Projective integration of expensive stochastic processes

Detailed microscale simulation is typically too computationally expensive for the long time simulations necessary to explore macroscale dynamics. Projective integration uses bursts of the microscale simulator, on microscale time steps, and then computes an approximation to the system over a macroscale time step by extrapolation. Projective integration has the potential to be an effective method to compute the long time dynamic behaviour of multiscale systems. However, many multiscale systems are significantly influenced by noise. By a maximum likelihood estimation, we fit a linear stochastic differential equation to short bursts of data. The analytic solution of the linear stochastic differential equation then estimates the solution over a macroscale, projective integration, time step. We explore how the noise affects the projective integration in two different methods. Monte Carlo simulation suggests design parameters offering stability and accuracy for the algorithms. The algorithms developed here may be applied to compute the long time dynamic behaviour of multiscale systems with noise and to exploit parallel computation. References Yacine Ait-Sahalia. Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica, 70(1):223--262, 2002. Yacine Ait-Sahalia. Likelihood inference for diffusions: a survey. In Jianqing Fan and Hira L. Koul, editors, Frontiers in Statistics: in Honor of Peter J. Bickel's 65th Birthday, chapter 17, pages 369--405. Imperial College Press, 2006. Yacine Ait-Sahalia and Robert Kimmel. Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics, 83(2):413--452, 2007. doi:10.1016/j.jfineco.2005.10.006 C. P. Calderon. Local diffusion models for stochastic reacting systems: estimation issues in equation-free numerics. Molecular Simulation, 33(9--10):713--731, August 2007. doi:10.1080/08927020701344740 Radek Erban, Ioannis G. Kevrekidis, and Hans G. Othmer. An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal. Physica D: Nonlinear Phenomena, 215(1):1--24, 2006. doi:10.1016/j.physd.2006.01.008 C. W. Gear, I. G. Kevrekidis, and C. Theodoropoulos. `Coarse' integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods. Computers and Chemical Engrg, 26:941--963, 2002. C. W. Gear and Ioannis G. Kevrekidis. Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum. SIAM Journal on Scientific Computing, 24(4):1091--1106, 2003. doi:10.1137/S1064827501388157 Dror Givon and Ioannis G. Kevrekidis. Coarse-grained projective schemes for certain singularly perturbed stochastic problems. Technical report, Princeton University, 2008. Ioannis G. Kevrekidis and Giovanni Samaey. Equation-free multiscale computation: Algorithms and applications. Annu. Rev. Phys. Chem., 60:321--44, 2009. doi:10.1146/annurev.physchem.59.032607.093610 P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations, volume 23 of Applications of Mathematics. Springer--Verlag, 1992. C. I. Siettos, M. D. Graham, and I. G. Kevrekidis. Coarse brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation. J. Chemical Physics, 118(22):10149--10156, 2003. doi:10.1063/1.157245