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We propose two methods to find a proper shift parameter in theshift-and-invert method for computing matrix exponential matrix-vectorproducts. These methods are useful in the case of matrix exponential action hasto be computed for a number of vectors. The first method is based on thezero-order optimization of the mean residual norm for a given number of initialvectors. The second method processes the vectors one-by-one and estimates, foreach vector, the derivative of the residual norm as a function of the shiftparameter. The estimated derivative value is then used to update the shiftvalue for the next vector. To demonstrate the performance of the proposedmethods we perform numerical experiments for two-dimensional non-stationaryconvection-diffusion equation with discontinuous coefficients andtwo-dimensional anisotropic diffusion equation.

Practical shift choice in the shift-and-invert Krylov subspace evaluations of the matrix exponential