55,813 research outputs found
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
On the one hand, the explicit Euler scheme fails to converge strongly to the
exact solution of a stochastic differential equation (SDE) with a superlinearly
growing and globally one-sided Lipschitz continuous drift coefficient. On the
other hand, the implicit Euler scheme is known to converge strongly to the
exact solution of such an SDE. Implementations of the implicit Euler scheme,
however, require additional computational effort. In this article we therefore
propose an explicit and easily implementable numerical method for such an SDE
and show that this method converges strongly with the standard order one-half
to the exact solution of the SDE. Simulations reveal that this explicit
strongly convergent numerical scheme is considerably faster than the implicit
Euler scheme.Comment: Published in at http://dx.doi.org/10.1214/11-AAP803 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Faster Methods for Contracting Infinite 2D Tensor Networks
We revisit the corner transfer matrix renormalization group (CTMRG) method of
Nishino and Okunishi for contracting two-dimensional (2D) tensor networks and
demonstrate that its performance can be substantially improved by determining
the tensors using an eigenvalue solver as opposed to the power method used in
CTMRG. We also generalize the variational uniform matrix product state (VUMPS)
ansatz for diagonalizing 1D quantum Hamiltonians to the case of 2D transfer
matrices and discuss similarities with the corner methods. These two new
algorithms will be crucial to improving the performance of variational infinite
projected entangled pair state (PEPS) methods.Comment: 20 pages, 5 figures, V. Zauner-Stauber previously also published
under the name V. Zaune
An example of optimal field cut in lattice gauge perturbation theory
We discuss the weak coupling expansion of a one plaquette SU(2) lattice gauge
theory. We show that the conventional perturbative series for the partition
function has a zero radius of convergence and is asymptotic. The average
plaquette is discontinuous at g^2=0. However, the fact that SU(2) is compact
provides a perturbative sum that converges toward the correct answer for
positive g^2. This alternate methods amounts to introducing a specific coupling
dependent field cut, that turns the coefficients into g-dependent quantities.
Generalizing to an arbitrary field cut, we obtain a regular power series with a
finite radius of convergence. At any order in the modified perturbative
procedure, and for a given coupling, it is possible to find at least one (and
sometimes two) values of the field cut that provide the exact answer. This
optimal field cut can be determined approximately using the strong coupling
expansion. This allows us to interpolate accurately between the weak and strong
coupling regions. We discuss the extension of the method to lattice gauge
theory on a D-dimensional cubic lattice.Comment: 9 pages, 11 figs., uses revtex4, modified presentatio
Jackknife resampling technique on mocks: an alternative method for covariance matrix estimation
We present a fast and robust alternative method to compute covariance matrix
in case of cosmology studies. Our method is based on the jackknife resampling
applied on simulation mock catalogues. Using a set of 600 BOSS DR11 mock
catalogues as a reference, we find that the jackknife technique gives a similar
galaxy clustering covariance matrix estimate by requiring a smaller number of
mocks. A comparison of convergence rates show that 7 times fewer
simulations are needed to get a similar accuracy on variance. We expect this
technique to be applied in any analysis where the number of available N-body
simulations is low.Comment: 11 pages, 11 figures, 2 table
High order analysis of the limit cycle of the van der Pol oscillator
We have applied the Lindstedt-Poincaré method to study the limit cycle of the van der Pol oscillator, obtaining the numerical coefficients of the series for the period and for the amplitude to order 859. Hermite-Padé approximants have been used to extract the location of the branch cut of the series with unprecedented accuracy (100 digits). Both series have then been resummed using an approach based on Padé approximants, where the exact asymptotic behaviors of the period and the amplitude are taken into account. Our results improve drastically all previous results obtained on this subject.Fil: Amore, Paolo. Universidad de Colima; MéxicoFil: Boyd, John P.. University of Michigan; Estados UnidosFil: Fernández, Francisco Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas; Argentin
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