A Kernel-Density-Estimator Minimizing Movement Scheme for Diffusion Equations

TL;DR

This paper introduces a new variational approximation scheme based on kernel density estimation for solving complex diffusion equations of second and fourth order, providing a novel computational approach.

Contribution

It develops a mathematical theory for a kernel-density-estimator-based minimization scheme tailored for general second and fourth order diffusion equations.

Findings

Establishes convergence of the scheme for specific PDE classes

Provides a rigorous variational framework for the approximation method

Demonstrates effectiveness through theoretical analysis

Abstract

The mathematical theory of a novel variational approximation scheme for general second and fourth order partial differential equations \begin{equation}\label{eq: A} \partial_t u - \nabla\cdot\Big(u\nabla\frac{\delta\phi}{\delta u}(u)\Big|\nabla\frac{\delta\phi}{\delta u}(u)\Big|^{q-2}\Big) \ = \ 0, \quad\quad u\geq0, \end{equation} $q\in(1, +\infty)$, is developed.