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Monte Carlo Techniques - One of the best experts on this subject based on the ideXlab platform.
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38. Monte Carlo techniques 1
2011Co-Authors: Monte Carlo TechniquesAbstract:Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 38.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator, ” which generates uniform statistically independent values on the half open Interval [0, 1); for reviews see, e.g., [1,2]. Uniform random number generators are available in software libraries such as CERNLIB [3], CLHEP [4], and ROOT [5]. For example, in addition to a basic congruential generator TRandom (see below), ROOT provides three more sophisticated routines: TRandom1 implements the RANLUX generator [6] based on the method by Lüscher, and allows the user to select different quality levels, trading off quality with speed; TRandom2 is based on the maximally equidistributed combined Tausworthe generator by L’Ecuyer [7]; the TRandom3 generator implements the Mersenne twister algorithm of Matsumoto an
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34. MONTE CARLO TECHNIQUES
2011Co-Authors: Monte Carlo TechniquesAbstract:Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 34.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator, ” which generates uniform statistically independent values on the half open Interval [0, 1); for reviews see, e.g., [1,2]. Uniform random number generators are available in software libraries such as CERNLIB [3], CLHEP [4], and ROOT [5]. For example, in addition to a basic congruential generator TRandom (see below), ROOT provides three more sophisticated routines: TRandom1 implements the RANLUX generator [6] based on the method by Lüscher, and allows the user to select different quality levels, trading off quality with speed; TRandom2 is based on the maximally equidistributed combined Tausworthe generator by L’Ecuyer [7]; the TRandom3 generator implements the Mersenne twister algorithm of Matsumoto an
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39. Monte Carlo techniques 1
2011Co-Authors: Monte Carlo TechniquesAbstract:Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 39.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator, ” which generates uniform statistically independent values on the half open Interval [0, 1); for reviews see, e.g., [1,2]. Uniform random number generators are available in software libraries such as CERNLIB [3], CLHEP [4], and ROOT [5]. For example, in addition to a basic congruential generator TRandom (see below), ROOT provides three more sophisticated routines: TRandom1 implements the RANLUX generator [6] based on the method by Lüscher, and allows the user to select different quality levels, trading off quality with speed; TRandom2 is based on the maximally equidistributed combined Tausworthe generator by L’Ecuyer [7]; the TRandom3 generator implements the Mersenne twister algorithm of Matsumoto an
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37. MONTE CARLO TECHNIQUES
2011Co-Authors: Monte Carlo TechniquesAbstract:Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 37.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator, ” which generates uniform statistically independent values on the half open Interval [0, 1); for reviews see, e.g., [1,2]. Uniform random number generators are available in software libraries such as CERNLIB [3], CLHEP [4], and ROOT [5]. For example, in addition to a basic congruential generator TRandom (see below), ROOT provides three more sophisticated routines: TRandom1 implements the RANLUX generator [6] based on the method by Lüscher, and allows the user to select different quality levels, trading off quality with speed; TRandom2 is based on the maximally equidistributed combined Tausworthe generator by L’Ecuyer [7]; the TRandom3 generator implements the Mersenne twister algorithm of Matsumoto an
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33. MONTE CARLO TECHNIQUES
2007Co-Authors: Monte Carlo TechniquesAbstract:Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 33.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator, ” which generates uniform statistically independent values on the half open Interval [0, 1); for reviews see, e.g.,[1, 2]. Uniform random number generators are available in software libraries such as CERNLIB [3], CLHEP [4], and ROOT [5]. For example, in addition to a basic congruential generator TRandom (see below) ROOT provides three more sophisticated routines: TRandom1 implements the RANLUX generator [6] based on the method by Lüscher, and allows the user to select different quality levels, trading off quality with speed; TRandom2 is based on the maximally equidistributed combined Tausworthe generator by L’Ecuyer [7]; the TRandom3 generator implements the Mersenne twister algorithm of Matsumoto an
Maria Roginskaya - One of the best experts on this subject based on the ideXlab platform.
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reducing conjugacy in the full diffeomorphism group of ℝ to conjugacy in the subgroup of orientation preserving maps
Journal of Mathematical Sciences, 2009Co-Authors: Anthony G Ofarrell, Maria RoginskayaAbstract:Let Diffeo = Diffeo(ℝ) denote the group of infinitely differentiable diffeomorphisms of the real line ℝ, under the operation of composition, and let Diffeo+ be the subgroup of diffeomorphisms of degree +1, i.e., orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements f, g Diffeo are conjugate in Diffeo to associated conjugacy problems in the subgroup Diffeo+. The main result concerns the case when f and g have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in Diffeo+, in order to ensure that f is conjugated to g by an element of Diffeo+. The methods involve formal power series and results of Kopell on centralisers in the diffeomorphism group of a Half-Open Interval.
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reducing conjugacy in the full diffeomorphism group of r to conjugacy in the subgroup of orientation preserving maps
arXiv: Dynamical Systems, 2008Co-Authors: Anthony G Ofarrell, Maria RoginskayaAbstract:Let $\Diffeo=\Diffeo(\R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\R$, under the operation of composition, and let $\Diffeo^+$ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements $f,g\in \Diffeo$ are conjugate in $\Diffeo$ to associated conjugacy problems in the subgroup $\Diffeo^+$. The main result concerns the case when $f$ and $g$ have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in $\Diffeo^+$, in order to ensure that $f$ is conjugated to $g$ by an element of $\Diffeo^+$. The methods involve formal power series, and results of Kopell on centralisers in the diffeomorphism group of a Half-Open Interval.
G I Shishkin - One of the best experts on this subject based on the ideXlab platform.
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difference scheme for an initial boundary value problem for a singularly perturbed transport equation
Computational Mathematics and Mathematical Physics, 2017Co-Authors: G I ShishkinAbstract:An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter e multiplying the spatial derivative is considered on the set Ḡ = G ∪ S, where Ḡ = D × [0 ≤ t ≤ T], D = {0 ≤ x ≤ d}, S = S l ∪ S, and S l and S0 are the lateral and lower boundaries. The parameter e takes arbitrary values from the Half-Open Interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of e, this problem involves a boundary layer of width O(e) appearing in the neighborhood of S l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge e-uniformly in the maximum norm. Convergence occurs only if h=dN-1 ≪ e and N0-1 ≪ 1, where N and N0 are the numbers of grid Intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges e-uniformly in the maximum norm at an Ϭ(N−1 + N0−1) rate.
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a parameter robust finite difference method for singularly perturbed delay parabolic partial differential equations
Journal of Computational and Applied Mathematics, 2007Co-Authors: Ali R Ansari, Shaaban A Bakr, G I ShishkinAbstract:A Dirichlet boundary value problem for a delay parabolic differential equation is studied on a rectangular domain in the x-t plane. The second-order space derivative is multiplied by a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. A numerical method comprising a standard finite difference operator (centred in space, implicit in time) on a rectangular piecewise uniform fitted mesh of N"xxN"t elements condensing in the boundary layers is proved to be robust with respect to the small parameter, or parameter-uniform, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the Half-Open Interval (0,1]. More specifically, it is shown that the errors are bounded in the maximum norm by C(N"x^-^2ln^2N"x+N"t^-^1), where C is a constant independent not only of N"x and N"t but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the standard finite difference operator on a uniform mesh of N"xxN"t elements is not parameter-robust.
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approximation of the solutions of singularly perturbed boundary value problems with a parabolic boundary layer
Ussr Computational Mathematics and Mathematical Physics, 1991Co-Authors: G I ShishkinAbstract:Abstract Boundary-value problems for an equation of parabolic type, in which the coefficient of the highest-order derivatives involves a parameter varying in the Half-Open Interval (0,1], are considered. As the parameter approaches zero, parabolic boundary layers develop near the boundary of the domain. It is shown that the attempt to use adjustive methods to construct difference schemes that are uniformly convergent (with respect to the parameter) for such systems meets certain difficulties; in fact, for uniform grids there is no such adjustive scheme. A study is presented of two problems for a parabolic equation with mixed derivatives: a periodic boundary-value problem in a strip and the Dirichlet problem in a two-dimensional domain whose boundary is a smooth curve. In both cases it is possible to construct difference schemes that converge uniformly in the parameter throughout the domain.
Mohan K Kadalbajoo - One of the best experts on this subject based on the ideXlab platform.
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a singular perturbation approach to solve burgers huxley equation via monotone finite difference scheme on layer adaptive mesh
Communications in Nonlinear Science and Numerical Simulation, 2011Co-Authors: Vikas Gupta, Mohan K KadalbajooAbstract:Abstract This paper deals with a numerical method for solving one-dimensional unsteady Burgers–Huxley equation with the viscosity coefficient e . The parameter e takes any values from the half open Interval (0, 1]. At small values of the parameter e , an outflow boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem with a singular perturbation parameter e . Using singular perturbation analysis, asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. We construct a numerical scheme that comprises of implicit-Euler method to discretize in temporal direction on uniform mesh and a monotone hybrid finite difference operator to discretize the spatial variable with piecewise uniform Shishkin mesh. To obtain better accuracy, we use central finite difference scheme in the boundary layer region. Shishkin meshes are refined in the boundary layer region, therefore stability constraint is satisfied by proposed scheme. Quasilinearization process is used to tackle the non-linearity and it is shown that quasilinearization process converges quadratically. The method has been shown to be first order uniformly accurate in the temporal variable, and in the spatial direction it is first order parameter uniform convergent in the outside region of boundary layer, and almost second order parameter uniform convergent in the boundary layer region. Accuracy and uniform convergence of the proposed method is demonstrated by numerical examples and comparison of numerical results made with the other existing methods.
David T Redden - One of the best experts on this subject based on the ideXlab platform.
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evaluating the use of residential mailing addresses in a metropolitan household survey
Public Opinion Quarterly, 2003Co-Authors: Vincent Iannacchione, Jennifer M Staab, David T ReddenAbstract:On-site enumeration is generally regarded as the most comprehensive method for developing sampling frames for area household surveys. However, the time and expense associated with on-site enumeration often precludes it from being a viable option for many household surveys. Residential mailing lists provide an alternative that enables in-person surveys to be done cheaper and faster than is possible with on-site enumeration. The primary drawback of mailing lists is that the completeness of the lists is unknown. In this article, we evaluate the coverage of mailing addresses that were used as a sampling frame for a probability-based survey of 15,000 households in Dallas County, TX. The addresses were obtained from the Delivery Sequence File (DSF) offered by the U.S. Postal Service (USPS) through a nonexclusive license agreement with private companies. The DSF is a computerized file that contains all delivery point addresses serviced by the USPS, with the exception of general delivery. To evaluate the coverage of the mailing addresses, we used Kish's Half-Open Interval (HOI) procedure to search for missed housing units in the Interval between the selected address and the next address in delivery sequence order. A total of 46 missed addresses (1.9 percent) were found among the 2,380 HOIs randomly selected for examination. In addition, we discovered that the vast majority of persons who maintained a residential P.O. box also have mail delivered to their street address. Finally, the mailing addresses yielded a 90 percent occupancy rate, which is consistent with metropolitan household surveys that use on-site enumeration methods
