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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...
A unified framework for numerically inverting Laplace transforms
 INFORMS Journal on Computing
, 2006
"... We introduce and investigate a framework for constructing algorithms to numerically invert Laplace transforms. Given a Laplace transform ˆ f of a complexvalued function of a nonnegative realvariable, f, the function f is approximated by a finite linear combination of the transform values; i.e., w ..."
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Cited by 5 (1 self)
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We introduce and investigate a framework for constructing algorithms to numerically invert Laplace transforms. Given a Laplace transform ˆ f of a complexvalued function of a nonnegative realvariable, f, the function f is approximated by a finite linear combination of the transform values; i.e., we use the inversion formula f(t) ≈ fn(t) ≡ 1 t n� ωk ˆ f k=0 αk
Dynamics of steps along a martensitic phase boundary I: semianalytical solution
"... We study the motion of steps along a martensitic phase boundary in a cubic lattice. To enable analytical calculations, we assume antiplane shear deformation and consider a phase transforming material with a stressstrain law that is piecewise linear with respect to one component of shear strain and ..."
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Cited by 3 (0 self)
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We study the motion of steps along a martensitic phase boundary in a cubic lattice. To enable analytical calculations, we assume antiplane shear deformation and consider a phase transforming material with a stressstrain law that is piecewise linear with respect to one component of shear strain and linear with respect to another. Under these assumptions we derive a semianalytical solution describing a steady sequential motion of the steps under an external loading. Our analysis yields kinetic relations between the driving force, the velocity of the steps and other characteristic parameters of the motion. These are studied in detail for the twostep and threestep configurations. We show that the kinetic relations are significantly affected by the material anisotropy. Our results indicate the existence of multiple solutions exhibiting sequential step motion. Key words: lattice model, phase boundary, interphase step, sequential motion 1
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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Cited by 2 (1 self)
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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.
AN ALGEBRAIC MULTIGRID BASED PRECONDITIONER FOR THE LAPLACE TRANSFORMATION METHOD. ∗
"... Abstract. Due to the exponential order convergence of the Laplace transformation method for a parabolic type PDE, it has been considered as an alternative of the traditional time marching algorithms such as backwardEuler and CrankNicolson. Although the Laplace transformation method has been welld ..."
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Abstract. Due to the exponential order convergence of the Laplace transformation method for a parabolic type PDE, it has been considered as an alternative of the traditional time marching algorithms such as backwardEuler and CrankNicolson. Although the Laplace transformation method has been welldeveloped in theoretical as well as computational aspect, the computation of the Laplace transformed equations, which is the most time consuming part, has not been payed much attention. In this paper, we have focused on the speedup for solving the Laplace transformed equations. Based on an operator shifting and the algebraic multigrid method, an efficient preconditioner is developed. The preconditioner quickly decreases the condition number of the linear system, and thus a small number of iterations is required to achieve a given tolerance compared to another preconditioner such as incomplete LU preconditioner. Spectral analysis is performed in a simple case, and two practical problems are shown. Key words. Laplace transformation, Laplace inversion, preconditioner, algebraic multigrid, AMG, parabolic equation, PDE, BiCGSTAB AMS subject classifications.
Solving Linear Recurrences on Hybrid GPU Accelerated Manycore Systems
"... Abstract—The aim of this paper is to show that linear recurrence systems with constant coefficients can be efficiently solved on hybrid GPU accelerated manycore systems with modern Fermi GPU cards. The main idea is to use the recently developed divideandconquer algorithm which can be expressed in t ..."
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Abstract—The aim of this paper is to show that linear recurrence systems with constant coefficients can be efficiently solved on hybrid GPU accelerated manycore systems with modern Fermi GPU cards. The main idea is to use the recently developed divideandconquer algorithm which can be expressed in terms of Level 2 and 3 BLAS operations. The results of experiments performed on hybrid system with Intel Core i7 and NVIDIA Tesla C2050 are also presented and discussed. I.