On approximate inverse of Hermite and Laguerre collocation differentiation matrices and new collocation schemes in unbounded domains

In this paper, we provide an explicit, stable and fast means to compute the approximate inverse of Hermite/Laguerre collocation differentiation matrices, and also the approximate inverse of the Hermite/Laguerre collocation matrices of a second-order differential operator. The latter offers optimal preconditioners for developing well-conditioned Hermite/Laguerre collocation schemes. We apply the new approaches to various partial differential equations in unbounded domains and demonstrate the advantages over the usual collocation methods.