Ricerc@Sapienza

We propose a fast method for high order approximation of potentials of the Helm-holtz type operator (Delta+kappa^2) over hyper-rectangles in (R^n). By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of

We derive new formulas for the high dimensional biharmonic potential acting on Gaussians or Gaussians times special polynomials. These formulas can be used to construct accurate cubature formulas of an arbitrary high order which are fast and effective also in very high dimensions. Numerical tests show that the formulas are accurate and provide the predicted approximation rate O(h^8) up to the dimension 10^7.

We report here on some recent results obtained in collaboration with V. Maz'ya and G. Schmidt cite{LMS2017}.
We derive semi-analytic cubature formulas for the solution of the Cauchy problem for the Schrödinger equation which are fast and accurate also if the space dimension is greater than or equal to 3.
We follow ideas of the method of approximate approximations, which provides high-order semi-analytic cubature formulas for many important integral operators of mathematical physics.

In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations.