Exact Regional Monte Carlo Weight Cancellation for Second Eigenfunction Calculations

Recently, we proposed a modified power iteration method that simultaneously determines the dominant and subdominant eigenvalues and eigenfunctions of a matrix or a continuous operator. One advantage of this method is the convergence rate to the dominant eigenfunction being [vertical bar]k3[vertical bar]/k1 instead of [vertical bar]k2[vertical bar]/k1, a potentially significant acceleration. One challenge for a Monte Carlo implementation of this method is that the second eigenfunction is represented by particles of both positive and negative weights that somehow must sum (cancel) to estimate the second eigenfunction faithfully. Our previous Monte Carlo work has demonstrated the improved convergence rate by using a point flux estimator method and a binning method to effect this cancellation. This paper presents an exact method that cancels over a region instead of at points or in small bins and has the potential of being significantly more efficient than the other two.