Preconditioning Lanczos approximations to the matrix exponential

The Lanczos method is an iterative procedure to compute an orthogonal basis for the Krylov subspace generated by a symmetric matrix $A$ and a starting vector $v$. An interesting application of this method is the computation of the matrix exponential $\exp(-\tau A)v$. This vector plays an important role in the solution of parabolic equations where $A$ results from some form of discretization of an elliptic operator. In the present paper we will argue that for these applications the convergence behavior of this method can be unsatisfactory. We will propose a modified method that resolves this by a simple preconditioned transformation at the cost of an inner-outer iteration. A priori error bounds are presented that are independent of the norm of $A$. This shows that the worst case convergence speed is independent of the mesh width in the spatial discretization of the elliptic operator. We discuss, furthermore, a posteriori error estimation and the tuning of the coupling between the inner and outer iteration. We conclude with several numerical experiments with the proposed method.