Fractal dimension analysis for run length diagnosis of Monte Carlo criticality calculation

Abstract

In Monte Carlo criticality calculation (MCCC), each output quantity of interest is a series of tallies under autocorrelation. As a consequence from the functional central limit theorem, the stepwise interpolation of standardized tallies (SIST) converges in distribution to Brownian bridge (BB). Here, the standardization of tallies is a functional version of the statistic in the central limit theorem with the sample mean at each generation and the true mean replaced by the sample mean at the final generation. Fractional Brownian motion (FBM) is a family of stochastic processes including Brownian motion and assumes a unique value of the box-counting dimension (BCD) in fractal geometry. This work shows that the BCD of SIST is an effective diagnostic measure for the run length of MCCC. The sufficiency of the number of generations run can be judged by relating the BCD of FBM to that of BB. Numerical results are presented for the tallies of representative autocorrelation characteristics in a three-dimensional model of a pressurized water reactor and the effective multiplication factor (k eff) tallies of the criticality problem by D. Mennerdahl.

A power spectrum approach to tally convergence in Monte Carlo criticality calculation

“…On the other hand, one cannot exclude the possibility that the real relative error / ( ) n   is practically small enough when n is not so large as to ensure ( ) . It follows from (a) and (b) in (FBM) that [18]…”

“…In Section 3, fractional Brownian motion (FBM) is introduced as a generalization of Brownian motion. The analysis of power spectrum is then proposed as a theoretically founded tally convergence assessment tool alternative to the ad-hoc run length diagnosis with fractal dimension [18]. The use of power spectrum is also argued for identifying the pre-CID phenomenon.…”

“…In fractal geometry[23], FBM is a stochastic model for continuous and non-differentiable lines with dimensions 1 through 2 and Brownian motion has a dimension of 1.5. These theories indicate that Brownian motion is a case of midway nature in Further motivation for the spectral analysis with power spectrum comes from the review of a theoretically less-founded part of the run length diagnosis of MC criticality calculation using fractal dimension analysis[18]. FBM ( ( ) and thus the path of ISTS has the dimension 2- dimension analysis was applied to compute the index  in order to see if (dimension of BM(t)).…”

Predicting correlation coefficients for Monte Carlo eigenvalue simulations with multitype branching process

“…Sutton [10] applied the discretized phase space (DPS) approach to predict the underestimation ratio but the method cannot predict the ratio when one neutron generates offspring in different phase space regions or generates a random number of offspring. Ueki [11] developed variance estimator with orthonormally weighted standardized time series (OWSTS). The estimator is based on the convergence of step-wise interpolation of standardized tallies (SIST) to Brownian bridge.…”