Any-order propagation of the nonlinear Schroedinger equation

We derive an exact propagation scheme for nonlinear Schroedinger equations. This scheme is entirely analogous to the propagation of linear Schroedinger equations. We accomplish this by defining a special operator whose algebraic properties ensure the correct propagation. As applications, we provide a simple proof of a recent conjecture regarding higher-order integrators for the Gross-Pitaevskii equation, extend it to multi-component equations, and to a new class of integrators.Comment: 10 pages, no figures, submitted to Phys. Rev.

Mixtures of Spatial Spline Regressions

We present an extension of the functional data analysis framework for univariate functions to the analysis of surfaces: functions of two variables. The spatial spline regression (SSR) approach developed can be used to model surfaces that are sampled over a rectangular domain. Furthermore, combining SSR with linear mixed effects models (LMM) allows for the analysis of populations of surfaces, and combining the joint SSR-LMM method with finite mixture models allows for the analysis of populations of surfaces with sub-family structures. Through the mixtures of spatial splines regressions (MSSR) approach developed, we present methodologies for clustering surfaces into sub-families, and for performing surface-based discriminant analysis. The effectiveness of our methodologies, as well as the modeling capabilities of the SSR model are assessed through an application to handwritten character recognition

Enhanced dispersion interaction in confined geometry

The dispersion interaction between two point-like particles confined in a dielectric slab between two plates of another dielectric medium is studied within a continuum (Lifshitz) theory. The retarded (Casimir-Polder) interaction at large inter-particle distances is found to be strongly enhanced as the mismatch between the dielectric permittivities of the two media is increased. The large-distance interaction is multiplied due to confinement by a factor of $(33\gamma^{5/2}+13\gamma^{-3/2})/46$ at zero temperature, and by $(5\gamma^2+\gamma^{-2})/6$ at finite temperature, \gamma=\ein(0)/\eout(0) being the ratio between the static dielectric permittivities of the inner and outer media. This confinement-induced amplification of the dispersion interaction can reach several orders of magnitude.Comment: 4 page

Numerical Implementation of Gradient Algorithms

A numerical method for computational implementation of gradient dynamical systems is presented. The method is based upon the development of geometric integration numerical methods, which aim at preserving the dynamical properties of the original ordinary differential equation under discretization. In particular, the proposed method belongs to the class of discrete gradients methods, which substitute the gradient of the continuous equation with a discrete gradient, leading to a map that possesses the same Lyapunov function of the dynamical system, thus preserving the qualitative properties regardless of the step size. In this work, we apply a discrete gradient method to the implementation of Hopfield neural networks. Contrary to most geometric integration methods, the proposed algorithm can be rewritten in explicit form, which considerably improves its performance and stability. Simulation results show that the preservation of the Lyapunov function leads to an improved performance, compared to the conventional discretization.Spanish Government project no. TIN2010-16556 Junta de Andalucía project no. P08-TIC-04026 Agencia Española de Cooperación Internacional para el Desarrollo project no. A2/038418/1

Quantum Statistical Calculations and Symplectic Corrector Algorithms

The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher order splitting will result in a higher order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Greens function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically give rises to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second order forward propagators can have fourth order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second order propagators and their minimal correctors.Comment: 9 pages, no figure, corrected typos, mostly missing right bracket

Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities

Since the kinetic and the potential energy term of the real time nonlinear Schr\"odinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where $k_{max}=\pi/\Delta x$.Comment: 10 pages, 8 figures, submitted to Phys. Rev.

Pseudo-High-Order Symplectic Integrators

Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly with order in timestep, rendering higher-order methods impractical. However, symplectic integrators are often applied to systems in which perturbations between bodies are a small factor of the force due to a dominant central mass. In this case, it is possible to create optimized symplectic algorithms that require fewer substeps per timestep. This is achieved by only considering error terms of order epsilon, and neglecting those of order epsilon^2, epsilon^3 etc. Here we devise symplectic algorithms with 4 and 6 substeps per step which effectively behave as 4th and 6th-order integrators when epsilon is small. These algorithms are more efficient than the usual 2nd and 4th-order methods when applied to planetary systems.Comment: 14 pages, 5 figures. Accepted for publication in the Astronomical Journa

Higher Order Effects in the Dielectric Constant of Percolative Metal-Insulator Systems above the Critical Point

The dielectric constant of a conductor-insulator mixture shows a pronounced maximum above the critical volume concentration. Further experimental evidence is presented as well as a theoretical consideration based on a phenomenological equation. Explicit expressions are given for the position of the maximum in terms of scaling parameters and the (complex) conductances of the conductor and insulator. In order to fit some of the data, a volume fraction dependent expression for the conductivity of the more highly conductive component is introduced.Comment: 4 pages, Latex, 4 postscript (*.epsi) files submitted to Phys Rev.