The field of numerical methods for partial differential equations and stochastic processes is rapidly advancing, with a focus on developing innovative and efficient algorithms for solving complex problems. Recent developments have centered around improving the stability and accuracy of numerical methods, such as inf-sup stable space-time discretization and stochastic gradient descent based variational inference. Additionally, there has been a growing interest in understanding the geometric regularity of deterministic sampling dynamics in diffusion-based generative models. Noteworthy papers in this area include:

• A paper on solving partial differential equations in participating media, which proposes a novel approach using volumetric walk on spheres and volumetric walk on stars algorithms.
• A paper on a simple analysis of discretization error in diffusion models, which presents a simplified theoretical framework for analyzing the Euler--Maruyama discretization of variance-preserving SDEs.
• A paper on geometric regularity in deterministic sampling of diffusion-based generative models, which reveals a striking geometric regularity in the deterministic sampling dynamics and proposes a dynamic programming-based scheme to improve image generation performance.