7,098 research outputs found
Quadrature methods for highly oscillatory singular integrals
We address the evaluation of highly oscillatory integrals, with power-law and logarithmic singularities. Such problems arise in numerical methods in engineering. Notably, the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering, and many of these exhibit singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion achieves an error that decays faster for increasing frequency. Based on the asymptotic analysis, a Filon-type method is constructed to approximate the integral. Unlike an asymptotic expansion, the Filon method achieves high accuracy for both small and large frequency. Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared. However, while it achieves higher accuracy for the same number of function evaluations, it requires significant additional cost of computation of orthogonal polynomials and their zeros
Asymptotic expansions and fast computation of oscillatory Hilbert transforms
In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table
Fast Computation of Fourier Integral Operators
We introduce a general purpose algorithm for rapidly computing certain types
of oscillatory integrals which frequently arise in problems connected to wave
propagation and general hyperbolic equations. The problem is to evaluate
numerically a so-called Fourier integral operator (FIO) of the form at points given on
a Cartesian grid. Here, is a frequency variable, is the
Fourier transform of the input , is an amplitude and
is a phase function, which is typically as large as ;
hence the integral is highly oscillatory at high frequencies. Because an FIO is
a dense matrix, a naive matrix vector product with an input given on a
Cartesian grid of size by would require operations.
This paper develops a new numerical algorithm which requires operations, and as low as in storage space. It operates by
localizing the integral over polar wedges with small angular aperture in the
frequency plane. On each wedge, the algorithm factorizes the kernel into two components: 1) a diffeomorphism which is
handled by means of a nonuniform FFT and 2) a residual factor which is handled
by numerical separation of the spatial and frequency variables. The key to the
complexity and accuracy estimates is that the separation rank of the residual
kernel is \emph{provably independent of the problem size}. Several numerical
examples demonstrate the efficiency and accuracy of the proposed methodology.
We also discuss the potential of our ideas for various applications such as
reflection seismology.Comment: 31 pages, 3 figure
Windowed Green Function Method for Nonuniform Open-Waveguide Problems
This contribution presents a novel Windowed Green Function (WGF) method for
the solution of problems of wave propagation, scattering and radiation for
structures which include open (dielectric) waveguides, waveguide junctions, as
well as launching and/or termination sites and other nonuniformities. Based on
use of a "slow-rise" smooth-windowing technique in conjunction with free-space
Green functions and associated integral representations, the proposed approach
produces numerical solutions with errors that decrease faster than any negative
power of the window size. The proposed methodology bypasses some of the most
significant challenges associated with waveguide simulation. In particular the
WGF approach handles spatially-infinite dielectric waveguide structures without
recourse to absorbing boundary conditions, it facilitates proper treatment of
complex geometries, and it seamlessly incorporates the open-waveguide character
and associated radiation conditions inherent in the problem under
consideration. The overall WGF approach is demonstrated in this paper by means
of a variety of numerical results for two-dimensional open-waveguide
termination, launching and junction problems.Comment: 16 Page
Boundary integral methods in high frequency scattering
In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources
- ā¦