-125. doi:10.1137/s1064827596300073Morgan, R. B. (1991). Computing interior eigenvalues of large matrices. Linear Algebra and its Applications, 154-156, 289-309. doi:10.1016/0024-3795(91)90381-6Paige, C. C., Parlett, B. N., & van der Vorst, H. A. (1995). Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications, 2(2), 115-133. doi:10.1002/nla.1680020205Stathopoulos, A., Saad, Y., & Wu, K. (1998). Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods. SIAM Journal on Scientific Computing, 19(1), 227-245. doi:10.1137/s1064827596304162Sleijpen, G. L. G., Booten, A. G. L., Fokkema, D. R., & van der Vorst, H. A. (1996). Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numerical Mathematics, 36(3), 595-633. doi:10.1007/bf01731936Balay S Buschelman K Eijkhout V Gropp W Kaushik D Knepley M McInnes LC Smith B Zhang H PETSc users manual 2010Hernandez, V., Roman, J. E., & Tomas, A. (2007). Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Computing, 33(7-8), 521-540. doi:10.1016/j.parco.2007.04.004Dannert, T., & Jenko, F. (2005). Gyrokinetic simulation of collisionless trapped-electron mode turbulence. Physics of Plasmas, 12(7), 072309. doi:10.1063/1.1947447Roman, J. E., Kammerer, M., Merz, F., & Jenko, F. (2010). Fast eigenvalue calculations in a massively parallel plasma turbulence code. Parallel Computing, 36(5-6), 339-358. doi:10.1016/j.parco.2009.12.001Merz, F., & Jenko, F. (2010). Nonlinear interplay of TEM and ITG turbulence and its effect on transport. Nuclear Fusion, 50(5), 054005. doi:10.1088/0029-5515/50/5/054005Simoncini, V., & Szyld, D. B. (2002). Flexible Inner-Outer Krylov Subspace Methods. SIAM Journal on Numerical Analysis, 40(6), 2219-2239. doi:10.1137/s0036142902401074Morgan, R. B. (2002). GMRES with Deflated Restarting. SIAM Journal on Scientific Computing, 24(1), 20-37. doi:10.1137/s106482759936465

Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems : part I

In this paper we will show how the Jacobi-Davidson iterative method can be used to solve generalized eigenproblems. Similar ideas as for the standard eigenproblem are used, but the projections, that are required to reduce the given problem to a small manageable size, need more attention. We show that by proper choices for the projection operators quadratic convergence can be achieved. The advantage of our approach is that none of the involved operators needs to be inverted. It turns out that similar projections can be used for the iterative approximation of selected eigenvalues and eigenvectors of polynomial eigenvalue equations. This approach has already been used with great success for the solution of quadratic eigenproblems associated with acoustic problems

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods

A parallel Jacobi-Davidson method for solving generalized eigenvalue problems in linear magnetohydrodynamics

We study the solution of generalized eigenproblems generated by a model which is used for stability investigation of tokamak plasmas. The eigenvalue problems are of the form $A x = lambda B x$, in which the complex matrices $A$ and $B$ are block tridiagonal, and $B$ is Hermitian positive definite. The Jacobi-Davidson method appears to be an excellent method for parallel computation of a few selected eigenvalues, because the basic ingredients are matrix-vector products, vector updates and inner products. The method is based on solving projected eigenproblems of order typically less than 30. The computation of an approximate solution of a large system of linear equations is usually the most expensive step in the algorithm. By using a suitable preconditioner, only a moderate number of steps of an inner iteration is required in order to retain fast convergence for the JD process. Several preconditioning techniques are discussed. It is shown, that for our application, a proper preconditioner is a complete block LU decomposition, which can be used for the computation of several eigenpairs. Reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelizable preconditioning technique. Results obtained on 64 processing elements of both a Cray T3D and a T3E will be shown