arXiv subjects

Thoa Thieu

Publications and source records attributed to Thoa Thieu.

8 recordsLinked to original sources

arXiv2026

A splitting mixed finite element method for a stochastic Keller-Segel system with multiplicative noise

In this paper, we propose and analyze a splitting mixed finite element method for a stochastic Keller--Segel system with logistic growth driven by multiplicative noise. By introducing an auxiliary variable representing the chemical gradient together with a time-lagged splitting strategy, the proposed method decouples the original coupled system into a sequence of simpler subproblems. Consequently, it eliminates the Ladyzhenskaya--Babu\v{s}ka--Brezzi stability constraint, permits the use of continuous piecewise linear finite element spaces for all unknowns, and avoids solving a fully coupled nonlinear system at each time step, thereby significantly reducing the computational cost. Combined with an implicit Euler time discretization, the proposed approach yields a fully discrete numerical scheme for the stochastic Keller--Segel system. Using a localization technique together with suitable stochastic stability arguments, we establish optimal strong error estimates for the fully discrete approximations and prove convergence in probability with explicit convergence rates. Numerical experiments verify the theoretical convergence rates and demonstrate that the proposed method successfully captures the global boundedness induced by the logistic growth term as well as the influence of multiplicative noise on chemotactic aggregation.

math.NA↗

arXiv2026

Finite element approximations of the stochastic Benjamin-Bona-Mahony equation with multiplicative noise

This paper is devoted to the numerical analysis of a fully discrete finite element approximation for the stochastic Benjamin-Bona-Mahony equation driven by multiplicative noise. We first establish the existence and uniqueness of solutions to the stochastic BBM equation within an appropriate variational framework and derive several stability estimates for the continuous problem, including an exponential stability result. For the numerical approximation, a conforming finite element method is employed for spatial discretization and is coupled with the implicit Euler-Maruyama scheme for time integration. The convergence of the fully discrete scheme is investigated under two different classes of multiplicative noise. When the noise coefficient is bounded, we obtain optimal strong error estimates in full expectation by combining exponential stability properties of both the stochastic BBM solution and its fully discrete counterpart with a stochastic Gronwall inequality. In the case of general multiplicative noise, where boundedness assumptions are no longer valid, a localization technique based on high-probability events in the sample space is introduced, leading to sub-optimal convergence rates in probability. Finally, numerical experiments are presented to corroborate the theoretical results and to demonstrate the performance of the proposed method.

math.NA↗

arXiv2026

Diffusion approximations for interacting stochastic systems with reflection and control

We study diffusion approximations for a class of interacting stochastic systems with reflection and control. Motivated by interacting stochastic dynamics subject to feedback mechanisms and boundary constraints, we consider diffusion-scaled stochastic processes incorporating stochastic fluctuations, state-dependent interactions, and reflection. Under suitable assumptions, we establish convergence in distribution of the scaled processes to systems of interacting reflected stochastic differential equations of Ornstein-Uhlenbeck type. The limiting dynamics capture key features of constrained multi-agent systems, including mean-reverting behavior, interaction effects, and confinement within bounded domains through Skorokhod reflection. The analysis combines diffusion-scaling arguments, stability estimates, and continuity properties of the Skorokhod map to connect discrete stochastic systems with their reflected diffusion limits. To illustrate the framework, we present numerical examples motivated by crowd dynamics and neural population dynamics. The simulations demonstrate qualitative agreement between the finite stochastic systems and the corresponding reflected diffusion models and illustrate how diffusion approximations can provide tractable descriptions of interacting stochastic systems with constraints.

math.PR↗

arXiv2025

A Stochastic Nonlinear Dynamical System for Smoothing Noisy Eye Gaze Data

In this study, we address the challenges associated with accurately determining gaze location on a screen, which is often compromised by noise from factors such as eye tracker limitations, calibration drift, ambient lighting changes, and eye blinks. We propose the use of an extended Kalman filter (EKF) to smooth the gaze data collected during eye-tracking experiments, and systematically explore the interaction of different system parameters. Our results demonstrate that the EKF significantly reduces noise, leading to a marked improvement in tracking accuracy. Furthermore, we show that our proposed stochastic nonlinear dynamical model aligns well with real experimental data and holds promise for applications in related fields.

math.NA↗

arXiv2025

Nonlinear Dynamical Systems for Automatic Face Annotation in Head Tracking and Pose Estimation

Facial landmark tracking plays a vital role in applications such as facial recognition, expression analysis, and medical diagnostics. In this paper, we consider the performance of the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) in tracking 3D facial motion in both deterministic and stochastic settings. We first analyze a noise-free environment where the state transition is purely deterministic, demonstrating that UKF outperforms EKF by achieving lower mean squared error (MSE) due to its ability to capture higher-order nonlinearities. However, when stochastic noise is introduced, EKF exhibits superior robustness, maintaining lower mean square error (MSE) compared to UKF, which becomes more sensitive to measurement noise and occlusions. Our results highlight that UKF is preferable for high-precision applications in controlled environments, whereas EKF is better suited for real-world scenarios with unpredictable noise. These findings provide practical insights for selecting the appropriate filtering technique in 3D facial tracking applications, such as motion capture and facial recognition.

cs.CV↗

arXiv2024

Spike-timing-dependent plasticity and random inputs shape interspike interval regularity of model STN neurons

Neuronal oscillations are closely related to the symptoms of Parkinson's disease (PD). In this study, we explore how random fluctuations (or "stochastic inputs") affect these oscillations in brain states, which reflect the collective activity of interconnected neurons. These random inputs are modeled in the context of the subthalamic nucleus (STN), a brain region implicated in PD, and their interaction with synaptic dynamics and spike-timing-dependent plasticity (STDP) in both healthy and PD-affected neurons. Specifically, we investigate the effects of random synaptic inputs and their correlations on the membrane potential of STN neurons. Our results show that these random inputs significantly influence the firing patterns of STN neurons, both in healthy cells and in those affected by PD under deep brain stimulation (DBS) treatment. We also find that STDP increases the regularity of the interspike intervals (ISI) in spike trains of output neurons. However, the introduction of random refractory periods and fluctuating input currents can induce greater irregularity in the spike trains. Furthermore, when random inputs and STDP are combined, the correlation between the activity of different neurons increases. These findings suggest that the stochastic dynamics of STN neurons, in conjunction with STDP, could offer insights into the mechanisms underlying PD symptoms and their potential management.

q-bio.NC↗

arXiv2023

Social human collective decision-making and its applications with brain network models

In this chapter, we consider probabilistic drift-diffusion models and Bayesian inference frameworks to address this issue, assisting better social human decision-making. We provide details of the models, as well as representative numerical examples, and discuss the decision-making process with a representative example of the escape route decision-making phenomena by further developing the drift-diffusion models and Bayesian inference frameworks. In the latter context, we also give a review of recent developments in human collective decision-making and its applications with brain network models. Furthermore, we provide illustrative numerical examples to discuss the role of neuromodulation, reinforcement learning in decision-making processes. Finally, we call attention to existing challenges, open problems, and promising approaches in studying social dynamics and collective human decision-making, including those arising from nonequilibrium considerations of the associated processes.

physics.soc-ph↗

arXiv2018

Modelling interactions between active and passive agents moving through heterogeneous environments

We study the dynamics of interacting agents from two distinct inter-mixed populations: One population includes active agents that follow a predetermined velocity field, while the second population contains exclusively passive agents, i.e. agents that have no preferred direction of motion. The orientation of their local velocity is affected by repulsive interactions with the neighboring agents and environment. We present two models that allow for a qualitative analysis of these mixed systems. We show that the residence times of this type of systems containing mixed populations is strongly affected by the interplay between these two populations. After showing our modeling and simulation results, we conclude with a couple of mathematical aspects concerning the well-posedness of our models.

math.NA↗