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22
The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy ..."
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Cited by 149 (51 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 25 (8 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
Parallel computation of response time densities and quantiles in large Markov and semiMarkov models
, 2004
"... Response time quantiles reflect userperceived quality of service more accurately than mean or average response time measures. Consequently, online transaction processing benchmarks, telecommunications Service Level Agreements and emergency services legislation all feature stringent 90th percenti ..."
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Cited by 14 (8 self)
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Response time quantiles reflect userperceived quality of service more accurately than mean or average response time measures. Consequently, online transaction processing benchmarks, telecommunications Service Level Agreements and emergency services legislation all feature stringent 90th percentile response time targets. This thesis presents techniques and tools for extracting response time densities, quantiles and moments from largescale models of reallife systems. This work expands the applicability, capacity and specification power of prior work, which was hitherto focused on the analysis of Markov models which only support exponential delays. Response time densities or cumulative distribution functions of interest are computed by calculating and subsequently numerically inverting their Laplace transforms. We develop techniques for the extraction of response time measures from Generalised Stochastic Petri Nets (GSPNs) and SemiMarkov Stochastic Petri Nets (SMSPNs). The latter is our proposed modelling formalism for the highlevel specification of semiMarkov models which support generallydistributed delays.
Parabolic and hyperbolic contours for computing the Bromwich integral
 Math. Comp
"... Abstract. Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. H ..."
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Cited by 13 (2 self)
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Abstract. Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented. 1.
COMPUTING A α, log(A) AND RELATED MATRIX FUNCTIONS BY CONTOUR INTEGRALS
, 2007
"... Abstract. New methods are proposed for the numerical evaluation of f(A) or f(A)b, where f(A) is a function such as A 1/2 or log(A) with singularities in (−∞, 0] and A is a matrix with eigenvalues on or near (0, ∞). The methods are based on combining contour integrals evaluated by the periodic trapez ..."
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Cited by 10 (1 self)
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Abstract. New methods are proposed for the numerical evaluation of f(A) or f(A)b, where f(A) is a function such as A 1/2 or log(A) with singularities in (−∞, 0] and A is a matrix with eigenvalues on or near (0, ∞). The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of f(A)b is typically reduced to one or two dozen linear system solves, which can be carried out in parallel. Key words. Cauchy integral, conformal map, contour integral, matrix function, quadrature, rational approximation, trapezoid rule AMS subject classifications. 65F30, 65D30 1. Introduction. It
Fast and oblivious convolution quadrature
 SIAM J. Sci. Comput
, 2005
"... Abstract. We give an algorithm to compute N steps of a convolution quadrature approximation to a continuous temporal convolution using only O(N log N) multiplications and O(log N) active memory. The method does not require evaluations of the convolution kernel, but instead O(log N) evaluations of it ..."
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Cited by 7 (4 self)
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Abstract. We give an algorithm to compute N steps of a convolution quadrature approximation to a continuous temporal convolution using only O(N log N) multiplications and O(log N) active memory. The method does not require evaluations of the convolution kernel, but instead O(log N) evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the stable numerical solution with quasioptimal complexity of linear and nonlinear integral and integrodifferential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions.
Fast RungeKutta approximation of inhomogeneous parabolic equations
 NUMER. MATH
, 2005
"... The result after N steps of an implicit RungeKutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy ε, by solving only O log N log 1 ε linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm. ..."
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Cited by 6 (5 self)
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The result after N steps of an implicit RungeKutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy ε, by solving only O log N log 1 ε linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm.
Nonreflecting boundary conditions for Maxwell’s equations
 Computing
, 2003
"... A new discrete nonreflecting boundary condition for the timedependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite differenc ..."
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Cited by 5 (2 self)
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A new discrete nonreflecting boundary condition for the timedependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.
Adaptive, fast and oblivious convolution in evolution equations with memory
, 2006
"... Abstract. To approximate convolutions which occur in evolution equations with memory terms, a variablestepsize algorithm is presented for which advancing N steps requires only O(N log N) operations and O(log N) active memory, in place of O(N 2) operations and O(N) memory for a direct implementation ..."
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Cited by 3 (0 self)
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Abstract. To approximate convolutions which occur in evolution equations with memory terms, a variablestepsize algorithm is presented for which advancing N steps requires only O(N log N) operations and O(log N) active memory, in place of O(N 2) operations and O(N) memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blowup example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.