59 research outputs found
On the limit configuration of four species strongly competing systems
We analysed some qualitative properties of the limit configuration of the
solutions of a reaction-diffusion system of four competing species as the
competition rate tends to infinity. Large interaction induces the spatial
segregation of the species and only two limit configurations are possible:
either there is a point where four species concur, a 4-point, or there are two
points where only three species concur. We characterized, for a given datum,
the possible 4-point configuration by means of the solution of a Dirichlet
problem for the Laplace equation
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
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
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
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
On the computation of high-dimensional potentials of advection-diffusion operators
We study a fast method for computing potentials of advection-diffusion operators
with and over rectangular boxes in R^n. By combining high-order cubature formulas with modern methods of structured tensor product approximations we derive an approximation of the potentials which is accurate and provides approximation formulas of high-order. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. The action of volume potentials on the basis functions allows one-dimensional integral representations with separable integrands i.e. a product of functions depending only on one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Since only one-dimensional operations are used, the resulting method is effective also in high-dimensional case
Geometry of the limiting solution of a strongly competing system
We report on known results on the geometry of the limiting solutions of a reaction-diffusion system in any number of competing species k as the competition rate m tends to infinity. The case k=8 is studied in detail. We provide numerical simulations of solutions of the system for k=4,6,8 and large competition rate. Thanks to FreeFEM++ software, we obtain nodal partitions showing the predicted limiting configurations
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A fast solution method for time dependent multidimensional Schrödinger equations
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
A fast solution method for time dependent multidimensional Schrödinger equations
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
Approximation of solutions to multidimensional parabolic equations by approximate approximations
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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