When a Monte Carlo algorithm is used to evaluate a physical observable A, it is possible to slightly modify the algorithm so that it evaluates simultaneously A and the derivatives ∂ ς A of A with respect to each problem-parameter ς. The principle is the following: Monte Carlo considers A as the expectation of a random variable, this expectation is an integral, this integral can be derivated as function of the problem-parameter to give a new integral, and this new integral can in turn be evaluated using Monte Carlo. The two Monte Carlo computations (of A and ∂ ς A) are simultaneous when they make use of the same random samples, i.e. when the two integrals have the exact same structure. It was proven theoretically that this was always possible, but nothing insures that the two estimators have the same convergence properties: even when a large enough sample-size is used so that A is evaluated very accurately, the evaluation of ∂ ς A using the same sample can remain inaccurate. We discuss here such a pathological example:null-collision algorithms are very successful when dealing with radiative transfer in heterogeneous media, but they are sources of convergence difficulties as soon as sensitivity-evaluations are considered. We analyse theoretically these convergence difficulties and propose an alternative solution.first practical benefit is that the next collision event can be sampled as if the medium was homogenous. Then the choice is made to select a true-collision or a virtual-collision as function of their local respective-amounts and this is how the spatial information is recovered. But several other benefits were recently foreseen in [1] and practically tested in [5,6,7,8,9,10,11,12,13,14,15,16], mainly for radiative-transfer applications. The main idea is that null-collision algorithms transform the non-linearity of Beer-extinction into a linear-recursive problem that Monte Carlo handles without approximation [14]. This was for instance used in [5] to deal with absorption-spectra of molecular gases combining very numerous transitions: the summation over all transitions could be treated by the Monte Carlo algorithm itself, which was previously assumed impossible because this summation was inside the exponential of Beer-extinction. Similarly, the vanishing of the exponential allowed the extension of implicit Monte Carlo algorithms for inversion of absorption and scattering coefficients from intensity measurements [6]. Outside radiative transfer, a very similar idea was used to solve Electromagnetic Maxwell equations for energy propagation in particle-ensembles of statistically-distributed shapes despite of the nonlinearity associated to the square of the electric field [13]. Again similar is the algorithm proposed in [14] solving Boltzmann equation for micro-fluidics applications despite of the nonlinearity of the collision operator.Back to radiative-transfer applications, the ideas suggested in [1] have motivated significant developments in the computer-graphics community for the cinema industry. Here the benef...

We address the question of evaluating shape derivatives of objective functions for radiative-transfer engineering involving semi-transparent media. After recalling the standard Monte-Carlo approach to sensitivity estimation and its current limitations, a new method is presented for the specific case of geometrical sensitivities. This method is then tested on configurations with multiple-scattering and absorbing (nonemitting) semi-transparent medium. A new geometrical sensitivity algorithm is presented with full details in order to extend, on several examples, its implementation in complex geometries.

It was recently shown that radiation, conduction and convection can be combined within a single Monte Carlo algorithm and that such an algorithm immediately benefits from state-of-the-art computer-graphics advances when dealing with complex geometries. The theoretical foundations that make this coupling possible are fully exposed for the first time, supporting the intuitive pictures of continuous thermal paths that run through the different physics at work. First, the theoretical frameworks of propagators and Green’s functions are used to demonstrate that a coupled model involving different physical phenomena can be probabilized. Second, they are extended and made operational using the Feynman-Kac theory and stochastic processes. Finally, the theoretical framework is supported by a new proposal for an approximation of coupled Brownian trajectories compatible with the algorithmic design required by ray-tracing acceleration techniques in highly refined geometry.

We address the question of evaluating shape derivatives of objective functions for radiative-transfer engineering involving semi-transparent media. After recalling the standard Monte-Carlo approach to sensitivity estimation and its current limitations, a new method is presented for the specific case of geometrical sensitivities. This method is then tested in a square cavity filled by a multiple-scattering and absorbing (non-emitting) semi-transparent medium, irradiated by an emissive cylinder. A new geometrical sensitivity algorithm is presented with full genericity in order to allow its future implementation in complex geometries.