We present recent developments in crowd dynamics models (e.g. pedestrian flow problems). Our formulation is given by a mean-field game (MFG) with congestion. We start by reviewing earlier models and results. Next, we develop our model. We establish new a priori estimates that give partial regularity of the solutions. Finally, we discuss numerical results.

The dynamics of biochemical reactive systems with small copy numbers of one or more reactant molecules is dominated by stochastic effects. For those systems, discrete state-space and stochastic simulation approaches were proved to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slowtimescales, the existing discrete space-state stochastic path simulation methods such as the stochastic simulation algorithm (SSA) and the explicit tauleap method can be very slow. Implicit approximations were developed in the literature to improve numerical stability and provide efficient simulation algorithms for those systems. In this work, we propose an efficient Multilevel Monte Carlo method in the spirit of the work by Anderson and Higham (2012) that uses drift-implicit tau-leap approximations at levels where the explicit tauleap method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.

In this work, we present a statistical treatment of stress-life (S-N) data drawn from a collection of records of fatigue experiments that were performed on 75S-T6 aluminum alloys. Our main objective is to predict the fatigue life of materials by providing a systematic approach to model calibration, model selection and model ranking with reference to S-N data. To this purpose, we consider fatigue-limit models and random fatigue-limit models that are specially designed to allow the treatment of the run-outs (right-censored data). We first fit the models to the data by maximum likelihood methods and estimate the quantiles of the life distribution of the alloy specimen. We then compare and rank the models by classical measures of fit based on information criteria. We also consider a Bayesian approach that provides, under the prior distribution of the model parameters selected by the user, their simulation-based posterior distributions.

In [5], we present a novel multilevel Monte Carlo method for kinetic simulation of stochastic reaction networks (SRNs) specifically designed for systems in which the set of reaction channels can be adaptively partitioned into two subsets characterized by either high or low activity. To estimate expected values of observables of the system, our method bounds the global computational error to be below a prescribed tolerance, TOL, within a given confidence level. This is achieved with a computational complexity of order O(TOL-2). We also present a novel control variate technique which may dramatically reduce the variance of the coarsest level at a negligible computational cost.

Markovian pure jump processes can model many phenomena, e.g. chemical reactions at molecular level, protein transcription and translation, spread of epidemics diseases in small populations and in wireless communication networks, among many others. In this work [6] we present a novel multilevel algorithm for the Chernoff-based hybrid tauleap algorithm. This variance reduction technique allows us to: (a) control the global exit probability of any simulated trajectory, (b) obtain accurate and computable estimates for the expected value of any smooth observable of the process with minimal computational work.

This work embeds a multilevel Monte Carlo (MLMC) sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF). In terms of computational cost vs. approximation error the asymptotic performance of the multilevel ensemble Kalman filter (MLEnKF) is superior to the EnKF s.