Monte Carlo solution of the integral equation

[myurkin]

Monte Carlo integration has been applied to volume integral equation in the scattering-order formulation (see Charon - https://www.giss.nasa.gov/staff/mmishchenko/ELS-XVI/Contributed/Charon.pdf ). While by itself it is just another iterative method (probably, not very efficient in comparison with others - #24) it is potentially very efficient in handling cases of multi-dimensional averaging (size, orientation, shape, etc.) - see #35, #54, #121. The main advantage is that the required number of paths is almost independent of the overall dimensionality.

The only immediate problem is that the Monte-Carlo scheme explicitly uses scattering-order formulation (and its convergence) - #45 . It is not clear if it will break down, when this series does not converge. Is it possible to modify it somehow to ensure fine behavior for dense highly-refractive media? In principle, any iterative solution is a polynomial of the original interaction matrix (hence, a multi-dimensional integral), so it should allow for Monte-Carlo evaluation, however, the coefficients of that polynomial are not known a priori.

[myurkin]

Here are a couple of relevant papers:

1. Charon J., Blanco S., Cornet J.-F., Dauchet J., El Hafi M., Fournier R., Abboud M.K., and Weitz S. Monte Carlo implementation of Schiff׳s approximation for estimating radiative properties of homogeneous, simple-shaped and optically soft particles: Application to photosynthetic micro-organisms, J. Quant. Spectrosc. Radiat. Transfer 172, 3–23 (2016).
2. Dauchet J., Charon J., Blanco S., Brunel L., Cornet J.-F., Coustet C., Hafi M.E., Eymet V., Forest V., Fournier R., Gros F., Piaud B., Terrée G., and Vourc’h T. Wave-scattering processes: path-integrals designed for the numerical handling of complex geometries, Opt. Lett. 48, 4909–4912 (2023).

The latter describes application of this approach to the infinite Born series. However, only a few results are shown and the convergence is slow, even for small particles with m=1.1 or 1.2.