ETD

• drift, diffusion,partial,differential,equations
• interacting, particle,systems
• kernel, density,estimation
• mean, field,approximation
• nonlinear, diffusion,equations
• numerical, simulations
• particle, based,methods

This project is devoted to the study of particle-based approximation methods for a class of nonlinear drift--diffusion partial differential equations (PDEs). Such equations arise naturally in the modeling of systems across a wide range of scientific disciplines, including physics, biology, and engineering. In particular, nonlinear diffusion equations generalize the classical linear diffusion equation, which plays a central role in probability theory and in modern applications such as diffusion-based generative models and sampling algorithms. Depending on the value of the nonlinearity parameter, these equations can exhibit markedly different behaviors, ranging from slow diffusion (porous medium regime) to fast diffusion, each characterized by distinct regularity properties and long-time dynamics.

The primary focus of this work is on PDEs of drift--diffusion type, where a nonlinear diffusion term is coupled with a transport term induced by a velocity field. These equations can also be interpreted as Fokker--Planck equations associated with stochastic processes or as continuity equations associated with deterministic processes: in both cases, the coefficients depend on the evolving density itself. This perspective establishes a natural bridge between deterministic PDEs and interacting particle systems, providing both a conceptual and computational framework for their approximation.

The central goal of the project is to develop, analyze, and compare two different particle-based approaches for approximating the solutions of such nonlinear PDEs. The first approach is stochastic in nature and relies on systems of interacting particles evolving according to stochastic differential equations (SDEs) with density-dependent diffusion coefficients. The second approach is deterministic and is based on rewriting the PDE as a continuity equation with an effective drift that incorporates both transport and diffusion effects. In both cases, the key idea is to approximate the underlying continuous density by the empirical distribution of a finite number of particles, smoothed via kernel density estimation. A major challenge in this context is that the particle systems are inherently nonlinear and involve interactions through the empirical measure. As a consequence, classical analytical tools for independent particle systems do not directly apply.

On the theoretical side, the project develops a rigorous analysis of the error between the particle-based approximation and the target PDE solution. This is achieved through entropy methods, which provide a robust framework for quantifying convergence. By deriving suitable estimates on this quantity, it is possible to obtain explicit convergence rates that depend on the number of particles and on additional parameters introduced in the approximation. Furthermore, a crucial ingredient in the construction of the particle systems is the use of kernel density estimation to approximate the continuous density from the discrete particle configuration. This introduces a smoothing parameter, known as the bandwidth, which must be carefully tuned. The analysis reveals a nontrivial trade-off between the number of particles and the bandwidth: one of the main theoretical contributions of the project is the identification of an optimal scaling of the bandwidth as function of the number of particles, leading to optimal convergence rates for both the stochastic and deterministic systems.

In addition to the theoretical analysis, the project includes a comprehensive numerical investigation aimed at validating the theoretical predictions and exploring the practical performance of the proposed methods. A series of computational experiments is conducted across different regimes of the nonlinearity parameter, dimensions, and choices of kernel bandwidth. These experiments provide insight into the accuracy of the solutions.


Overall, this project aims to provide a unified and systematic study of particle-based approximation methods for nonlinear diffusion equations. By combining probabilistic techniques, analytical tools, and numerical simulations, it contributes to a deeper understanding of the interplay between particle systems and PDEs. The results obtained highlight both the strengths and limitations of different approximation strategies and offer guidance for their practical implementation.