Abstract
We propose a time-exact Krylov-subspace-based method for solving large linear inhomogeneous systems of ODE (ordinary differential equations). The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the inhomogeneous source term, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method for the matrix exponential. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations. This conference paper is based on the preprint (Botchev, A block Krylov subspace time-exact solution method for linear ODE systems, Memorandum 1973, Department of Applied Mathematics, University of Twente, Enschede, 2012, http://eprints.eemcs.utwente.nl/21277/).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Botchev, M.A.: A block Krylov subspace time-exact solution method for linear ODE systems. Memorandum 1973, Department of Applied Mathematics, University of Twente, Enschede (2012). http://eprints.eemcs.utwente.nl/21277/
Botchev, M.A., Grimm, V., Hochbruck, M.: Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013)
Druskin, V.L., Knizhnerman, L.A.: Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Math. Math. Phys. 29(6), 112–121 (1989)
Druskin, V.L., Knizhnerman, L.A.: Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Linear Algebra Appl. 2, 205–217 (1995)
Druskin, V.L., Greenbaum, A., Knizhnerman, L.A.: Using nonorthogonal Lanczos vectors in the computation of matrix functions. SIAM J. Sci. Comput. 19(1), 38–54 (1998). doi:10.1137/S1064827596303661
Eiermann, M., Ernst, O.G., Güttel, S.: Deflated restarting for matrix functions. SIAM J. Matrix Anal. Appl. 32(2), 621–641 (2011). http://dx.doi.org/10.1137/090774665
Enright, W.H.: Continuous numerical methods for ODEs with defect control. J. Comput. Appl. Math. 125(1–2), 159–170 (2000). doi:10.1016/S0377-0427(00)00466-0. Numerical analysis 2000, vol. VI, Ordinary differential equations and integral equations
Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997)
Hochbruck, M., Niehoff, J.: Approximation of matrix operators applied to multiple vectors. Math. Comput. Simul. 79(4), 1270–1283 (2008)
Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010). doi:10.1017/S0962492910000048
in ’t Hout, K.J., Weideman, J.A.C.: A contour integral method for the Black-Scholes and Heston equations. SIAM J. Sci. Comput. 33(2), 763–785 (2011). doi:10.1137/090776081. http://dx.doi.org/10.1137/090776081
Lubich, C.: From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008). doi:10.4171/067. http://dx.doi.org/10.4171/067
Saad, Y.: Iterative Methods for Sparse Linear Systems. Book out of print (2000). www-users.cs.umn.edu/~saad/books.html
Shampine, L.F.: Solving ODEs and DDEs with residual control. Appl. Numer. Math. 52(1), 113–127 (2005)
Tal-Ezer, H.: Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989)
Tal-Ezer, H.: On restart and error estimation for Krylov approximation of w = f(A)v. SIAM J. Sci. Comput. 29(6), 2426–2441 (2007). doi:10.1137/040617868. http://dx.doi.org/10.1137/040617868
van den Eshof, J., Hochbruck, M.: Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006)
van der Vorst, H.A.: An iterative solution method for solving f(A)x = b, using Krylov subspace information obtained for the symmetric positive definite matrix A. J. Comput. Appl. Math. 18, 249–263 (1987)
van der Vorst, H.A.: Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, Cambridge (2003)
Acknowledgements
This work was supported by the Russian Federal Program “Scientific and scientific-pedagogical personnel of innovative Russia”, Grant 8500.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Botchev, M.A. (2014). Time-Exact Solution of Large Linear ODE Systems by Block Krylov Subspace Projections. In: Fontes, M., Günther, M., Marheineke, N. (eds) Progress in Industrial Mathematics at ECMI 2012. Mathematics in Industry(), vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-05365-3_55
Download citation
DOI: https://doi.org/10.1007/978-3-319-05365-3_55
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05364-6
Online ISBN: 978-3-319-05365-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)