52,830 research outputs found
Fast cubature of volume potentials over rectangular domains
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 up to dimension
.Comment: 17 page
Accurate computation of the high dimensional diffraction potential over hyper-rectangles
We propose a fast method for high order approximation of potentials of the
Helmholtz 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 one-dimensional
integrals with separable integrands. Then a separated representation of the
density, combined with a suitable quadrature rule, leads to a tensor product
representation of the integral operator. Numerical tests show that these
formulas are accurate and provide approximations of order 6 up to dimension 100
and kappa^2=100
Tensor product approximations of high dimensional potentials
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.Comment: 20 page
Fast cubature of high dimensional biharmonic potential based on Approximate Approximations
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
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
- …