Long-Run Accuracy of Variational Integrators in the Stochastic Context

This paper presents a Lie-Trotter splitting for inertial Langevin equations (Geometric Langevin Algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial Langevin to within the accuracy of the variational integrator in representing the Hamiltonian. In particular, if the variational integrator admits no energy error, then the method samples the invariant measure of inertial Langevin without error. Numerical validation is provided using explicit variational integrators with first, second, and fourth order accuracy.Comment: 30 page

Completing the market orientation matrix: The impact of proactive competitor orientation on innovation and firm performance

The concept of market orientation comprises four components: customer and competitor orientations, each with a proactive and responsive dimension. Studies have considered both responsive and proactive customer orientation. Competitor orientation, however, has been investigated more narrowly. Research has focused specifically on its responsive dimension, a firm's posture of quickly responding to its competitors' actions and their offerings; but has largely disregarded proactive competitor orientation, a firm's posture towards altering the market's competitive behavior in its favor. This study investigates the role of responsive and proactive competitor orientation on influencing innovation and firm performance, as well as the mediating effects of technology and learning orientation. Utilizing a unique dataset that combines primary and time-lagged secondary data from 306 firms, we find that both responsive and proactive competitor orientation are observable drivers of performance in the market, but in notably different ways. Proactive competitor orientation drives innovation performance, directly and through technology orientation. Responsive competitor orientation, instead, enhances firm performance through learning orientation. By providing insights about the proactive side of competitor orientation, this study supplements and completes the so called “market orientation matrix”. This framework provides guidance for leaders to develop and manage a practical application of, and future research on market orientation

Langevin Thermostat for Rigid Body Dynamics

We present a new method for isothermal rigid body simulations using the quaternion representation and Langevin dynamics. It can be combined with the traditional Langevin or gradient (Brownian) dynamics for the translational degrees of freedom to correctly sample the NVT distribution in a simulation of rigid molecules. We propose simple, quasi-symplectic second-order numerical integrators and test their performance on the TIP4P model of water. We also investigate the optimal choice of thermostat parameters.Comment: 15 pages, 13 figures, 1 tabl

On the terminal velocity of sedimenting particles in a flowing fluid

The influence of an underlying carrier flow on the terminal velocity of sedimenting particles is investigated both analytically and numerically. Our theoretical framework works for a general class of (laminar or turbulent) velocity fields and, by means of an ordinary perturbation expansion at small Stokes number, leads to closed partial differential equations (PDE) whose solutions contain all relevant information on the sedimentation process. The set of PDE's are solved by means of direct numerical simulations for a class of 2D cellular flows (static and time dependent) and the resulting phenomenology is analysed and discussed.Comment: 13 pages, 2 figures, submitted to JP

Convergence of the stochastic Euler scheme for locally Lipschitz coefficients

Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.Comment: Published at http://www.springerlink.com/content/g076w80730811vv3 in the Foundations of Computational Mathematics 201

Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise

A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order Runge-Kutta scheme for weak approximation, the new class of methods affords less random variable evaluations and is also applicable to SDEs with multidimensional noise. Order conditions up to order three are calculated and coefficients of a four stage third order method are given. This method has deterministic order four and minimized error constants, and needs in addition less function evaluations than the method of Platen. Applied to some examples, the new method is compared numerically with Platen's method and some well known second order methods and yields very promising results.Comment: Two further examples added, small correction

Ergodic properties of quasi-Markovian generalized Langevin equations with configuration dependent noise and non-conservative force

We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted $L^{\infty}$ spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force

A generic materials and operations planning approach for inventory turnover optimization in the chemical industry

Chemical industries usually involve continuous and large-scale production processes that require demanding inventory control systems. This paper aims to show the results of the implementation of a mixed-integer programming model (MIP) based on the Generic Materials and Operations Planning Problem (GMOP) for optimizing the inventory turnover in a fertilizer company. Results showed significant improvements for Inventory Turnover Ratios and overall costs when compared with an empirical production planning method