Category Archives: Stochastic Differential Equations

Superlinear Drift in Consensus-Based Optimization with Condensation Phenomena

by Mattia Zanella

J. Franceschi, L. Pareschi, M. Zanella

Preprint arXiv, 2025

Consensus-based optimization (CBO) is a class of metaheuristic algorithms designed for global optimization problems. In the many-particle limit, classical CBO dynamics can be rigorously connected to mean-field equations that ensure convergence toward global minimizers under suitable conditions.

In this work, we draw inspiration from recent extensions of the Kaniadakis–Quarati model for indistinguishable bosons to develop a novel CBO method governed by a system of SDEs with superlinear drift and nonconstant diffusion. The resulting mean-field formulation in one dimension exhibits condensation-like phenomena, including finite-time blow-up and loss of L2-regularity. To avoid the curse of dimensionality a marginal based formulation which permits to leverage the one-dimensional results to multiple dimensions is proposed. We support our approach with numerical experiments that highlight both its consistency and potential performance improvements compared to classical CBO methods.

Measure-valued death state and local sensitivity analysis for Winfree models with uncertain high-order couplings

by Mattia Zanella

S.-Y. Ha, M. Kang, J. Yoon, M. Zanella

Communications in Mathematical Sciences, 23(6):1583-1629, 2025. (Preprint arXiv)

We study the measure-valued death state and local sensitivity analysis of the Winfree model and its mean-field counterpart with uncertain high-order couplings. The Winfree model is the first mathematical model for synchronization, and it can cast as the effective approximation of the pulse-coupled model for synchronization, and it exhibits diverse asymptotic patterns depending on system parameters and initial data. For the proposed models, we present several frameworks leading to oscillator death in terms of system parameters and initial data, and the propagation of regularity in random space. We also present several numerical tests and compare them with analytical results.

Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties

by Mattia Zanella

José Antonio Carrillo, Mattia Zanella

Vietnam Journal of Mathematics, 47(4):931-954, 2019. (Preprint arXiv)

In this paper we introduce and discuss numerical schemes for the approximation of kinetic equations for flocking behavior with phase transitions that incorporate uncertain quantities. This class of schemes here considered make use of a Monte Carlo approach in the phase space coupled with a stochastic Galerkin expansion in the random space. The proposed methods naturally preserve the positivity of the statistical moments of the solution and are capable to achieve high accuracy in the random space. Several tests on a kinetic alignment model with self propulsion validate the proposed methods both in the homogeneous and inhomogeneous setting, shading light on the influence of uncertainties in phase transition phenomena driven by noise such as their smoothing and confidence bands.

 

Stochastic modeling and simulation of ion transport through channels

by Mattia Zanella

Fig6d_grandeDaniela Morale, Mattia Zanella, Vincenzo Capasso, Willi Jäger                                                                                            

SIAM Journal on Multiscale Modeling & Simulation, 14(1): 113-137, 2016. (Preprint arXiv)

Ion channels are of major interest and form an area of intensive
research in the fields of biophysics and medicine since they control many vital physiological functions. The aim of this work is on one hand to propose a fully stochastic and discrete model describing the main characteristics of a multiple channel system. The movement of the ions is coupled, as usual, with a Poisson equation for the electrical field; we have considered, in addition, the influence of exclusion forces. On the other hand, we have discussed about the nondimensionalization of the stochastic system by using real physical parameters, all supported by numerical simulations. The specific features of both cases of micro- and nanochannels have been taken in due consideration with particular attention to the latter case in order to show that it is necessary to consider a discrete and stochastic model for ions movement inside the channels.

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