Fast collocation

Abstract

In this paper a new method to compute in a fast and reliable way the collocation solution is presented. In order to speed up the numerical procedures, some restrictions on input data are needed.

The basic assumption is that data are gridded and homogeneous; this implies that the autocovariance matrix entering in the collocation formula is of Toeplitz type. In particular, if observations are placed on a two dimensional planar grid, the autocovariance matrix is a symmetric block Toeplitz matrix and each block is itself a symmetric Toeplitz matrix (Toeplitz/Toeplitz structure). The analysis can be extended to a regular geographical grid, considered as a generalization of the planar one, taking into account the distortions on the Toeplitz/Toeplitz structure induced by the convergence of the meridians. The devised method is based on a combined application of the Preconditioned Conjugate Gradient Method and of the Fast Fourier Transform. This allows a proper exploitation of the Toeplitz/Toeplitz structure of the autocovariance matrix in computing the collocation solution.

The numerical tests proved that the application of this algorithm leads to a relevant decrease in CPU time if compared with standard methods used to solve a collocation problem (Cholesky, Levinson).