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Author: Lanzara, Flavia ×

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    Approximation of solutions to multidimensional parabolic equations by approximate approximations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lanzara, Flavia; Mazya, Vladimir; Schmidt, Gunther
    We propose a fast method for high order approximations of the solution of n-dimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, makes our method effective also in very high dimensions.We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 also in high dimensions.
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    Tensor product approximations of high dimensional potentials
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Lanzara, Flavia; Mazʾya, Vladimir; Schmidt, Gunther
    The paper is devoted to the efficient computation of high-order cubature formulas for volume potentials obtained within the framework of approximate approximations. We combine this approach with modern methods of structured tensor product approximations. Instead of performing high-dimensional discrete convolutions the cubature of the potentials can be reduced to a certain number of one-dimensional convolutions leading to a considerable reduction of computing resources. We propose one-dimensional integral representions of high-order cubature formulas for n-dimensional harmonic and Yukawa potentials, which allow low rank tensor product approximations
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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.