483,640 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
Chord distribution functions of three-dimensional random media: Approximate first-passage times of Gaussian processes
The main result of this paper is a semi-analytic approximation for the chord
distribution functions of three-dimensional models of microstructure derived
from Gaussian random fields. In the simplest case the chord functions are
equivalent to a standard first-passage time problem, i.e., the probability
density governing the time taken by a Gaussian random process to first exceed a
threshold. We obtain an approximation based on the assumption that successive
chords are independent. The result is a generalization of the independent
interval approximation recently used to determine the exponent of persistence
time decay in coarsening. The approximation is easily extended to more general
models based on the intersection and union sets of models generated from the
iso-surfaces of random fields. The chord distribution functions play an
important role in the characterization of random composite and porous
materials. Our results are compared with experimental data obtained from a
three-dimensional image of a porous Fontainebleau sandstone and a
two-dimensional image of a tungsten-silver composite alloy.Comment: 12 pages, 11 figures. Submitted to Phys. Rev.
A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations
A scheme for approximating the kernel of the fractional -integral
by a linear combination of exponentials is proposed and studied. The scheme is
based on the application of a composite Gauss-Jacobi quadrature rule to an
integral representation of . This results in an approximation of in an
interval , with , which converges rapidly in the number
of quadrature nodes associated with each interval of the composite rule.
Using error analysis for Gauss-Jacobi quadratures for analytic functions, an
estimate of the relative pointwise error is obtained. The estimate shows that
the number of terms required for the approximation to satisfy a prescribed
error tolerance is bounded for all , and that is bounded
for , , and
Optimal approximation of convex curves by functions which are piecewise linear
AbstractIn this paper an efficient method is presented for solving the problem of approximation of convex curves by functions that are piecewise linear, in such a manner that the maximum absolute value of the approximation error is minimized. The method requires the curves to be convex on the approximation interval only. The boundary values of the approximation function can be either free or specified. The method is based on the property of the optimal solution to be such that each linear segment approximates the curve on its interval optimally while the optimal error is uniformly distributed among the linear segments of the approximation function. Using this method the optimal solution can be determined analytically to the full extent in certain cases, as it was done for functions x2 and x12. In general, the optimal solution has to be computed numerically following the procedure suggested in the paper. Using this procedure, optimal solutions were computed for functions sin x, tg x, and arc tg x. Optimal solutions to these functions were used in practical applications
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