3,950 research outputs found
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
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
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
Integral methods for conical diffraction
The paper is devoted to the scattering of a plane wave obliquely
illuminating a periodic surface. Integral equation methods lead to a system
of singular integral equations over the profile. Using boundary integral
techniques we study the equivalence of these equations to the electromagnetic
formulation, the existence and uniqueness of solutions under general
assumptions on the permittivity and permeability of the materials. In
particular, new results for materials with negative permittivity or
permeability are established
Scattering of general incident beams by diffraction gratings
The paper is devoted to the electromagnetic scattering of arbitrary time-harmonic fields by periodic structures. The Floquet-Fourier transform converts the full space Maxwell problem to a two-parameter family of diffraction problems with quasiperiodic incidence waves, for which conventional grating methods become applicable. The inverse transform is given by integrating with respect to the parameters over a infinite strip in ℝ². For the computation of the scattered fields we propose an algorithm, which extends known adaptive methods for the approximate calculation of multiple integrals. The novel adaptive approach provides autonomously the expansion of the incident field into quasiperiodic waves in order to approximate the scattered fields within a prescribed error tolerance. Some application examples are numerically examined
Integral equations for conical diffraction by coated gratings
The paper is devoted to integral formulations for the scattering of plane waves by diffraction gratings under oblique incidence. For the case of coated gratings Maxwell's equations can be reduced to a system of four singular integral equations on the piecewise smooth interfaces between different materials. We study analytic properties of the integral operators for periodic diffraction problems and obtain existence and uniqueness results for solutions of the systems corresponding to electromagnetic fields with locally finite energy
Conical diffraction by multilayer gratings: A recursive integral equations approach
In this paper we consider an integral equation algorithm to study
the scattering of plane waves by multilayer diffraction gratings under oblique incidence.
The scattering problem is described by a system of Helmholtz equations with piecewise
constant coefficients in coupled by special transmission conditions at the interfaces between different
layers. Boundary integral methods lead to a system of singular
integral equations, containing at least two equations for each interface.
To deal with an arbitrary number of material layers we present the extension of
a recursive procedure developed by Maystre for normal incidence, which transforms the problem
to a sequence of equations with operator matrices on each interface.
Necessary and sufficient conditions for the applicability of the algorithm are derived
Approximate Approximations and their Applications
This paper gives a survey of an approximation method which was proposed by V. Maz'ya as underlying procedure for numerical algorithms to solve initial and boundary value problems of mathematical physics. Due to a greater flexibility in the choice of approximating functions it allows efficient approximations of multi-dimensional integral operators often occuring in applied problems. Its application especially in connection with integral equation methods is very promising, which has been proved already for different classes of evolution equations. The survey describes some basic results concerning error estimates for quasi-interpolation and cubature of integral operators with singular kernels as well as a multiscale and wavelet approach to approximate those operators over bounded domains. Finally a general numerical method for solving nonlocal nonlinear evolution equations is presented
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