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Convergence issues in derivatives of Monte Carlo null-collision integral formulations: A solution

•Monte Carlo algorithm is used to evaluate simultaneously the observable and its Jacobian.•Null-collision algorithm (NCA) are very successful when dealing with linear-physics in heterogeneous media.•We showed that NCA are sources of convergence difficulties to sensitivity-evaluations of optical para...

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Published in:	Journal of computational physics 2020-07, Vol.413, p.109463-20/109463, Article 109463
Main Authors:	Tregan, J.-M., Blanco, S., Dauchet, J., El Hafi, M., Fournier, R., Ibarrart, L., Lapeyre, P., Villefranque, N.
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Language:	English
Subjects:	Algorithms Computational physics Convergence Derivatives Direct derivatives Engineering Sciences Integral formulation Integrals Monte Carlo method Null-collision algorithm Parameters Radiative transfer Random variables Sensitivity
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creator	Tregan, J.-M. Blanco, S. Dauchet, J. El Hafi, M. Fournier, R. Ibarrart, L. Lapeyre, P. Villefranque, N.
description	•Monte Carlo algorithm is used to evaluate simultaneously the observable and its Jacobian.•Null-collision algorithm (NCA) are very successful when dealing with linear-physics in heterogeneous media.•We showed that NCA are sources of convergence difficulties to sensitivity-evaluations of optical parameter.•We propose an alternative solution. 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 ∂ςA 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 ∂ςA) 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 ensures 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 ∂ςA 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.
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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 ∂ςA 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 ∂ςA) 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 ensures 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 ∂ςA using the same sample can remain inaccurate. 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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 ∂ςA 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 ∂ςA) 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 ensures 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 ∂ςA using the same sample can remain inaccurate. 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subjects	Algorithms Computational physics Convergence Derivatives Direct derivatives Engineering Sciences Integral formulation Integrals Monte Carlo method Null-collision algorithm Parameters Radiative transfer Random variables Sensitivity
title	Convergence issues in derivatives of Monte Carlo null-collision integral formulations: A solution
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