Complex moment-based methods for differential eigenvalue problems

This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on a "solve-then-discretize" paradigm, an operator analogue of the FEAST method has been proposed for DEPs without discretization of the coefficient operators. Compared to conventional "discretize-then-solve" approaches that discretize the operators and solve the resulting matrix problem, the operator analogue of FEAST exhibits much higher accuracy; however, it involves solving a large number of ordinary differential equations (ODEs). In this paper, to reduce the computational costs, we propose operation analogues of Sakurai-Sugiura-type complex moment-based eigensolvers for DEPs using higher-order complex moments and analyze the error bound of the proposed methods. We show that the number of ODEs to be solved can be reduced by a factor of the degree of complex moments without degrading accuracy, which is verified by numerical results. Numerical results demonstrate that the proposed methods are over five times faster compared with the operator analogue of FEAST for several DEPs while maintaining almost the same high accuracy. This study is expected to promote the "solve-then-discretize" paradigm for solving DEPs and contribute to faster and more accurate solutions in real-world applications.Comment: 26 pages, 9 figure

Strategies for spectrum slicing based on restarted Lanczos methods

In the context of symmetric-definite generalized eigenvalue problems, it is often required to compute all eigenvalues contained in a prescribed interval. For large-scale problems, the method of choice is the so-called spectrum slicing technique: a shift-and-invert Lanczos method combined with a dynamic shift selection that sweeps the interval in a smart way. This kind of strategies were proposed initially in the context of unrestarted Lanczos methods, back in the 1990's. We propose variations that try to incorporate recent developments in the field of Krylov methods, including thick restarting in the Lanczos solver and a rational Krylov update when moving from one shift to the next. We discuss a parallel implementation in the SLEPc library and provide performance results. © 2012 Springer Science+Business Media, LLC.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under grant TIN2009-07519.Campos González, MC.; Román Moltó, JE. (2012). Strategies for spectrum slicing based on restarted Lanczos methods. Numerical Algorithms. 60(2):279-295. https://doi.org/10.1007/s11075-012-9564-z279295602Amestoy, P.R, Duff, I.S., L’Excellent, J.Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184(2–4), 501–520 (2000)Balay, S., Brown, J., 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 3.2, Argonne National Laboratory (2011)Ericsson, T., Ruhe, A.: The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comput. 35(152), 1251–1268 (1980)Grimes, R.G., Lewis, J.G., Simon, H.D.: A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems. SIAM J. Matrix Anal. Appl. 15(1), 228–272 (1994)Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31(3), 351–362 (2005)Hernandez, V., Roman, J.E., Tomas, A.: Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33(7–8), 521–540 (2007)Marques, O.A.: BLZPACK: description and user’s guide. Tech. Rep. TR/PA/95/30, CERFACS, Toulouse, France (1995)Meerbergen, K.: Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem. Numer. Linear Algebra Appl. 8(1), 33–52 (2001)Meerbergen, K., Scott, J.: The design of a block rational Lanczos code with partial reorthogonalization and implicit restarting. Tech. Rep. RAL-TR-2000-011, Rutherford Appleton Laboratory (2000)Nour-Omid, B., Parlett, B.N., Ericsson, T., Jensen, P.S.: How to implement the spectral transformation. Math. Comput. 48(178), 663–673 (1987)Olsson, K.H.A., Ruhe, A.: Rational Krylov for eigenvalue computation and model order reduction. BIT Numer. Math. 46, 99–111 (2006)Ruhe, A.: Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl. 58, 391–405 (1984)Ruhe, A.: Rational Krylov subspace method. In: 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, pp. 246–249. Philadelphia (2000)Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)Stewart, G.W.: A Krylov–Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601–614 (2001)Vidal, AM., Garcia, V.M., Alonso, P., Bernabeu, M.O.: Parallel computation of the eigenvalues of symmetric Toeplitz matrices through iterative methods. J. Parallel Distrib. Comput. 68(8), 1113–1121 (2008)Wu, K., Simon, H.: Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22(2), 602–616 (2000)Zhang, H., Smith, B., Sternberg, M., Zapol, P.: SIPs: Shift-and-invert parallel spectral transformations. ACM Trans. Math. Softw. 33(2), 1–19 (2007