Efficient uncertainty propagation for photonics: Combining Implicit Semi-analog Monte Carlo (ISMC) and Monte Carlo generalised Polynomial Chaos (MC-gPC)

Abstract

In this paper, we build wellposed intrusive generalised Polynomial Chaos (gPC) based reduced models for uncertain photonics. We solve the reduced models with a Monte-Carlo (MC) scheme. Care is taken to highlight under which condition a reduced model (gPC based or not) is wellposed. The analysis is carried out thanks to an analogy between the construction of reduced models for uncertainty quantification and the construction of reduced models for kinetic equations. In order to enforce the aforementioned wellposedness conditions, several strategies, inspired from the hyperbolicity-preserving ones [1–8] are reviewed, adapted and analysed. The resolution of the reduced models is performed thanks to an astute combination of the Implicit Semi-analog MC (ISMC, see [9]) scheme for photonics and of MC-gPC (see [10]) for uncertainty propagation. This work demonstrates that MC-gPC can be efficiently applied to a stiff nonlinear set of partial derivative equations if the MC resolution allows a fast convergence with respect to both the time and spatial discretisations (the latter properties being allowed by ISMC). Several benchmarks are investigated in the last section, they allow illustrating important aspects of the new ISMC-gPC solver for uncertain photonics.