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Author: Lanzara, Flavia × Subject: higher dimensions ×

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    Fast cubature of volume potentials over rectangular domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Lanzara, Flavia; Maz' ya, Vladimir; Schmidt, Gunther
    In the present paper we study high-order cubature formulas for the computation of advection-diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order O(h6) up to dimension 108.
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    A fast solution method for time dependent multidimensional Schrödinger equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Lanzara, Flavia; Mazya, Vladimir; Schmidt, Gunther
    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. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.