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

## TL;DR

This paper investigates convergence issues in Monte Carlo derivatives of null-collision integral formulations, especially in radiative transfer, and proposes a solution to improve the accuracy of sensitivity evaluations.

## Contribution

It identifies convergence problems in Monte Carlo sensitivity analysis for null-collision algorithms and offers a theoretical analysis and an alternative method to address these issues.

## Key findings

- Null-collision algorithms cause convergence difficulties in sensitivity evaluations.
- Theoretical analysis reveals why simultaneous Monte Carlo estimators may diverge.
- An alternative solution improves the accuracy of derivative estimations.

## Abstract

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 $\partial$ $\varsigma$ A of A with respect to each problem-parameter $\varsigma$. 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 $\partial$ $\varsigma$ 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 $\partial$ $\varsigma$ 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.