Abstract

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows us to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.

Keywords

  1. generalized eigenproblem
  2. FEAST
  3. quadrature
  4. Zolotarev
  5. filter design
  6. load balancing

MSC codes

  1. 65F15
  2. 41A20
  3. 65Y05

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References

1.
M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions, Verlag Harri Deutsch, Thun, Switzerland, 1984.
2.
N. I. Akhiezer, Elements of the Theory of Elliptic Functions, AMS, Providence, RI, 1990.
3.
A. P. Austin, P. Kravanja, and L. N. Trefethen, Numerical algorithms based on analytic function values at roots of unity, SIAM J. Numer. Anal., 52 (2014), pp. 1795--1821.
4.
K. J. Bathe, Convergence of subspace iteration, in Formulations and Computational Algorithms in Finite Element Analysis, K. J. Bathe, J. T. Oden, and W. Wunderlich, eds., MIT Press, Cambridge, MA, 1977, pp. 575--598.
5.
B. Beckermann, D. Kressner, and C. Tobler, An error analysis of Galerkin projection methods for linear systems with tensor product structure, SIAM J. Numer. Anal., 51 (2013), pp. 3307--3326.
6.
H. Blinchikoff and A. Zverev, Filtering in the Time and Frequency Domains, John Wiley & Sons, New York, 1976.
7.
W. Cauer, Ein Interpolationsproblem mit Funktionen mit positivem Realteil, Math. Z., 38 (1934), pp. 1--44.
8.
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, New York, 1984.
9.
L. Delves and J. Lyness, A numerical method for locating the zeros of an analytic function, Math. Comp., 21 (1967), pp. 543--560.
10.
V. Druskin, S. Güttel, and L. Knizhnerman, Near-optimal perfectly matched layers for indefinite Helmholtz problems, SIAM Rev., to appear.
11.
E. Di Napoli, E. Polizzi, and Y. Saad, Efficient estimation of eigenvalue counts in an interval, preprint, arXiv:1308.4275, 2013.
12.
FEAST Eigenvalue Solver, 2009--2015. http://www.feast-solver.org/ (2015).
13.
M. Galgon, L. Krämer, and B. Lang, The FEAST algorithm for large eigenvalue problems, PAMM, 11 (2011), pp. 747--748.
14.
M. Galgon, L. Krämer, J. Thies, A. Basermann, and B. Lang, On the Parallel Iterative Solution of Linear Systems Arising in the FEAST Algorithm for Computing Inner Eigenvalues, Technical report BUW-IMACM 14/35, Bergische Universität Wuppertal, Germany, 2014.
15.
B. Gavin and E. Polizzi, Non-linear eigensolver-based alternative to traditional SCF methods, J. Chem. Phys., 138 (2013), 194101.
16.
S. Güttel, Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection, GAMM-Mitt. Ges. Angew. Math. Mech., 36 (2013), pp. 8--31.
17.
R. W. Hamming, Digital Filters, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1983.
18.
T. Ikegami and T. Sakurai, Contour integral eigensolver for non-Hermitian systems: A Rayleigh--Ritz-type approach, Taiwanese J. Math., 14 (2010), pp. 825--837.
19.
T. Ikegami, T. Sakurai, and U. Nagashima, A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method, J. Comput. Appl. Math., 233 (2008), pp. 1927--1936.
20.
D. Ingerman, V. Druskin, and L. Knizhnerman, Optimal finite difference grids and rational approximations of the square root. I. Elliptic problems, Commun. Pure Appl. Anal., 53 (2000), pp. 1039--1066.
21.
L. Krämer, E. Di Napoli, M. Galgon, B. Lang, and P. Bientinesi, Dissecting the FEAST algorithm for generalized eigenproblems, J. Comput. Appl. Math., 244 (2013), pp. 1--9.
22.
V. I. Lebedev, On a Zolotarev problem in the method of alternating directions, Comput. Math. Math. Phys., 17 (1977), pp. 58--76.
23.
A. Levin, D. Zhang, and E. Polizzi, Feast fundamental framework for electronic structure calculations: Reformulation and solution of the muffin-tin problem, Comput. Phys. Comm., 183 (2012), pp. 2370--2375.
24.
J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, FL, 2003.
25.
A. A. Medovikov and V. I. Lebedev, Variable time steps optimization of $L_\omega$ stable Crank--Nicolson method, Russian J. Numer. Anal. Math. Modelling, 20 (2005), pp. 283--304.
26.
H. Murakami, Experiments of Filter Diagonalization Method for Real Symmetric Definite Generalized Eigenproblems by the Use of Elliptic Filters, Technical report 2010-HPC-125, Information Processing Society of Japan, 2010 (in Japanese).
27.
H. Murakami, Filter designs for the symmetric eigenproblems to solve eigenpairs whose eigenvalues are in the specified interval, IPSJ Trans. Adv. Comput. Systems, 3 (2010), pp. 1--21 (in Japanese).
28.
P. P. Petrushev and V. A. Popov, Rational Approximation of Real Functions, Encyclopedia Math. Appl. 28, Cambridge University Press, Cambridge, 2010.
29.
E. Polizzi, Density-matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B (3), 79 (2009), 115112.
30.
E. Polizzi and J. Kestyn, FEAST Eigenvalue Solver v3.0 User Guide, preprint, arXiv:1203.4031, 2015.
31.
Y. Saad, Numerical Methods for Large Eigenvalue Problems, Halsted Press, New York, 1992.
32.
J. Sabino, Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method, Ph.D. thesis, Rice University, Houston, TX, 2006.
33.
T. Sakurai, Y. Futamura, and H. Tadano, Efficient parameter estimation and implementation of a contour integral-based eigensolver, Algorithms Comput. Technol., 7 (2013), pp. 249--270.
34.
T. Sakurai, Y. Kodaki, H. Tadano, D. Takahashi, M. Sato, and U. Nagashima, A parallel method for large sparse generalized eigenvalue problems using a GridRPC system, Future Gener. Comp. Systems, 24 (2008), pp. 613--619.
35.
T. Sakurai and H. Sugiura, A projection method for generalized eigenvalue problems using numerical integration, J. Comput. Appl. Math., 159 (2003), pp. 119--128.
36.
T. Sakurai and H. Tadano, CIRR: A Rayleigh--Ritz type method with contour integral for generalized eigenvalue problems, Hokkaido Math. J., 36 (2007), pp. 745--757.
37.
P. T. P. Tang and E. Polizzi, FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 354--390.
38.
J. Todd, Applications of transformation theory: A legacy from Zolotarev (1847--1878), in Approximation Theory and Spline Functions, S. P. Singh, ed., D. Reidel, Dordrecht, Netherlands, 1984, pp. 207--245.
39.
L. N. Trefethen and J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Rev., 56 (2014), pp. 385--458.
40.
M. Van Valkenburg, Analog Filter Design, Holt, Rinehart and Winston, New York, 1982.
41.
G. Viaud, The FEAST Method, M.Sc. dissertation, University of Oxford, Oxford, England, 2012.
42.
E. L. Wachspress, The ADI minimax problem for complex spectra, Appl. Math. Lett., 1 (1988), pp. 311--314.
43.
D. Zhang and E. Polizzi, Efficient modeling techniques for atomistic-based electronic density calculations, J. Comput. Electron, 7 (2008), pp. 427--431.
44.
E. I. Zolotarev, Application of elliptic functions to questions of functions deviating least and most from zero, Zap. Imp. Akad. Nauk St. Petersburg, 30 (1877), pp. 1--59 (in Russian).

Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A2100 - A2122
ISSN (online): 1095-7197

History

Submitted: 30 July 2014
Accepted: 8 June 2015
Published online: 18 August 2015

Keywords

  1. generalized eigenproblem
  2. FEAST
  3. quadrature
  4. Zolotarev
  5. filter design
  6. load balancing

MSC codes

  1. 65F15
  2. 41A20
  3. 65Y05

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