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

Numerical Analysis of the Monte-Carlo Noise for the Resolution of the Deterministic and Uncertain Linear Boltzmann Equation (Comparison of Non-Intrusive gPC and MC-gPC)

“…This may be considered a drawback. But, on another hand, important computational gains obtained with MC-gPC have been observed on many (linear [1,3] or nonlinear [4,5,6,7]) applications. Its use allows performing studies (uncertainty propagation, sensitivity analysis etc.)…”

“…Its use allows performing studies (uncertainty propagation, sensitivity analysis etc.) which, up to now, were out of reach [1,7,6]. In this article, we are interested in the MC resolution of the gPC based reduced model of the uncertain linear transport equation ∂ t u(x, t, v, X) + v • ∇ x u(x, t, v, X) = − vσ t (x, v, X)u(x, t, v, X)…”

“…mean, variance, histogram, sensitivity indices [17] etc.) at specified locations x ∈ D, times t ∈ [0, T ] and velocities v ∈ V. The uncertain transport equation is of importance in many physical domains such as neutronics [18,19,20,7], photonics [21,22,23,24,25,6], biology [26], socio-economics [5,27,3], epidemiology [28] etc. In neutronics [7] or photonics [6] for example, equation (1) must be solved at each iteration/time step.…”

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Monte Carlo-generalised Polynomial Chaos (MC-gPC) has already been thoroughly studied in the literature [1,2,3,4,5,6,7]. MC-gPC both builds a gPC based reduced model of a partial differential equation (PDE) of interest and solves it with an intrusive MC scheme in order to propagate uncertainties.

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“…But, on another hand, important computational gains obtained with MC-gPC have been observed on many (linear [1,3] or nonlinear [4,5,6,7]) applications. The MC-gPC resolution of Boltzmann equation has been investigated in many different ways: the wellposedness of the gPC based reduced model has been studied in [2,6], the convergence with respect to the truncation order P has been theoretically and numerically studied [2, 1], the coupling to nonlinear physics has been performed in [7,6]. But the study of the MC noise remains, to our knowledge, to be done.…”