Convergence of Multi-Level Hybrid Monte Carlo Methods for 1-D Particle Transport Problems

We present in this paper a hybrid, Multi-Level Monte Carlo (MLMC) method for solving the neutral particle transport equation. MLMC methods, originally developed to solve parametric integration problems, work by using a cheap, low fidelity solution as a base solution and then solves for additive correction factors on a sequence of computational grids. The proposed algorithm works by generating a scalar flux sample using a Hybrid Monte Carlo method based on the low-order Quasidiffusion equations. We generate an initial number of samples on each grid and then calculate the optimal number of samples to perform on each level using MLMC theory. Computational results are shown for a 1-D slab model to demonstrate the weak convergence of considered functionals. The analyzed functionals are integrals of the scalar flux solution over either the whole domain or over a specific subregion. We observe the variance of the correction factors decreases faster than increase in the cost of generating a MLMC sample grows. The variance and costs of the MLMC solution are driven by the coarse grid calculations. Therefore, we should be able to add additional computational levels at minimal cost since fewer samples would be needed to converge estimates of the correction factors on subsequent levels.