Kristian Debrabant

Kristian Debrabant, Giovanni Samaey

We present and analyze a micro/macro acceleration technique for the Monte Carlo simulation of stochastic differential equations (SDEs) in which there is a separation between the (fast) time-scale on which individual trajectories of the SDE need to be simulated and the (slow) time-scale on which we want to observe the (macroscopic) function of interest. The method performs short bursts of microscopic simulation using an ensemble of SDE realizations, after which the ensemble is restricted to a number of macroscopic state variables. The resulting macroscopic state is then extrapolated forward in time and the ensemble is projected onto the extrapolated macroscopic state. We provide a first analysis of its convergence in terms of extrapolation time step and number of macroscopic state variables. The effects of the different approximations on the resulting error are illustrated via numerical experiments.

Adrien Busnot Laurent, Kristian Debrabant, Anne Kværnø

The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong approximations and adapt them for the weak approximation by replacing the iterated stochastic integrals by appropriate random variables. The methods obtained this way are sub-optimal in their number of function evaluations and the analysis of order conditions is unnecessarily complicated. We provide in this paper a novel approach, relying on well-chosen sets of random Runge-Kutta coefficients, that greatly reduce the number of order conditions. The approach is successfully applied to the creation of a collection of new stochastic Runge-Kutta methods of second weak order with an optimal number of function evaluations and a smaller number of random variables. The efficiency of the new methods is confirmed with numerical experiments and a modern algebraic approach using Hopf algebras is provided for the derivation and the study of the order conditions.

Nicklas Sindlev Andersen, Marco Chiarandini, Kristian Debrabant

Given the counters of vehicles that traverse the roads of a traffic network, we reconstruct the travel demand that generated them expressed in terms of the number of origin-destination trips made by users. We model the problem as a bi-level optimization problem. At the inner-level, given a tentative demand, we solve a Dynamic Traffic Assignment (DTA) problem to decide the routing of the users between their origins and destinations. Finally, we adjust the number of trips and their origins and destinations at the outer-level to minimize the discrepancy between the counters generated at the inner-level and the given vehicle counts measured by sensors in the traffic network. We solve the DTA problem by employing a mesoscopic model implemented by the traffic simulator SUMO. Thus, the outer problem becomes an optimization problem that minimizes a black-box Objective Function (OF) determined by the results of the simulation, which is a costly computation. We study different approaches to the outer-level problem categorized as gradient-based and derivative-free approaches. Among the gradient-based approaches, we look at an assignment matrix-based approach and an assignment matrix-free approach that uses the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm. Among the derivative-free approaches, we investigate Machine Learning (ML) algorithms to learn a model of the simulator that can then be used as a surrogate OF in the optimization problem. We compare these approaches computationally on an artificial network. The gradient-based approaches perform the best in terms of solution quality and computational requirements. In contrast, the results obtained by the ML approach are currently less satisfactory but provide an interesting avenue for future research.