Results 1  10
of
33
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 ..."
Abstract

Cited by 149 (51 self)
 Add to MetaCart
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...
Computer Experiments
, 1996
"... Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, a ..."
Abstract

Cited by 67 (5 self)
 Add to MetaCart
Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, and so on. Some of the most widely used computer models, and the ones that lead us to work in this area, arise in the design of the semiconductors used in the computers themselves. A process simulator starts with a data structure representing an unprocessed piece of silicon and simulates the steps such as oxidation, etching and ion injection that produce a semiconductor device such as a transistor. A device simulator takes a description of such a device and simulates the flow of current through it under varying conditions to determine properties of the device such as its switching speed and the critical voltage at which it switches. A circuit simulator takes a list of devices and the
Variance Reduction via Lattice Rules
 Management Science
, 2000
"... This is a review article on lattice methods for multiple integration over the unit hypercube, with a variancereduction viewpoint. It also contains some new results and ideas. The aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation ..."
Abstract

Cited by 51 (13 self)
 Add to MetaCart
This is a review article on lattice methods for multiple integration over the unit hypercube, with a variancereduction viewpoint. It also contains some new results and ideas. The aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of Management Science. These models can usually be reformulated as integration problems over the unit hypercube with a large (sometimes infinite) number of dimensions. We examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected lowdimensional subspaces. The criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators. Numerical examples illustrate the effectiveness of the approach.
Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems
 Statistical Science
"... This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain method ..."
Abstract

Cited by 32 (4 self)
 Add to MetaCart
This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...
On a high order numerical method for functions with singularities
 Math. Comp
, 1998
"... Abstract. By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various t ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
Abstract. By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented. 1.
Evaluating Padé Approximants Of The Matrix Logarithm
, 2000
"... . The inverse scaling and squaring method for evaluating the logarithm of a matrix takes repeated square roots to bring the matrix close to the identity, computes a Pad'e approximant, and then scales back. We analyze several methods for evaluating the Pad'e approximant, including Horner's method (us ..."
Abstract

Cited by 18 (13 self)
 Add to MetaCart
. The inverse scaling and squaring method for evaluating the logarithm of a matrix takes repeated square roots to bring the matrix close to the identity, computes a Pad'e approximant, and then scales back. We analyze several methods for evaluating the Pad'e approximant, including Horner's method (used in some existing codes), suitably customized versions of the Paterson Stockmeyer method and Van Loan's variant, and methods based on continued fraction and partial fraction expansions. The computational cost, storage, and numerical accuracy of the methods are compared. We find the partial fraction method to be the best method overall and illustrate the benefits it brings to a transformationfree form of the inverse scaling and squaring method recently proposed by Cheng, Higham, Kenney and Laub. We comment briefly on how the analysis carries over to the matrix exponential. Key words. matrix logarithm, Pad'e approximation, inverse scaling and squaring method, Horner's method, PatersonS...
Fast evaluation of quadrature formulae on the sphere
 Math. Comput
, 2006
"... Abstract. Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard L 2 � S 2 �orthonormal spherical harmonics at arbitrary nodes on the sphere S 2 has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75–98, 2 ..."
Abstract

Cited by 12 (11 self)
 Add to MetaCart
Abstract. Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard L 2 � S 2 �orthonormal spherical harmonics at arbitrary nodes on the sphere S 2 has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75–98, 2003]. The aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules. We give a formulation in matrixvector notation and an explicit factorisation of the spherical Fourier matrix based on the former algorithm. Starting from this, we obtain the corresponding factorisation of the adjoint spherical Fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere. We provide results of numerical tests showing the stability of the obtained algorithm using as examples classical GaußLegendre and ClenshawCurtis quadrature rules as well as the HEALPix pixelation scheme and an equidistribution. 1.
Cubature Formulae and Orthogonal Polynomials
, 2000
"... The connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 years. The article J. Radon published about 50 years ago has been very inuential. In this text we describe some of the results that were obtained during ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
The connection between orthogonal polynomials and cubature formulae for the approximation of multivariate integrals has been studied for about 100 years. The article J. Radon published about 50 years ago has been very inuential. In this text we describe some of the results that were obtained during the search for answers to questions raised by his article.
Stieltjes polynomials and the error of GaussKronrod quadrature formulas
, 1998
"... . The GaussKronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the error of the GaussKronrod scheme. An ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
. The GaussKronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the error of the GaussKronrod scheme. An essential progress was made only recently, based on new bounds and asymptotic properties for the Stieltjes polynomials. The purpose of this paper is to give a survey on these results. In particular, the quality of the GaussKronrod formula for smooth and for nonsmooth functions is investigated and compared with other quadrature formulas. 1. Introduction The numerical evaluation of definite integrals is an important step in many applications of mathematics. From a practical point of view, great interest lies in automatic routines, whose only input are the integrand function, the domain of integration, and a tolerance for the error. In applications, such routines are expected to give quick and...