Efficient computation of the exponential operator for large, sparse, symmetric matrices

In this paper we compare Krylov subspace methods with Chebyshev series expansion for approximating the matrix exponential operator on large, sparse, symmetric matrices. Experimental results upon negative-definite matrices with very large size, arising from (2D and 3D) FE and FD spatial discretization of linear parabolic PDEs, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques, especially when memory bounds do not allow the storage of all Ritz vectors. We discuss also sensitivity of Chebyshev convergence to extreme eigenvalue approximation, as well as reliability of various a priori and a posteriori error estimates for both methods.