422 research outputs found
The Cuba Library
Concepts and implementation of the Cuba library for multidimensional
numerical integration are elucidated.Comment: 6 pages. Talk given at the X International Workshop on Advanced
Computing and Analysis Techniques in Physics Research, ACAT 2005,
DESY-Zeuthen, Germany, 22-27 May 200
Notes on higher-dimensional partitions
We show the existence of a series of transforms that capture several
structures that underlie higher-dimensional partitions. These transforms lead
to a sequence of triangles whose entries are given combinatorial
interpretations as the number of particular types of skew Ferrers diagrams. The
end result of our analysis is the existence of a triangle, that we denote by F,
which implies that the data needed to compute the number of partitions of a
given positive integer is reduced by a factor of half. The number of spanning
rooted forests appears intriguingly in a family of entries in the triangle F.
Using modifications of an algorithm due to Bratley-McKay, we are able to
directly enumerate entries in some of the triangles. As a result, we have been
able to compute numbers of partitions of positive integers <= 25 in any
dimension.Comment: 36 pages; Mathematica file attached; See
http://www.physics.iitm.ac.in/~suresh/partitions.html to generate numbers of
partition
Error in Monte Carlo, quasi-error in Quasi-Monte Carlo
While the Quasi-Monte Carlo method of numerical integration achieves smaller
integration error than standard Monte Carlo, its use in particle physics
phenomenology has been hindered by the abscence of a reliable way to estimate
that error. The standard Monte Carlo error estimator relies on the assumption
that the points are generated independently of each other and, therefore, fails
to account for the error improvement advertised by the Quasi-Monte Carlo
method. We advocate the construction of an estimator of stochastic nature,
based on the ensemble of pointsets with a particular discrepancy value. We
investigate the consequences of this choice and give some first empirical
results on the suggested estimators.Comment: 41 pages, 19 figure
Optimization Under Uncertainty Using the Generalized Inverse Distribution Function
A framework for robust optimization under uncertainty based on the use of the
generalized inverse distribution function (GIDF), also called quantile
function, is here proposed. Compared to more classical approaches that rely on
the usage of statistical moments as deterministic attributes that define the
objectives of the optimization process, the inverse cumulative distribution
function allows for the use of all the possible information available in the
probabilistic domain. Furthermore, the use of a quantile based approach leads
naturally to a multi-objective methodology which allows an a-posteriori
selection of the candidate design based on risk/opportunity criteria defined by
the designer. Finally, the error on the estimation of the objectives due to the
resolution of the GIDF will be proven to be quantifiableComment: 20 pages, 25 figure
SINGINT: Automatic numerical integration of singular integrands
We explore the combination of deterministic and Monte Carlo methods to facilitate efficient automatic numerical computation of multidimensional integrals with singular integrands. Two adaptive algorithms are presented that employ recursion and are runtime and memory optimised, respectively. SINGINT, a C implementation of the algorithms, is introduced and its utilisation in the calculation of particle scattering amplitudes is exemplified
M. R. Flecknoe, Gillian Bratley, Josephine Hinstock, Alan Cooper, Sandra Hillcock to Dear Sir (5 October 1962)
https://egrove.olemiss.edu/mercorr_pro/2005/thumbnail.jp
Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers
This is basically a review of the field of Quasi-Monte Carlo intended for
computational physicists and other potential users of quasi-random numbers. As
such, much of the material is not new, but is presented here in a style
hopefully more accessible to physicists than the specialized mathematical
literature. There are also some new results: On the practical side we give
important empirical properties of large quasi-random point sets, especially the
exact quadratic discrepancies; on the theoretical side, there is the exact
distribution of quadratic discrepancy for random point sets.Comment: 51 pages. Full paper, including all figures also available at:
ftp://ftp.nikhef.nl/pub/preprints/96-017.ps.gz Accepted for publication in
Comp.Phys.Comm. Fixed some typos, corrected formula 108,figure 11 and table
Discrepancy-based error estimates for Quasi-Monte Carlo. I: General formalism
We show how information on the uniformity properties of a point set employed
in numerical multidimensional integration can be used to improve the error
estimate over the usual Monte Carlo one. We introduce a new measure of
(non-)uniformity for point sets, and derive explicit expressions for the
various entities that enter in such an improved error estimate. The use of
Feynman diagrams provides a transparent and straightforward way to compute this
improved error estimate.Comment: 23 pages, uses axodraw.sty, available at
ftp://nikhefh.nikhef.nl/pub/form/axodraw Fixed some typos, tidied up section
3.
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