Functional Expansion Tallies of Matrix Operators for Prediction for Integrated Autocorrelation Time in Batch Monte Carlo: an Analytic 2D Scattering Chain Benchmark
We investigate functional expansion tallies as a reduced-basis representation for predicting inter-cycle correlations in Monte Carlo transport. Using an analytic two-dimensional isotropic scattering-chain benchmark with reflective boundaries, we compare a conventional discrete-cell Markov-chain estimator with a Galerkin reduced-order model built directly from Monte Carlo tallies of basis-function products. The reduced model estimates integrated autocorrelation time without first constructing a large discrete transition matrix. For the benchmark problem, the cosine basis converges rapidly to the exact result, while polynomial bases show systematic convergence with increasing order. Compared with discrete binning, the reduced-basis approach achieves lower bias at comparable or lower solve cost, suggesting that functional-expansion representations can provide an efficient path toward correlation prediction, uncertainty quantification, and future variance-reduction methods in Monte Carlo criticality calculations.