Flavia Lanzara

ON THE FAST COMPUTATION OF HIGH DIMENSIONAL VOLUME POTENTIALS

A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate ...

Approximate approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.

Approximate Hermite quasi-interpolation

In this article, we derive approximate quasi-interpolants when the values of a function u and of some of its derivatives are prescribed at the points of a uniform grid. As a byproduct of these formulas we obtain very simple approximants, which provide high-order approximations for solutions to elliptic differential equations with constant coefficients.

Approximate approximations with data on a perturbed uniform grid

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to the case that the data are given on a perturbed uniform grid. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible.

Cubature of Integral Operators by Approximate Quasi-interpolation

In this paper we report on some recent results concerning Hermite quasi-interpolation on uniform grids with interesting applications to the approximation of solutions to elliptic PDE, quasi-interpolation on nonuniform grids and the cubature of convolutions with radial kernel functions based on an approximation method proposed by V. Maz’ya.