9,677 research outputs found
Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules
We show how to obtain a fast component-by-component construction algorithm
for higher order polynomial lattice rules. Such rules are useful for
multivariate quadrature of high-dimensional smooth functions over the unit cube
as they achieve the near optimal order of convergence. The main problem
addressed in this paper is to find an efficient way of computing the worst-case
error. A general algorithm is presented and explicit expressions for base~2 are
given. To obtain an efficient component-by-component construction algorithm we
exploit the structure of the underlying cyclic group.
We compare our new higher order multivariate quadrature rules to existing
quadrature rules based on higher order digital nets by computing their
worst-case error. These numerical results show that the higher order polynomial
lattice rules improve upon the known constructions of quasi-Monte Carlo rules
based on higher order digital nets
Appropriate SCF basis sets for orbital studies of galaxies and a `quantum-mechanical' method to compute them
We address the question of an appropriate choice of basis functions for the
self-consistent field (SCF) method of simulation of the N-body problem. Our
criterion is based on a comparison of the orbits found in N-body realizations
of analytical potential-density models of triaxial galaxies, in which the
potential is fitted by the SCF method using a variety of basis sets, with those
of the original models. Our tests refer to maximally triaxial Dehnen
gamma-models for values of in the range 0<=gamma<=1. When an N-body
realization of a model is fitted by the SCF method, the choice of radial basis
functions affects significantly the way the potential, forces, or derivatives
of the forces are reproduced, especially in the central regions of the system.
We find that this results in serious discrepancies in the relative amounts of
chaotic versus regular orbits, or in the distributions of the Lyapunov
characteristic exponents, as found by different basis sets. Numerical tests
include the Clutton-Brock and the Hernquist-Ostriker (HO) basis sets, as well
as a family of numerical basis sets which are `close' to the HO basis set. The
family of numerical basis sets is parametrized in terms of a quantity
which appears in the kernel functions of the Sturm-Liouville (SL)
equation defining each basis set. The HO basis set is the member
of the family. We demonstrate that grid solutions of the SL equation yielding
numerical basis sets introduce large errors in the variational equations of
motion. We propose a quantum-mechanical method of solution of the SL equation
which overcomes these errors. We finally give criteria for a choice of optimal
value of and calculate the latter as a function of the value of
gamma.Comment: 22 pages, 13 figures, Accepted in MNRA
Testing an Optimised Expansion on Z_2 Lattice Models
We test an optimised hopping parameter expansion on various Z_2 lattice
scalar field models: the Ising model, a spin-one model and lambda (phi)^4. We
do this by studying the critical indices for a variety of optimisation
criteria, in a range of dimensions and with various trial actions. We work up
to seventh order, thus going well beyond previous studies. We demonstrate how
to use numerical methods to generate the high order diagrams and their
corresponding expressions. These are then used to calculate results numerically
and, in the case of the Ising model, we obtain some analytic results. We
highlight problems with several optimisation schemes and show for the best
scheme that the critical exponents are consistent with mean field results to at
least 8 significant figures. We conclude that in its present form, such
optimised lattice expansions do not seem to be capturing the non-perturbative
infra-red physics near the critical points of scalar models.Comment: 47 pages, some figures in colour but will display fine in B
Fast Converging Path Integrals for Time-Dependent Potentials I: Recursive Calculation of Short-Time Expansion of the Propagator
In this and subsequent paper arXiv:1011.5185 we develop a recursive approach
for calculating the short-time expansion of the propagator for a general
quantum system in a time-dependent potential to orders that have not yet been
accessible before. To this end the propagator is expressed in terms of a
discretized effective potential, for which we derive and analytically solve a
set of efficient recursion relations. Such a discretized effective potential
can be used to substantially speed up numerical Monte Carlo simulations for
path integrals, or to set up various analytic approximation techniques to study
properties of quantum systems in time-dependent potentials. The analytically
derived results are numerically verified by treating several simple models.Comment: 29 pages, 5 figure
The construction of good lattice rules and polynomial lattice rules
A comprehensive overview of lattice rules and polynomial lattice rules is
given for function spaces based on semi-norms. Good lattice rules and
polynomial lattice rules are defined as those obtaining worst-case errors
bounded by the optimal rate of convergence for the function space. The focus is
on algebraic rates of convergence for
and any , where is the decay of a series representation
of the integrand function. The dependence of the implied constant on the
dimension can be controlled by weights which determine the influence of the
different dimensions. Different types of weights are discussed. The
construction of good lattice rules, and polynomial lattice rules, can be done
using the same method for all ; but the case is special
from the construction point of view. For the
component-by-component construction and its fast algorithm for different
weighted function spaces is then discussed
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