The idea of preconditioning is usually associated with solution techniques for solving linear systems or eigenvalue problems. It refers to a general method by which the original system is transformed into one which admits the same solution but which is easier to solve. Following this principle we consider in this paper techniques for preconditioning the matrix exponential operator, e A y 0, using different approximations of the matrix A. These techniques are based on using generalized Runge Kutta type methods. Preconditioners based on the sparsity structure of the matrix, such as diagonal, block diagonal, and least-squares tensor sum approximations are presented. Numerical experiments are reported to compare the quality of the schemes introduced.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Castillo, P., and Saad, Y. (1997). Tensor Sum Approximation Preconditioners. Proc. Eighth SIAM Conference on Parallel Processing for Scientific Computing.
Chartier, P., and Philippe, B. (1997). Solution of Markov Processes by Waveform Relaxation Methods. Technical Report, IRISA, University of Rennes, France.
Ehle, B. L., and Lawson, J. D. (1975). Generalized Runge-Kutta processes for stiff initial-value problems. J. Inst. Maths. Appl. 16, 11-21.
Friesner, R. A., Tuckerman, L. S., Dornblaser, B. C., and Russo, T. U. (1989). A method for exponential propagation of large systems of stiff nonlinear differential equations. J. Sci. Comput. 4, 327-354.
Gallopoulos, E., and Saad, Y. (1992). Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput. 13(5), 1236-1264.
Hairer, E., Norsett, S. P., and Wanner, G. (1993). Solving Ordinary Differential Equations I, 2nd Ed., Springer-Verlag, Berlin.
Hochbruck, M., and Lubich, C. (1997). On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911-1925.
Hochbruck, M., Lubich, C., and Selhofer, H. (1998). Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552-1574.
Knizhnerman, L. A. (1991). Computations of functions of unsymmetric matrices by means of Arnoldi's method. J. Comput. Math. Math. Phys. 31(1), 5-16.
Lawson, J. D. (1967). Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4(3), 372-380.
Lawson, J. D. (1972). Some Numerical Methods for Stiff Ordinary and Partial Differential Equations. Proc. Sec. Manitoba Conference on Numerical Mathematics, pp. 27-34.
Lawson, J. D., and Swayne, D. A. (1980). High-order near best uniform approximations to the solution of heat conduction problems. Proc. IFIP Cong. 80, 741-746.
Lawson, J. D., and Swayne, D. A. (1986). Reduction of matrix factorizations in solvers for stiff ordinary differential equations. Cong. Numer. (52), 147-152.
Nour-Omid, B. (1989). Applications of the Lanczos algorithm. Comput. Phys. Commun. (53), 153-168.
Moler, C. B., and VanLoan, C. F. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20(4), 801-836.
Saad, Y. (1992). Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209-228.
Saad, Y. (1990). SPARSKIT: A Basic Tool Kit for Sparse Matrix Computations. Technical Report 90-20, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffet Field, California.
About this article
Cite this article
Castillo, P., Saad, Y. Preconditioning the Matrix Exponential Operator with Applications. Journal of Scientific Computing 13, 275–302 (1998). https://doi.org/10.1023/A:1023219016301
- Exponential operator
- generalized Runge Kutta methods