Low-rank variance reduction for uncertain radiative transfer with control variates

TL;DR

This paper introduces a low-rank Monte Carlo estimator combined with control variates based on multi-fidelity approximations to efficiently reduce variance in solving uncertain radiative transfer equations, improving computational efficiency.

Contribution

It proposes a novel low-rank Monte Carlo estimator with control variates for variance reduction in uncertain radiative transfer problems, integrating dynamical low-rank approximation techniques.

Findings

Joint rank and grid size balancing is essential.

The combined approach enhances estimator efficiency.

Numerical results confirm variance reduction benefits.

Abstract

The radiative transfer equation models various physical processes ranging from plasma simulations to radiation therapy. In practice, these phenomena are often subject to uncertainties. Modeling and propagating these uncertainties requires accurate and efficient solvers for the radiative transfer equations. Due to the equation's high-dimensional phase space, fine-grid solutions of the radiative transfer equation are computationally expensive and memory-intensive. In recent years, dynamical low-rank approximation has become a popular method for solving kinetic equations due to the development of computationally inexpensive, memory-efficient and robust algorithms like the augmented basis update \& Galerkin integrator. In this work, we propose a low-rank Monte Carlo estimator and combine it with a control variate strategy based on multi-fidelity low-rank approximations for variance reduction. We investigate the error analytically and numerically and find that a joint approach to balance rank and grid size is necessary. Numerical experiments further show that the efficiency of estimators can be improved using dynamical low-rank approximation, especially in the context of control variates.