Open Access

Margarita Kostré

• Particle-based models are crucial for capturing stochastic dynamics in a diverse range of fields, including biochemistry, social dynamics, and optimization. These systems often exhibit behavior across multiple spatial or temporal scales, requiring methods that can adapt to multiscales. However, particle-based models lack a rigorous theoretical foundation due to their complexity. In biochemistry, a key challenge in particle-based modeling is developing simulations that operate consistently across the microscopic and macroscopic scale while remaining computationally feasible. This thesis establishes a mathematically consistent coupling between particle-based simulations and reactiondiffusion partial differential equations (PDEs). To this end, we derive the Kurtz representation of the reaction-diffusion master equation (RDME) and we extend the chemical diffusion master equation (CDME) to an open setting. Then, we connect thermodynamic properties across scales and incorporate cross-scale interactions by matching the mean-fields of the particle-based models (RDME and CDME) with the PDE-based dynamics. Two new hybrid numerical schemes are developed to efficiently couple particle-based domains with a concentration reservoir. Like in simulation, a key challenge in particle-based optimization is adapting effectively to multiscale landscapes. This thesis develops a novel parameter selection framework for the Particle Swarm Optimization (PSO) algorithm to navigate a multiscale loss landscape. We define new parameter regions based on the moments of particles, which characterize the exploration and exploitation of the optimization algorithm, along with suitable measures for exploration and exploitation. Building on the parameter selection framework, we show that our PSO framework identifies a greater variety of solutions when inferring cultural diffusion of Roman influence.