Spectral methods for multiscale stochastic differential equations

This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented

Spectral Methods for Multiscale Stochastic Differential Equations

This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented

Monte Carlo approximations of the Neumann problem

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied

Corrections to Einstein's relation for Brownian motion in a tilted periodic potential

In this paper we revisit the problem of Brownian motion in a tilted periodic potential. We use homogenization theory to derive general formulas for the effective velocity and the effective diffusion tensor that are valid for arbitrary tilts. Furthermore, we obtain power series expansions for the velocity and the diffusion coefficient as functions of the external forcing. Thus, we provide systematic corrections to Einstein's formula and to linear response theory. Our theoretical results are supported by extensive numerical simulations. For our numerical experiments we use a novel spectral numerical method that leads to a very efficient and accurate calculation of the effective velocity and the effective diffusion tensor.Comment: 29 pages, 7 figures, submitted to the Journal of Statistical Physic

Bayesian Estimation of Hardness Ratios: Modeling and Computations

A commonly used measure to summarize the nature of a photon spectrum is the so-called Hardness Ratio, which compares the number of counts observed in different passbands. The hardness ratio is especially useful to distinguish between and categorize weak sources as a proxy for detailed spectral fitting. However, in this regime classical methods of error propagation fail, and the estimates of spectral hardness become unreliable. Here we develop a rigorous statistical treatment of hardness ratios that properly deals with detected photons as independent Poisson random variables and correctly deals with the non-Gaussian nature of the error propagation. The method is Bayesian in nature, and thus can be generalized to carry out a multitude of source-population--based analyses. We verify our method with simulation studies, and compare it with the classical method. We apply this method to real world examples, such as the identification of candidate quiescent Low-mass X-ray binaries in globular clusters, and tracking the time evolution of a flare on a low-mass star.Comment: 43 pages, 10 figures, 3 tables; submitted to Ap

On characterising the variability properties of X-ray light curves from active galaxies

We review some practical aspects of measuring the amplitude of variability in `red noise' light curves typical of those from Active Galactic Nuclei (AGN). The quantities commonly used to estimate the variability amplitude in AGN light curves, such as the fractional rms variability amplitude, F_var, and excess variance, sigma_XS^2, are examined. Their statistical properties, relationship to the power spectrum, and uses for investigating the nature of the variability processes are discussed. We demonstrate that sigma_XS^2 (or similarly F_var) shows large changes from one part of the light curve to the next, even when the variability is produced by a stationary process. This limits the usefulness of these estimators for quantifying differences in variability amplitude between different sources or from epoch to epoch in one source. Some examples of the expected scatter in the variance are tabulated for various typical power spectral shapes, based on Monte Carlo simulations. The excess variance can be useful for comparing the variability amplitudes of light curves in different energy bands from the same observation. Monte Carlo simulations are used to derive a description of the uncertainty in the amplitude expected between different energy bands (due to measurement errors). Finally, these estimators are used to demonstrate some variability properties of the bright Seyfert 1 galaxy Markarian 766. The source is found to show a strong, linear correlation between rms amplitude and flux, and to show significant spectral variability.Comment: 14 pages. 12 figures. Accepted for publication in MNRA

On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D