Projective integration of expensive stochastic processes

Xiaopeng Chen, Anthony J. Roberts, Ioannis Kevrekidis


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.

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stochastic differential equations, stochastic process, projective integration, maximum likelihood estimation

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