Addressing nonlinearities in Monte Carlo

TL;DR

This paper introduces a novel Monte Carlo method extension that effectively handles nonlinearities by projecting onto polynomial bases and increasing configuration space, enabling previously impractical complex and rare-event problems.

Contribution

The authors develop a Monte Carlo extension that manages nonlinearities without sacrificing model complexity or convergence rates, demonstrated through four real-world test cases.

Findings

Successfully applied to phytoplankton growth modeling

Enabled handling of rare events in particle interactions

Proven effective in four complex real-world scenarios

Abstract

Monte Carlo is famous for accepting model extensions and model refinements up to infinite dimension. However, this powerful incremental design is based on a premise which has severely limited its application so far: a state-variable can only be recursively defined as a function of underlying state-variables if this function is linear. Here we show that this premise can be alleviated by projecting nonlinearities onto a polynomial basis and increasing the configuration space dimension. Considering phytoplankton growth in light-limited environments, radiative transfer in planetary atmospheres, electromagnetic scattering by particles, and concentrated solar power plant production, we prove the real-world usability of this advance in four test cases which were previously regarded as impracticable using Monte Carlo approaches. We also illustrate an outstanding feature of our method when applied to acute problems with interacting particles: handling rare events is now straightforward. Overall, our extension preserves the features that made the method popular: addressing nonlinearities does not compromise on model refinement or system complexity, and convergence rates remain independent of dimension. Published: Dauchet J, Bezian J-J, Blanco S, Caliot C, Charon J, Coustet C, El Hafi M, Eymet V, Farges O, Forest V, Fournier R, Galtier M, Gautrais J, Khuong A, Pelissier L, Piaud B, Roger M, Terr\'ee G, Weitz S (2018) Addressing nonlinearities in Monte Carlo. Sci. Rep. 8: 13302, DOI:10.1038/s41598-018-31574-4