Advection, diffusion and linear transport in a single path-sampling Monte-Carlo algorithm : getting insensitive to geometrical refinement

We address the question of numerically simulating the coupling of diffusion, advection and one-speed linear transport with the specific objective of handling increases of the amount, the geometrical refinement and the accuracy level of input data. The computer graphics research community has succeeded in designing Monte Carlo algorithms simulating linear radiation transport in physically realistic scenes with numerical costs that are insensitive to geometrical refinement: adding more details to the scene description does not affect the computation time. The corresponding benefits in terms of engineering flexibility are already fully integrated in the cinema industry and are gradually inherited by the video game industry. We show here that the same insensitivity to the complexity of the geometrical description can also be achieved when considering one-speed linear transport not only alone but coupled with diffusion and advection. Pure linear-transport paths are replaced with advection-diffusion/linear-transport paths constituted of subpaths, each representing one of the three physical phenomena in a recursive manner. Illustration is made with a porous medium involving up to 10000 pores, the computation time being strictly independent of the number of pores