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 $\sim$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