1,626 research outputs found
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Computation of the expected value of a function of a chi-distributed random variable
We consider the problem of numerically evaluating the expected value of a
smooth bounded function of a chi-distributed random variable, divided by the
square root of the number of degrees of freedom. This problem arises in the
contexts of simultaneous inference, the selection and ranking of populations
and in the evaluation of multivariate t probabilities. It also arises in the
assessment of the coverage probability and expected volume properties of the
some non-standard confidence regions. We use a transformation put forward by
Mori, followed by the application of the trapezoidal rule. This rule has the
remarkable property that, for suitable integrands, it is exponentially
convergent. We use it to create a nested sequence of quadrature rules, for the
estimation of the approximation error, so that previous evaluations of the
integrand are not wasted. The application of the trapezoidal rule requires the
approximation of an infinite sum by a finite sum. We provide a new easily
computed upper bound on the error of this approximation. Our overall conclusion
is that this method is a very suitable candidate for the computation of the
coverage and expected volume properties of non-standard confidence regions
Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
In this work, we extend the fractional linear multistep methods in [C.
Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional
integral and derivative operators in the sense that the tempered fractional
derivative operator is interpreted in terms of the Hadamard finite-part
integral. We develop two fast methods, Fast Method I and Fast Method II, with
linear complexity to calculate the discrete convolution for the approximation
of the (tempered) fractional operator. Fast Method I is based on a local
approximation for the contour integral that represents the convolution weight.
Fast Method II is based on a globally uniform approximation of the trapezoidal
rule for the integral on the real line. Both methods are efficient, but
numerical experimentation reveals that Fast Method II outperforms Fast Method I
in terms of accuracy, efficiency, and coding simplicity. The memory requirement
and computational cost of Fast Method II are and ,
respectively, where is the number of the final time steps and is the
number of quadrature points used in the trapezoidal rule. The effectiveness of
the fast methods is verified through a series of numerical examples for
long-time integration, including a numerical study of a fractional
reaction-diffusion model
Fast spectral source integration in black hole perturbation calculations
This paper presents a new technique for achieving spectral accuracy and fast
computational performance in a class of black hole perturbation and
gravitational self-force calculations involving extreme mass ratios and generic
orbits. Called \emph{spectral source integration} (SSI), this method should see
widespread future use in problems that entail (i) point-particle description of
the small compact object, (ii) frequency domain decomposition, and (iii) use of
the background eccentric geodesic motion. Frequency domain approaches are
widely used in both perturbation theory flux-balance calculations and in local
gravitational self-force calculations. Recent self-force calculations in Lorenz
gauge, using the frequency domain and method of extended homogeneous solutions,
have been able to accurately reach eccentricities as high as . We
show here SSI successfully applied to Lorenz gauge. In a double precision
Lorenz gauge code, SSI enhances the accuracy of results and makes a factor of
three improvement in the overall speed. The primary initial application of
SSI--for us its \emph{raison d'\^{e}tre}--is in an arbitrary precision
\emph{Mathematica} code that computes perturbations of eccentric orbits in the
Regge-Wheeler gauge to extraordinarily high accuracy (e.g., 200 decimal
places). These high accuracy eccentric orbit calculations would not be possible
without the exponential convergence of SSI. We believe the method will extend
to work for inspirals on Kerr, and will be the subject of a later publication.
SSI borrows concepts from discrete-time signal processing and is used to
calculate the mode normalization coefficients in perturbation theory via sums
over modest numbers of points around an orbit. A variant of the idea is used to
obtain spectral accuracy in solution of the geodesic orbital motion.Comment: 15 pages, 7 figure
Finite element implementation of state variable-based viscoplasticity models
The implementation of state variable-based viscoplasticity models is made in a general purpose finite element code for structural applications of metals deformed at elevated temperatures. Two constitutive models, Walker's and Robinson's models, are studied in conjunction with two implicit integration methods: the trapezoidal rule with Newton-Raphson iterations and an asymptotic integration algorithm. A comparison is made between the two integration methods, and the latter method appears to be computationally more appealing in terms of numerical accuracy and CPU time. However, in order to make the asymptotic algorithm robust, it is necessary to include a self adaptive scheme with subincremental step control and error checking of the Jacobian matrix at the integration points. Three examples are given to illustrate the numerical aspects of the integration methods tested
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