Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach

arXiv (Cornell University)

Abstract

We propose a novel alternative approach to our previous work (Ben Hammouda et al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare event probabilities for stochastic reaction networks (SRNs). In the same spirit of (Ben Hammouda et al., 2023), an efficient path-dependent measure change is derived based on a connection between determining optimal importance sampling (IS) parameters within a class of probability measures and a stochastic optimal control formulation, corresponding to solving a variance minimization problem. In this work, we propose a novel approach to address the encountered curse of dimensionality by mapping the problem to a significantly lower-dimensional space via a Markovian projection (MP) idea. The output of this model reduction technique is a low-dimensional SRN (potentially even one dimensional) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained by solving a related optimization problem via a discrete $L^2$ regression. By solving the resulting projected Hamilton-Jacobi-Bellman (HJB) equations for the reduced-dimensional SRN, we obtain projected IS parameters, which are then mapped back to the original full-dimensional SRN system, resulting in an efficient IS-MC estimator for rare events probabilities of the full-dimensional SRN. Our analysis and numerical experiments reveal that the proposed MP-HJB-IS approach substantially reduces the MC estimator variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.

Authors 4

1. Chiheb Ben Hammouda

   Utrecht University

   Affiliation as printed

   Utrecht University , Mathematical Institute , 3584 CD Utrecht , The Netherlands

2. Nadhir Ben Rached

   University of Leeds

   Affiliation as printed

   University of Leeds , School of Mathematics , Woodhouse , Leeds LS2 9JT , UK

3. Raul F. Tempone Aachen

   Alexander von Humboldt Foundation · King Abdullah University of Science and Technology · RWTH Aachen University

   Affiliation as printed

   King Abdullah University of Science and Technology (KAUST) , Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE) , Thuwal 23955-6900 , Saudi Arabia

   RWTH Aachen University , Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification , 52062 Aachen , Germany

4. Sophia Wiechert Aachen Chair of Mathematics for Uncertainty Quantification

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University , Chair of Mathematics for Uncertainty Quantification , Pontdriesch 14-16 , 52062 Aachen , Germany