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Linear response theory and optimal control for a molecular system under nonequilibrium conditions
(2013)
Grand-canonical-like molecular-dynamics simulations by using an adaptive-resolution technique
(2013)
In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: We first show that if we start with an ergodic diffusion processes whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Zwanzig-Mori, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the ”equation-free” approach and the ”heterogeneous multiscale method” can be seen as special cases of our approach.
Markov jump processes are widely used to model natural and engineered processes. In the context of biological or chemical applications one typically refers to the chemical master equation (CME), which models the evolution of the probability mass of any copy-number combination of the interacting particles. When many interacting particles (“species”) are considered, the complexity of the CME quickly increases, making direct numerical simulations impossible. This is even more problematic when one aims at controlling the Markov jump processes defined by the CME.
In this work, we study both open loop and feedback optimal control problems of the Markov jump processes in the case that the controls can only be switched at fixed control stages. Based on Kurtz’s limit theorems, we prove the convergence of the respective control value functions of the underlying Markov decision problem as the copy numbers of the species go to infinity. In the case of the optimal control problem on a finite time-horizon, we propose a hybrid control policy algorithm to overcome the difficulties due to the curse of dimensionality when the copy number of the involved species is large. Two numerical examples demonstrate the suitability of both the analysis and the proposed algorithms.
Applications of the cross-entropy method to importance sampling and optimal control of diffusions
(2014)
We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method.