GPU accelerated Monte Carlo simulation of Brownian motors dynamics with CUDA

This work presents an updated and extended guide on methods of a proper acceleration of the Monte Carlo integration of stochastic differential equations with the commonly available NVIDIA Graphics Processing Units using the CUDA programming environment. We outline the general aspects of the scientific computing on graphics cards and demonstrate them with two models of a well known phenomenon of the noise induced transport of Brownian motors in periodic structures. As a source of fluctuations in the considered systems we selected the three most commonly occurring noises: the Gaussian white noise, the white Poissonian noise and the dichotomous process also known as a random telegraph signal. The detailed discussion on various aspects of the applied numerical schemes is also presented. The measured speedup can be of the astonishing order of about 3000 when compared to a typical CPU. This number significantly expands the range of problems solvable by use of stochastic simulations, allowing even an interactive research in some cases.Comment: 21 pages, 5 figures; Comput. Phys. Commun., accepted, 201

How close are time series to power tail L\'evy diffusions?

This article presents a new and easily implementable method to quantify the so-called coupling distance between the law of a time series and the law of a differential equation driven by Markovian additive jump noise with heavy-tailed jumps, such as $\alpha$-stable L\'evy flights. Coupling distances measure the proximity of the empirical law of the tails of the jump increments and a given power law distribution. In particular they yield an upper bound for the distance of the respective laws on path space. We prove rates of convergence comparable to the rates of the central limit theorem which are confirmed by numerical simulations. Our method applied to a paleoclimate time series of glacial climate variability confirms its heavy tail behavior. In addition this approach gives evidence for heavy tails in data sets of precipitable water vapor of the Western Tropical Pacific.Comment: 30 pages, 10 figure

Numerical computation of rare events via large deviation theory

An overview of rare events algorithms based on large deviation theory (LDT) is presented. It covers a range of numerical schemes to compute the large deviation minimizer in various setups, and discusses best practices, common pitfalls, and implementation trade-offs. Generalizations, extensions, and improvements of the minimum action methods are proposed. These algorithms are tested on example problems which illustrate several common difficulties which arise e.g. when the forcing is degenerate or multiplicative, or the systems are infinite-dimensional. Generalizations to processes driven by non-Gaussian noises or random initial data and parameters are also discussed, along with the connection between the LDT-based approach reviewed here and other methods, such as stochastic field theory and optimal control. Finally, the integration of this approach in importance sampling methods using e.g. genealogical algorithms is explored

Bayesian Estimation for Continuous-Time Sparse Stochastic Processes

We consider continuous-time sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal in-between (interpolation problem). By relying on tools from the theory of splines, we derive the joint a priori distribution of the samples and show how this probability density function can be factorized. The factorization enables us to tractably implement the maximum a posteriori and minimum mean-square error (MMSE) criteria as two statistical approaches for estimating the unknowns. We compare the derived statistical methods with well-known techniques for the recovery of sparse signals, such as the $\ell_1$ norm and Log ($\ell_1$-$\ell_0$ relaxation) regularization methods. The simulation results show that, under certain conditions, the performance of the regularization techniques can be very close to that of the MMSE estimator.Comment: To appear in IEEE TS