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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 remar ..."
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Cited by 212 (53 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...
On the Laguerre method for numerically inverting Laplace transforms
 INFORMS Journal on Computing
, 1996
"... The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 TricomiWidder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coeff ..."
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Cited by 43 (7 self)
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The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 TricomiWidder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (1) using our previously developed variant of the Fourierseries method to calculate the coefficients of the Laguerre generating function, (2) developing systematic methods for scaling, and (3) using Wynn’s ɛalgorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux’s theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples. Subject classifications: Mathematics, functions: Laplace transforms. Probability, distributions: calculation by transform inversion. Queues, algorithms: Laplace transform inversion.
2006), Groundwater age, life expectancy and transit time distributions in advectivedispersive systems: 1. Generalized reservoir theory, Adv
 Water Resour
"... We present a methodology for determining reservoir groundwater age and transit time probability distributions in a deterministic manner, considering advectivedispersive transport in steady velocity fields. In a first step, we propose to model the statistical distribution of groundwater age at aqui ..."
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Cited by 8 (3 self)
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We present a methodology for determining reservoir groundwater age and transit time probability distributions in a deterministic manner, considering advectivedispersive transport in steady velocity fields. In a first step, we propose to model the statistical distribution of groundwater age at aquifer scale by means of the classical advectiondispersion equation for a conservative and nonreactive tracer, associated to proper boundary conditions. The evaluated function corresponds to the density of probability of the random variable age, age being defined as the time elapsed since the water particles entered the aquifer. An adjoint backward model is introduced to characterize the life expectancy distribution, life expectancy being the time remaining before leaving the aquifer. By convolution of these two distributions, groundwater transit time distributions, from inlet to outlet, are fully defined for the entire aquifer domain. In a second step, an accurate and efficient method is introduced to simulate the transit time distribution at discharge zones. By applying the reservoir theory to advective–dispersive aquifer systems, we demonstrate that the discharge zone transit time distribution can be evaluated if the internal age probability distribution is known. The reservoir theory also applies to internal life expectancy probabilities yielding the recharge zone life expectancy distribution. Internal groundwater volumes are finally identified with respect to age and transit time. One and twodimensional theoretical examples are presented to illustrate the proposed mathematical models, and make inferences on the effect of aquifer structure and macro–dispersion on the distributions of age, life expectancy and transit time.
A Flexible Inverse Laplace Transform Algorithm and its Application
"... A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotientdifference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the qu ..."
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Cited by 7 (0 self)
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A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotientdifference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the quotientdifference schemes, the algorithm controls the dimension of the inverse Laplace transform approximation automatically. Application of the algorithm to the solute transport equations in porous media is explained in a general setting. Also, a numerical simulation is performed to show the accuracy and efficiency of the developed algorithm. Key words. Inverse Laplace transform, timeintegration, transport equation, porous media. AMS subject classfications. 65M60, 65Y20. 1
Asthenospheric viscosity and stress diffusion: A mechanism to explain correlated earthquakes and surface deformation in
 NE Japan, Geophys. J. Int
, 1990
"... A significant correlation is found, in both space and time, between the intraplate (land) and interplate (sea, thrust zone only) earthquakes in Tohoku, N E Japan that has persisted since the times of reliably reported events in AD 1600. The correlation peaks at a landlead of about 36 yr with an ave ..."
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Cited by 6 (1 self)
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A significant correlation is found, in both space and time, between the intraplate (land) and interplate (sea, thrust zone only) earthquakes in Tohoku, N E Japan that has persisted since the times of reliably reported events in AD 1600. The correlation peaks at a landlead of about 36 yr with an average correlation distance of 200 km, with the implication of an average strain migration rate of 5.6 km yrl. The correlation is highly significant (>99 per cent), both from formal statistics and from tests of random shuffles of the data. Additional analysis of the data, as a point process, confirms the results of the correlation analysis. The sharpness of the correlation peak, when compared to the individual times of occurrence of the land and sea events suggests a trigger mechanism. To explain the correlation, the general model of subductionrupturerebound is extended to include additional features; the buckling of the land plate from the force of the subducting slab, and the viscoelastic coupling of the plate to the underlying asthenosphere. A buckle produces a highstress region in the continental plate where earthquakes are more prone to occur, thus producing the spatial correlation
Inversion of the Laplace transform from the real axis using an adaptive iterative method
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Thermoelastic generation of ultrasound by linefocused laser irradiation
 Int. J. Solids Struct
"... A twodimensional theoretical model for the eld generated in the thermoelastic regime by linefocused laser illumination of a homogeneous, isotropic, linearly elastic halfspace is presented. The model accounts for the eects of thermal diusion and optical penetration, as well as the nite width and ..."
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Cited by 5 (2 self)
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A twodimensional theoretical model for the eld generated in the thermoelastic regime by linefocused laser illumination of a homogeneous, isotropic, linearly elastic halfspace is presented. The model accounts for the eects of thermal diusion and optical penetration, as well as the nite width and duration of the laser source. The model is obtained by solving the thermoelastic problem in plane strain, rather than by integrating available solutions for the pointsource, leading to a lower computational eort. The wellknown dipole model follows from appropriate limits. However, it is shown that, by simple elasticity arguments, the strength of the dipole can be related apriori to the heat input and certain material properties. The strength is found to be smaller than that of the dipoles equivalent to a buried source due to the eect of the free surface. This fact has been overlooked by some previous researchers. Excellent quantitative agreement with experimental observations provides validation for the model. Some representative results are presented to illustrate the generated eld and provide insight into the relevance of the dierent mechanisms taken into account in the model.
A Hybrid Finite ElementLaplace Transform Method for the Analysis of Transient Electromagnetic Scattering by anOverFilledCavity in theGround Plane
, 2008
"... Abstract. A hybrid finite elementLaplace transform method is implemented to analyze the time domain electromagnetic scattering induced by a 2D overfilled cavity embedded in the infinite ground plane. The algorithm divides the whole scattering domain into two, interior and exterior, subdomains. I ..."
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Abstract. A hybrid finite elementLaplace transform method is implemented to analyze the time domain electromagnetic scattering induced by a 2D overfilled cavity embedded in the infinite ground plane. The algorithm divides the whole scattering domain into two, interior and exterior, subdomains. In the interior subdomain which covers the cavity, the problem is solved via the finite element method. The problem is solved analytically in the exterior subdomain which slightly overlaps the interior subdomain and extends to the rest of the upper half plane. The use of the Laplace transform leads to an analytical link condition between the overlapping subdomains. The analytical link guides the selection of the overlapping zone and eliminates the need to use the conventional Schwartz iteration. This dramatically improves the efficiency for solving transient scattering problems. Numerical solutions are tested favorably against analytical ones for a canonical geometry. The perfect link over the artificial boundary between the finite element approximation in the interior and analytical solution in the exterior further indicates the reliability of the method. An error analysis is also performed.
2009), ‘Laplace transformation method for the Black–Scholes equations
 Int. J. Numer. Anal. Model
"... Abstract. In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the BlackScholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very ..."
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Abstract. In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the BlackScholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various option prices. Existence and uniqueness properties of the Laplace transformed BlackScholes equation are analyzed. Also a transparent boundary condition associated with the Laplace transformation method is proposed. Several numerical results for various options under various situations confirm the efficiency, convergence and parallelization property of the proposed scheme.
Approximating Response Time Distributions
 Performance Evaluation Review
, 1989
"... : The response time is the most visible performance index to users of computer systems. Endusers see individual response times, not the average. Therefore the distribution of response times is important in performance evaluation and capacity planning studies. However, the analytic results cannot be ..."
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: The response time is the most visible performance index to users of computer systems. Endusers see individual response times, not the average. Therefore the distribution of response times is important in performance evaluation and capacity planning studies. However, the analytic results cannot be obtained in practical cases. A new method is proposed to approximate the responsetime distribution. Unlike the previous methods the proposed one takes into account the servicetime distributions and routing behaviour. The reported results indicate that the method provides reasonable approximations in many cases. 1 Introduction Queueing network modelling is a popular tool in the performance evaluation of computer systems. It has been successfully used in various applications of modelling computer systems. In most applications only the mean values of performance indices, such as mean device queuelengths and the mean system responsetime, have been considered. The increasing usage of comput...