Spectral Convergence Properties and Optimal Performance of Monte Carlo Functional Expansion Tallies vs. the Histogram Approximation

We investigate asymptotic convergence properties of two methods for tallying responses in Monte Carlo particle transport. The first method is the histogram approximation, in which particle scores generated during a Monte Carlo simulation are projected onto a histogram representation of the domain. The second method employs functional expansion tallies, where scores are projected onto orthogonal basis functions, enabling a continuous representation of the response across the domain. For functional expansion tallies, previous work analysed the behaviour of expansion coefficients using an integration by parts framework, establishing upper bounds on the decay rates. We recast the spectral analysis in a more general Sobolev framework, obtaining both upper and lower bounds on coefficient decay under necessary and sufficient regularity conditions, applicable consistently across the bases considered. From these bounds, we derive bands for asymptotic convergence rates of truncation errors in the $L^2$ and $L^\infty$ norms. Moreover, we introduce a new optimality analysis that determines optimal configurations for both methods, namely the number of histogram bins or expansion orders that minimise total estimation error. The analysis yields explicit expressions for these optimal parameters, providing a practical and general guideline for configuring stochastic spectral estimators. We assess several orthogonal bases with respect to theoretical convergence and practical viability, and validate the theoretical framework on simple and representative numerical problems. The results are consistent with theoretical predictions and demonstrate that, depending on the smoothness of the target distribution, functional expansion tallies can yield more efficient estimates of the target response than the histogram approximation.