13,348 research outputs found
A semi-analytical scheme for highly oscillatory integrals over tetrahedra
This is the peer reviewed version of the following article: [Hospital-Bravo, R., Sarrate, J., and DÃez, P. (2017) A semi-analytical scheme for highly oscillatory integrals over tetrahedra. Int. J. Numer. Meth. Engng, 111: 703–723. doi: 10.1002/nme.5474], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5474/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.This paper details a semi-analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the Partition of Unity Method (PUM) enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non-oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared to standard Gauss-Legendre quadratures.Peer ReviewedPostprint (author's final draft
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
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
GMRES for oscillatory matrix-valued differential equations
We investigate the use of Krylov subspace methods to solve linear, oscillatory ODEs. When we apply a Krylov subspace method to a properly formulated equation, we retain the asymptotic accuracy of the asymptotic expansion whilst converging to the exact solution. We will demonstrate the effectiveness of this method by computing Error and Mathieu functions
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
High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing
Wave propagation and scattering problems in acoustics are often solved with
boundary element methods. They lead to a discretization matrix that is
typically dense and large: its size and condition number grow with increasing
frequency. Yet, high frequency scattering problems are intrinsically local in
nature, which is well represented by highly localized rays bouncing around.
Asymptotic methods can be used to reduce the size of the linear system, even
making it frequency independent, by explicitly extracting the oscillatory
properties from the solution using ray tracing or analogous techniques.
However, ray tracing becomes expensive or even intractable in the presence of
(multiple) scattering obstacles with complicated geometries. In this paper, we
start from the same discretization that constructs the fully resolved large and
dense matrix, and achieve asymptotic compression by explicitly localizing the
Green's function instead. This results in a large but sparse matrix, with a
faster associated matrix-vector product and, as numerical experiments indicate,
a much improved condition number. Though an appropriate localisation of the
Green's function also depends on asymptotic information unavailable for general
geometries, we can construct it adaptively in a frequency sweep from small to
large frequencies in a way which automatically takes into account a general
incident wave. We show that the approach is robust with respect to non-convex,
multiple and even near-trapping domains, though the compression rate is clearly
lower in the latter case. Furthermore, in spite of its asymptotic nature, the
method is robust with respect to low-order discretizations such as piecewise
constants, linears or cubics, commonly used in applications. On the other hand,
we do not decrease the total number of degrees of freedom compared to a
conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure
On the computation of confluent hypergeometric functions for large imaginary part of parameters b and z
The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-319-42432-3_30We present an efficient algorithm for the confluent hypergeometric functions when the imaginary part of b and z is large. The algorithm is based on the steepest descent method, applied to a suitable representation of the confluent hypergeometric functions as a highly oscillatory integral, which is then integrated by using various quadrature methods. The performance of the algorithm is compared with open-source and commercial software solutions with arbitrary precision, and for many cases the algorithm achieves high accuracy in both the real and imaginary parts. Our motivation comes from the need for accurate computation of the characteristic function of the Arcsine distribution or the Beta distribution; the latter being required in several financial applications, for example, modeling the loss given default in the context of portfolio credit risk.Peer ReviewedPostprint (author's final draft
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