Many Monte Carlo methods to solve a given Partial Differential Equation (PDE) are built by sampling the PDE's Green's function. E.g., for heat diffusion, diffusion-convection-reaction type of equations, and so on, have algorithms that can be derived directly from the PDE (i.e., through Ito calculus or stochastic integral). On the other hand, for the Radiative Transfer Equation (RTE), there is an Integral representation. However, the argument for explaining Monte Carlo Radiative Transfer (MCRT) ALWAYS revolves around the physical interpretation.
I even found a review article [1] that states on page 16: "Unlike traditional approaches to RT problems, MCRT calculations do not attempt to solve the RTE directly."
Is there really NO relation (discovered yet) between MCRT and the RTE? or is it just that no one has ever proven this?. I understand the physical interpretation; it is just that having mathematical foundations would also help teach it in class. Can anyone help me by directing me to a reference that derives this?.
[1] Noebauer, U. M., & Sim, S. A. (2019). Monte Carlo radiative transfer. Living Reviews in Computational Astrophysics, 5(1), 1-103.
It's incorrect to state that solving PDEs implies sampling the Green function. What happens with the diffusion equation is that it defines a probability measure and it is this mesure that is sampled by Monte Carlo simulations. It is possible to solve the diffusion equation by reconstructing this measure using Monte Carlo simulations that realize a random walk in the space of configurations.
The same can be done for the radiative transfer equation or any equation for that matter, by using the Hybrid Monte Carlo method. This consists in using the integrand of the action, whose extremization leads to the PDE, as the potential energy of a fictious field theory, generate the conjugate momenta from a Gaussian distribution, solve Hamilton's equations for a certain number of time steps and then accept/reject the resulting configuration with probability exp(-Delta H), where Delta H = Hfinal-Hinitial.
Stam Nicolis, I appreciate your insightful comment. Do you know of any reference (article/thesis/book) that presents this procedure in full detail, specifically for the Radiative Transfer Equation?.
These: http://www-star.st-and.ac.uk/~kw25/research/montecarlo/book.pdf
https://www.roe.ac.uk/ifa/postgrad/pedagogy/2009_forgan.pdf
might be a good place to start.
Also this one: Article Monte Carlo radiative transfer
Stam Nicolis I checked these references thoroughly. Actually, the document written by Duncan Forgan was the one I used to implement (probably) my first Radiative Monte Carlo code about 10 years ago. Still, I couldn't find what I was looking for. Maybe I am not making myself clear, but what I would like to know is if there is a document that clearly and explicitly explains the relation between the MCRT algorithm (i.e. steps involved and the iterative structure) and the Radiative Transfer Equation (e.g. the algorithm's relation to each of the operators present within the equation). I.e. a relation between MCRT and the RTE without resorting to the "imagine what happens to a photon in the physical scenario".
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