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Background

I'm in the process of writing a Monte Carlo simulation, which solves the radiative transfer problem in arbitrary anisotropic media with (in general) scattering and absorptive properties, where the emission is purely thermal. In order to improve computational efficiency, I solve the reverse problem as opposed to the forward problem (that is, I trace rays backward from the sensor/antenna/detector, compute the loss in intensity along the way, and the final intensity using the intensity at the endpoint and the loss). Solving the forward problem might otherwise include considering rays which never reach the sensor/antenna/detector. The code works as follows:

  1. Since the media are in general anisotropic (i.e. their refractive indices are in general functions of position $\langle x,y,z \rangle$), light does not take a straight path through the medium between two points. To determine the path taken, I take Runge-Kutta steps.
  2. At each step, I compute the probability of scattering/absorption. I then draw a uniformly distributed random number. If the resulting value is less than the probability of scattering/absorption, I compute the new direction via the scattering phase function. If it is greater than the probability of interaction, I repeat.

I do this over a range of $N$ initial directions (isotropically distributed in spherical coordinates $\theta$ and $\phi$), over some boundary (e.g. a 1m box surrounding the sensor/antenna/detector), over some set of frequencies. For each direction, I follow the stochastic path of $M$ rays, and average over the resultant intensity.

Problem

Currently, I simply set $N$ and $M$ to be quite large. I'd like to instead approach this more efficiently by monitoring some convergence condition, if possible, and terminating the Monte Carlo process upon satisfactory convergence.

It isn't clear to me, however, what convergence condition I could define that would be useful for arbitrary media with arbitrary refractive, absorptive, and scattering processes.

What test(s) might I consider running for convergence here?

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