A Functional Monte Carlo Method for k-Eigenvalue Problems

In Monte Carlo simulations of k-eigenvalue problems for optically thick fissile systems with a high dominance ratio, the eigenfunction is often poorly estimated because of the undersampling of the fission source. Although undersampling can be addressed by sufficiently increasing the number of particles per cycle, this can be impractical in difficult problems. Here, we present a new functional Monte Carlo (FMC) method that minimizes this difficulty for many problems and yields a more accurate estimate of the k-eigenvalue. In the FMC method, standard Monte Carlo techniques do not directly estimate the eigenfunction; instead, they directly estimate certain nonlinear functionals that depend only weakly on the eigenfunction. The functionals are then used to more accurately estimate the k-eigenfunction and the eigenvalue. Like standard Monte Carlo methods, the FMC method has only statistical errors that limit to zero as the number of particles per cycle and the number of cycles become large. We provide numerical results that illustrate the advantages and limitations of the new method.