1,058 research outputs found
Efficient computation of highly oscillatory integrals by using QTT tensor approximation
We propose a new method for the efficient approximation of a class of highly
oscillatory weighted integrals where the oscillatory function depends on the
frequency parameter , typically varying in a large interval. Our
approach is based, for fixed but arbitrary oscillator, on the pre-computation
and low-parametric approximation of certain -dependent prototype
functions whose evaluation leads in a straightforward way to recover the target
integral. The difficulty that arises is that these prototype functions consist
of oscillatory integrals and are itself oscillatory which makes them both
difficult to evaluate and to approximate. Here we use the quantized-tensor
train (QTT) approximation method for functional -vectors of logarithmic
complexity in in combination with a cross-approximation scheme for TT
tensors. This allows the accurate approximation and efficient storage of these
functions in the wide range of grid and frequency parameters. Numerical
examples illustrate the efficiency of the QTT-based numerical integration
scheme on various examples in one and several spatial dimensions.Comment: 20 page
Modified Filon-Clenshaw-Curtis rules for oscillatory integrals with a nonlinear oscillator
Filon-Clenshaw-Curtis rules are among rapid and accurate quadrature rules for
computing highly oscillatory integrals. In the implementation of the
Filon-Clenshaw-Curtis rules in the case when the oscillator function is not
linear, its inverse should be evaluated at some points. In this paper, we solve
this problem by introducing an approach based on the interpolation, which leads
to a class of modifications of the original Filon-Clenshaw-Curtis rules. In the
absence of stationary points, two kinds of modified Filon-Clenshaw-Curtis rules
are introduced. For each kind, an error estimate is given theoretically, and
then illustrated by some numerical experiments. Also, some numerical
experiments are carried out for a comparison of the accuracy and the efficiency
of the two rules. In the presence of stationary points, the idea is applied to
the composite Filon-Clenshaw-Curtis rules on graded meshes. An error estimate
is given theoretically, and then illustrated by some numerical experiments
A Survey Of Numerical Quadrature Methods For Highly Oscillatory Integrals
In this thesis, we examine the main types of numerical quadrature methods for a special subclass of one-dimensional highly oscillatory integrals. Along with a presentation of the methods themselves and the error bounds, the thesis contains implementations of the methods in Maple and Python. The implementations take advantage of the symbolic computational abilities of Maple and allow for a larger class of problems to be solved with greater ease to the user. We also present a new variation on Levin integration which uses differentiation matrices in various interpolation bases
An extended Filon--Clenshaw--Curtis method for high-frequency wave scattering problems in two dimensions
We study the efficient approximation of integrals involving Hankel functions
of the first kind which arise in wave scattering problems on straight or convex
polygonal boundaries. Filon methods have proved to be an effective way to
approximate many types of highly oscillatory integrals, however finding such
methods for integrals that involve non-linear oscillators and
frequency-dependent singularities is subject to a significant amount of ongoing
research. In this work, we demonstrate how Filon methods can be constructed for
a class of integrals involving a Hankel function of the first kind. These
methods allow the numerical approximation of the integral at uniform cost even
when the frequency is large. In constructing these Filon methods we
also provide a stable algorithm for computing the Chebyshev moments of the
integral based on duality to spectral methods applied to a version of Bessel's
equation. Our design for this algorithm has significant potential for further
generalisations that would allow Filon methods to be constructed for a wide
range of integrals involving special functions. These new extended Filon
methods combine many favourable properties, including robustness in regard to
the regularity of the integrand and fast approximation for large frequencies.
As a consequence, they are of specific relevance to applications in wave
scattering, and we show how they may be used in practice to assemble
collocation matrices for wavelet-based collocation methods and for hybrid
oscillatory approximation spaces in high-frequency wave scattering problems on
convex polygonal shapes
The numerical steepest descent path method for calculating physical optics integrals on smooth conducting quadratic surfaces
published_or_final_versio
- …