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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 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...
Fault rupture between dissimilar materials: Illposedness, regularization, and slippulse response
 J. Geophys. Res
, 2000
"... Faults often separate materials with different elastic properties. Nonuniform slip on such faults induces a change in normal stress. That suggests the possibility of selfsustained slip pulses [Weertman, 1980] propagating at the generalized Rayleigh wave speed even with a Coulomb constitutive law (i ..."
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Cited by 9 (6 self)
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Faults often separate materials with different elastic properties. Nonuniform slip on such faults induces a change in normal stress. That suggests the possibility of selfsustained slip pulses [Weertman, 1980] propagating at the generalized Rayleigh wave speed even with a Coulomb constitutive law (i.e., with a constant coefficient of friction) and a remote driving shear stress that is arbitrarily less than the corresponding frictional strength. Following Andrews and BenZion [1997] (ABZ), we study numerically, with a twodimensional (2D) plane strain geometry, the propagation of ruptures along such a dissimilar material interface. However, this problem has been shown to be illposed for a wide range of elastic material contrasts [Renardy, 1992; Martins and Simoes, 1995; Adams, 1995]. Ranjith and Rice [2000] (RR) showed that when the generalized Rayleigh speed exists, as is the case for the material contrast studied by ABZ, the problem is illposed for all values of the coefficient of friction, , whereas when it does not exist, the problem is illposed only for greater than a critical value. We illustrate the illposedness by showing that in the unstable range the numerical solutions do not converge through grid size reduction. By contrast, convergence is achieved in the stable range but, not unexpectedly, only dying pulses are then observed. RR showed that among other regularization procedures, use of an experimentally based law [Prakash and Clifton, 1993; Prakash, 1998], in which the shear strength in response to an abrupt change in normal stress evolves continuously with time or slip toward the corresponding Coulomb strength, provides a regularization. (Classical slipweakening or rate and statedependent constitutive laws having the same kind of abrupt respo...
Probabilistic scaling for the numerical inversion of nonprobability transforms
 INFORMS J. Computing
, 1997
"... Abstract It is known that probability density functions and probability mass functions can usually be calculated quite easily by numerically inverting their transforms (Laplace transforms and generating functions, respectively) with the Fourierseries method, but other more general functions can be ..."
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Cited by 6 (5 self)
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Abstract It is known that probability density functions and probability mass functions can usually be calculated quite easily by numerically inverting their transforms (Laplace transforms and generating functions, respectively) with the Fourierseries method, but other more general functions can be substantially more difficult to invert, because the aliasing and roundoff errors tend to be more difficult to control. In this paper we propose a simple new scaling procedure for nonprobability functions that is based on transforming the given function into a probability density function or a probability mass function and transforming the point of inversion to the mean. This new scaling is even useful for probability functions, because it enables us to compute very small values at large arguments with controlled relative error. Subject classifications: Mathematics, functions: scaling for numerical transform inversion. Queues, algorithms: scaling for numerical transform inversion.
Transient Behaviour of Queueing Systems with Correlated Traffic
 Journal of Stochastic Models
, 1996
"... In this paper, we present the timedependent solutions of various stochastic processes associated with a finite QuasiBirthDeath queueing system. These include the transient queueing solutions, the transient departure and loss intensity processes and certain transient cummulative measures associate ..."
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Cited by 3 (0 self)
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In this paper, we present the timedependent solutions of various stochastic processes associated with a finite QuasiBirthDeath queueing system. These include the transient queueing solutions, the transient departure and loss intensity processes and certain transient cummulative measures associated with the queueing system. The focus of our study is the effect of the arrival process correlation on the queueing system before it reaches steadystate. With the aid of numerous examples, we investigate the strong relationship between the time scales of variation of the arrival process and those of the transient queueing, loss and departure processes. These timedependent solutions require the computation of the exponential of the stochastic generator matrix G which may be of very large order. This precludes the use of known techniques to solve the matrix exponential such as the eigenvalue decomposition of G. We present a numerical technique based on the computation of the Laplace Transfor...
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 1 (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
On NonMonotone Solutions Of An Integrodifferential Equation In Linear Viscoelasticity
, 1996
"... . We consider the integrodifferential equation u(t; x) = R t 0 a(t \Gamma s)uxx (s; x)ds with initial and boundary conditions corresponding to the Rayleigh problem. The kernel has the form a(t) = a 0 + a1 t + R t 0 a 1 (s)ds, where a 0 0, a1 0, and a 1 2 L 1 loc (R+ ) is of positive type ..."
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. We consider the integrodifferential equation u(t; x) = R t 0 a(t \Gamma s)uxx (s; x)ds with initial and boundary conditions corresponding to the Rayleigh problem. The kernel has the form a(t) = a 0 + a1 t + R t 0 a 1 (s)ds, where a 0 0, a1 0, and a 1 2 L 1 loc (R+ ) is of positive type and satisfies the condition R 1 0 e \Gammafflt ja 1 (t)jdt ! 1 for every ffl ? 0. Solving the equation numerically and performing a careful error analysis we show that the solution u(t; x) need not be nondecreasing in t 0 for fixed x ? 0, if a 1 is nonnegative, nonincreasing, and convex. The same result is shown to hold under the assumption that a 1 is completely positive. This answers a question that remained unsolved in [J. Pruß, Math. Ann., 279 (1987), p. 330]. In the case where a1 is convex, piecewise linear, the solution is shown to be almost everywhere equal to a function which is discontinuous across infinitely many parallel lines. Key words. viscoelasticity, integrodifferentia...
DISCONTINUITIES IN AN AXISYMMETRIC GENERALIZED THERMOELASTIC PROBLEM
, 2005
"... This paper deals with discontinuities analysis in the temperature, displacement, and stress fields of a thick plate whose lower and upper surfaces are tractionfree and subjected to a given axisymmetric temperature distribution. The analysis is carried out under three thermoelastic theories. Potenti ..."
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This paper deals with discontinuities analysis in the temperature, displacement, and stress fields of a thick plate whose lower and upper surfaces are tractionfree and subjected to a given axisymmetric temperature distribution. The analysis is carried out under three thermoelastic theories. Potential functions together with Laplace and Hankel transform techniques are used to derive the solution in the transformed domain. Exact expressions for the magnitude of discontinuities are computed by using an exact method developed by Boley (1962). It is found that there exist two coupled waves, one of which is elastic and the other is thermal, both propagating with finite speeds with exponential attenuation, and a third which is called shear wave, propagating with constant speed but with no exponential attenuation. The Hankel transforms are inverted analytically. The inversion of the Laplace transforms is carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical results are presented graphically along with a comparison of the three theories of thermoelasticity. 1.
A Hybrid Laplace Transform Mixed Multiscale FiniteElement Method for Flow Model in Porous Media ⋆
"... This paper presents a hybrid Laplace transform Mixed Multiscale Finiteelement Method (MMsFEM) to solve partial differential equations of flow in porous media. First, the time term of parabolic equation with unknown pressure term is removed by the Laplace transform. Then, to obtain the numerical app ..."
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This paper presents a hybrid Laplace transform Mixed Multiscale Finiteelement Method (MMsFEM) to solve partial differential equations of flow in porous media. First, the time term of parabolic equation with unknown pressure term is removed by the Laplace transform. Then, to obtain the numerical approximation of pressure and velocity directly, the transformed equations on coarse mesh are solved by mixed multiscale FEM, which utilizes the effects of finescale heterogeneities through basis function formulations computed from local flow problems. Finally, the associated pressure and velocity transform are inverted by the method of numerical inversion of the Laplace transform to obtain the numerical solution.