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Multi-Level Hybrid Monte Carlo / Deterministic Methods for Particle Transport Problems

Vincent N. Novellino, Dmitriy Y. Anistratov

TL;DR

This work develops Multi-Level Hybrid Monte Carlo / Deterministic (MLHT) methods for neutral particle transport by coupling low-order moment closures with MC closures across a grid hierarchy. It introduces HQD and HSM HMCD schemes and formulates MLMC optimization to estimate functionals of the flux with reduced cost via the telescoping sum $E[F_L] = E[F_0] + \sum_{\ell=1}^L E[\Delta F_\ell]$. Numerical results in 1-D slab problems show the variance of the correction terms decreases and most work is concentrated on the coarse grid, achieving mean-squared error $< \varepsilon^2$ under the MLMC conditions. The study demonstrates the validity and potential impact of MLHT for unbiased functional estimation and outlines future work toward higher-order discretizations and additional variance reduction.

Abstract

This paper presents multi-level hybrid transport (MLHT) methods for solving the neutral particle Boltzmann transport equation. The proposed MLHT methods are formulated on a sequence of spatial grids using a multi-level Monte Carlo (MLMC) approach. The general MLMC algorithm is defined by the recursive estimation of the expected value of a solution functional's correction with respect to a neighboring grid. MLMC theory optimizes the total computational cost for estimating a functional to within a target accuracy. The proposed MLHT algorithms are based on the quasidiffusion (Variable Eddington Factor) and second-moment methods. For these methods, the low-order equations for the angular moments of the high-order transport solution are discretized in space. Monte Carlo techniques compute the closures for the low-order equations; then, the equations are solved, yielding a single realization of the global flux solution. The ensemble average of the realizations yields the level solution. The results for 1-D slab transport problems demonstrates weak convergence of the functionals considered. We observe that the variance of the correction factors decreases faster than the increase in computational costs of generating an MLMC sample. In the problems considered, the variance and costs of the MLMC solution are driven by the coarse grid calculations.

Multi-Level Hybrid Monte Carlo / Deterministic Methods for Particle Transport Problems

TL;DR

This work develops Multi-Level Hybrid Monte Carlo / Deterministic (MLHT) methods for neutral particle transport by coupling low-order moment closures with MC closures across a grid hierarchy. It introduces HQD and HSM HMCD schemes and formulates MLMC optimization to estimate functionals of the flux with reduced cost via the telescoping sum . Numerical results in 1-D slab problems show the variance of the correction terms decreases and most work is concentrated on the coarse grid, achieving mean-squared error under the MLMC conditions. The study demonstrates the validity and potential impact of MLHT for unbiased functional estimation and outlines future work toward higher-order discretizations and additional variance reduction.

Abstract

This paper presents multi-level hybrid transport (MLHT) methods for solving the neutral particle Boltzmann transport equation. The proposed MLHT methods are formulated on a sequence of spatial grids using a multi-level Monte Carlo (MLMC) approach. The general MLMC algorithm is defined by the recursive estimation of the expected value of a solution functional's correction with respect to a neighboring grid. MLMC theory optimizes the total computational cost for estimating a functional to within a target accuracy. The proposed MLHT algorithms are based on the quasidiffusion (Variable Eddington Factor) and second-moment methods. For these methods, the low-order equations for the angular moments of the high-order transport solution are discretized in space. Monte Carlo techniques compute the closures for the low-order equations; then, the equations are solved, yielding a single realization of the global flux solution. The ensemble average of the realizations yields the level solution. The results for 1-D slab transport problems demonstrates weak convergence of the functionals considered. We observe that the variance of the correction factors decreases faster than the increase in computational costs of generating an MLMC sample. In the problems considered, the variance and costs of the MLMC solution are driven by the coarse grid calculations.

Paper Structure

This paper contains 15 sections, 1 theorem, 77 equations, 10 figures, 13 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Hybrid Transport Methods Based on Low-Order Equations for Moments
  3. Hybrid Quasidiffusion/VEF Method
  4. Hybrid Second Moment Method
  5. MC Estimators of Closure Functionals for Low-Order Equations
  6. Basic Idea of MLMC
  7. Multilevel Hybrid Transport Methods
  8. Multi-Level Monte Carlo Optimization
  9. The MLHT algorithms with MLMC Optimization
  10. Numerical Results
  11. Comparison of HQD and HSM Schemes
  12. Global Numerical Solution of MLHT Algorithms
  13. Convergence of MLHT algorithms with MLMC Optimization for Functionals
  14. Accuracy of Functional Estimator
  15. Conclusion

Key Result

Theorem 5.1

Let $< \Delta F_{\ell}>= \frac{1}{N_{\ell}} \sum_{n=1}^{N_{\ell}} ( F_{\ell}(\omega_{n,\ell}) - F_{\ell -1}(\omega_{n,\ell}) )$ and assume that there are constants $\alpha>0$, $\beta>0$, $\gamma>0$ such that $\alpha \ge \frac{1}{2} min(\beta,\gamma)$, and Then, $\forall \varepsilon < e^{-1}$, there exits a value $L$ (and corresponding $I_{\ell})$ and a sequence $\{ N_{\ell}\}_{\ell=0}^L$ such th

Figures (10)

  • Figure 1: Test 1. $< \phi_0 >$ and $< \Delta \phi_{\ell} >$ obtained by the MLHQD algorithm with $L=3$ and $K_{\ell}=10^4$, $\ell=0,\ldots,L$.
  • Figure 2: Test 1. $< \phi_0 >$ and $< \Delta \phi_{\ell} >$ obtained by the MLHSM algorithm with $L=3$ and $K_{\ell}=10^4$, $\ell=0,\ldots,L$.
  • Figure 3: Test 2 with $c_2=0.1$. $< \phi_0 >$ and $< \Delta \phi_{\ell} >$ obtained by the MLMC-HQD algorithm with $L=3$ and $K_{\ell}=10^4$, $\ell=0,\ldots,L$.
  • Figure 4: Test 2 with $c_2=0.1$. $< \phi_0 >$ and $< \Delta \phi_{\ell} >$ obtained by the MLMC-HSM algorithm with $L=3$ and $K_{\ell}=10^4$, $\ell=0,\ldots,L$.
  • Figure 5: Test 2, $c_2=0.5$, $F=\mathcal{F}_D$. Data on convergence of $< F>$ computed by the MLMC-HQD algorithm with $K_{\ell}=10^4$ for $\varepsilon = 1 \times 10^{-3}$.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem 5.1