Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.07702

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

Vincent N. Novellino·Dmitriy Y. Anistratov
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vincent N. Novellino, Dmitriy Y. Anistratov. 2025-01-13. Convergence of Multi-Level Hybrid Monte Carlo Methods for 1-D Particle Transport Problems. https://doi.org/10.13182/xyz-47082

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗