Approximation of Covariance Functions by Non-Positive Definite Functions.

Abstract : In some applications of collocation we face two serious drawbacks, frequent calculations of linear functionals operating on the covariance function, and the inversion of a large matrix, both causing much computer time. The frame of this work is an investigation how to avoid calculations of the exact covariance function and to replace it by some approximations. Three different kinds of approximating functions are studied, all of them being finite elements: the step function, the piecewise linear function and the cubic spline function. After stating the essential properties of covariance functions, its approximations are discussed for the distance dependent covariance function and its extension into space. (Author)