Canceling Teleportation Error in Legacy IMC Code for Photonics (Without Tilts, With Simple Minimal Modifications)

Abstract

Monte Carlo (MC) schemes for photonics have been intensively studied throughout the past decades. The recent ISMC scheme presents many advantages (no teleportation error, converging behavior with respect to the spatial and time discretisations). But it is rather different from the IMC one (it is based on a different linearization and needs a slightly different code architecture). On another hand, legacy codes are often based on IMC implementations. For this reason, it remains important to be able to cancel the teleportation error within IMC codes. Canceling the teleportation error within the IMC framework is also important for fair comparisons between both the IMC and the ISMC linearisations. This paper aims at suggesting some simple corrections to apply to an IMC implementation to completely cancel the teleportation error.

Keywords:

• Transport
• Monte Carlo
• numerical scheme
• photonics
• IMC
• ISMC

Notes

1 Whose linearisation demands unaffordable time steps.

2 Leading to affordable time steps but introducing the teleportation error.

3 Tilts of different order or different nature, on-the-fly resampling etc.

4 By construction without teleportation error nor competing behavior between the spatial and time discretisations.

5 Equilibrium means T_m = T_r.

6 Diffusion refers to the presence of the second order spatial term in (2(2) $$\{\begin{array}{c}{\partial }_t({\Phi }_r(T_r)+E(T_r))-\nabla ·\left(\frac{c}{3{\sigma }_t}\nabla (\Phi (T_r)\right)=\mathcal{O}(\delta ),\hfill \\ {\Phi }_r(T_r)={\int }_{4\pi }\frac{I}{4\pi }d\omega =B(T_m)+\mathcal{O}(\delta ).\hfill \end{array}$$(2) ).

7 Explicit with respect to $\sigma ,\beta ,$ implicit with respect $\Phi .$

8 Usually, ${\Phi }_{\mathit{approx}}^n$ is based on a Taylor development and ${\Phi }_{i,1}^n$ is a first order spatial derivative.

9 i.e. we have ${\mathbf{1}}_{{\Omega }_i}(x)=1$ if $x\in {\Omega }_i,{\mathbf{1}}_{{\Omega }_i}(x)=0$ if $x\notin {\Omega }_i.$

10 i.e. no need for a tilt or any spatial reconstructions within the cells.

11 To go from the first to the second equation, the isotropy of the source term is used.

12 Convergence in law is also commonly called weak convergence and is defined as such: a sequence of probability measures ${( \mathrm{d}{\mu }_n)}_{n\in \mathbb{N}}$ is said to convergence in law toward measure $\mathrm{d}\mu$ if $\int f(x) \mathrm{d}{\mu }_n(x)$ converges toward $\int f(x) \mathrm{d}\mu (x)$ as n goes to infinity for all bounded Lipshitz function f. In our MC context, n echoes N_MC and $\mathrm{d}{\mu }_n={\sum }_{p=1}^n{i}_p(x,t,\omega )$ (the Dirac ${\delta }_x,{\delta }_{\omega }$ in (17(17) $$i_p(x,t,\omega )=w_p(t){\delta }_x(x_p(t)){\delta }_{\omega }({\omega }_p(t)),$$(17) ) are measures with our notations).

13 Introduce $F_{\tau }$ the integral of $f_{\tau },$ then by definition, see Saporta (2006), $\tau =F_{\tau }^{-1}(\mathcal{U})$ where $\mathcal{U}$ is uniformly distributed on $[0,1].$

14 Replication domain consists in replicating the geometry on several processors and tracking several MC particles populations with different initial seeds in every replicated domains. At the end of the time steps, the contribution of every processors are averaged. This parallel strategy is particularly well suited to MC codes, taking advantage of the independence of the MC particles.

15 In the next numerical Section 5, we do use the semi-analog MC scheme for (16(16) $$\begin{array}{cc}{\partial }_{t}I+c\omega ·\nabla I+c{\Sigma }_{t}I\hfill & ={\int }_{4\pi }\left(\frac{S}{U}+c{\Sigma }_s\right)I\frac{d\omega \prime }{4\pi },\hfill \\ \hfill & =c{\Sigma }_S{\int }_{4\pi }I\frac{d\omega \prime }{4\pi },\hfill \end{array}$$(16) ) for the practical reason previously exposed. But it is way easier comparing both non-analog MC schemes on the paper. This does have a sense as both semi-analog and non-analog MC scheme recover asymptocally the same solution: the comments on the non-analog nssIMC can be transposed, without loss of generality, to the semi-analog nssIMC one.

16 Still, it does cancel the teleportation error.

17 Note that the upperscripts are relative to the δ development whereas the lowerscripts are relative to the ϵ one.

18 The calculations are similar to the ones performed in Poëtte and Valentin (2020) and Densmore and Larsen (2004). The last equation is integrated with respect to ω.

19 We drop the ⁰ upperscripts for convenience.

20 ISMC is an implicitation of SMC, see Poëtte and Valentin (2020) and 8], which by construction is a teleporation error free solver.

21 Just as tilted IMC consider higher orders of ${\Phi }^n(x)={\Phi }_{\mathit{approx}}^n(x)+\mathcal{O}({\delta }_x^{k>1})$ with respect to δ_x.

22 One can check that in a closed cell, every time intervals become (only interval) $[0,t]$ and conservativity is ensured.

23 Note that there are N_x = 40 cells instead of only 20 as in Poëtte and Valentin (2020).

24 Note that the results from IMC are not displayed but can be found in Poëtte and Valentin (2020).

25 Note that the results tilted IMC are not displayed but can be found in Poëtte and Valentin (2020).

26 i.e. the last cell in which the radiative temperature T_r is different than the initial temperature.

27 Note that the results from IMC are not displayed but can be found in Poëtte and Valentin (2020).

28 Note that the results from tilted IMC are not displayed but can be found in Poëtte and Valentin (2020).

29 i.e. the last cell in which the radiative temperature T_r is different than the initial temperature.

30 The faster spatial convergence rate of ISMC with respect to nssIMC.

31 As soon as the modified Fleck factor is positive, the artificial opacities are positive and the MC scheme to discretise both photons and matter ensure the positiveness of both quantities for stable calculations, see Poëtte and Valentin (2020).

32 In the sense nssIMC converges as the time step goes to zero for a fixed grid whereas IMC presents a diverging behavior in the same conditions, see the definition in Remark 3.1 and the illustration in Section 3.