PhD defence of Damien Rioux-Lavoie – Flexible Monte Carlo Methods for Fluid and Light Transport Simulations

Abstract

The remarkable progress in graphics processing units over recent decades has given rise to a significant increase in the application of computer graphics across various industries, such as in cinema and video games. Consequently, there has been a surge in opportunities to push the boundaries of computer graphics algorithms, leading to the development of highly intricate fluid simulations and photorealistic rendering. These methods now serve as the foundation for AAA games, animation features, and blockbuster movies.

This thesis emphasizes the development and application of flexible Monte Carlo techniques for two specific subfields within computer graphics, namely physically based rendering and fluid simulations. The rationale behind this focus is the shared similarities between the equations and mathematical models governing both domains, presenting an opportunity to bridge the gap between them and formulate new and effective methods. Moreover, since the realism of a fluid simulation rendering is inherently tied to the simulation itself, and vice versa, advancements in both areas are crucial for achieving maximum impact.

First, we present a versatile two-stage mutation strategy based on the delayed rejection Markov chain Monte Carlo framework to generalize the Metropolis light transport algorithm. By generating multiple proposals informed by previous failures while maintaining Markov chain ergodicity, we can develop efficient strategies such as trying a cheap mutation first, followed by a more expensive one only upon failure. This approach allows for the optimal allocation of computational resources, which is critical when tackling complex scenes.

Drawing on the success of Monte Carlo methods in physically based rendering and, more recently, in discrete geometry processing, we propose a Monte Carlo approach for fluid simulations. Specifically, we employ the Feynman–Kac stochastic representation of the vorticity transport equation and devise a recursive Monte Carlo estimator of the

Biot-Savart law that can generate pointwise approximate solutions. We expand this method with a stream function formulation that enables us to manage free-slip boundary conditions using a Walk-on-Spheres algorithm. To our knowledge, this is the first time that Monte Carlo methods have been studied in the context of fluid simulations, opening the door to a new family of solvers. We offer an in-depth examination of several potential directions for future research based on this novel numerical simulation modality.