1,479 research outputs found
Fast, numerically stable computation of oscillatory integrals with stationary points
We present a numerically stable way to compute oscillatory integrals of the form . For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane
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
A numerical method for oscillatory integrals with coalescing saddle points
The value of a highly oscillatory integral is typically determined
asymptotically by the behaviour of the integrand near a small number of
critical points. These include the endpoints of the integration domain and the
so-called stationary points or saddle points -- roots of the derivative of the
phase of the integrand -- where the integrand is locally non-oscillatory.
Modern methods for highly oscillatory quadrature exhibit numerical issues when
two such saddle points coalesce. On the other hand, integrals with coalescing
saddle points are a classical topic in asymptotic analysis, where they give
rise to uniform asymptotic expansions in terms of the Airy function. In this
paper we construct Gaussian quadrature rules that remain uniformly accurate
when two saddle points coalesce. These rules are based on orthogonal
polynomials in the complex plane. We analyze these polynomials, prove their
existence for even degrees, and describe an accurate and efficient numerical
scheme for the evaluation of oscillatory integrals with coalescing saddle
points
Computing the Hilbert transform and its inverse
We construct a new method for approximating Hilbert transforms and their inverse throughout the complex plane. Both problems can be formulated as Riemann-Hilbert problems via Plemelj's lemma. Using this framework, we re-derive existing approaches for computing Hilbert transforms over the real line and unit interval, with the added benefit that we can compute the Hilbert transform in the complex plane. We then demonstrate the power of this approach by generalizing to the half line. Combining two half lines, we can compute the Hilbert transform of a more general class of functions on the real line than is possible with existing methods
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
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
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