A Dynamical Low-rank Multilevel Monte Carlo Estimator for High-Dimensional Kinetic Equations
This work addresses the challenge of uncertainty quantification in high-dimensional kinetic equations, which is critical for applications in astrophysics, nuclear engineering, and social sciences, by providing a computationally feasible method.
The authors propose a low-rank multilevel Monte Carlo estimator for high-dimensional kinetic equations, combining dynamical low-rank approximation with multilevel Monte Carlo to efficiently quantify uncertainties. Numerical experiments in radiation transport, therapy, and shallow water flow demonstrate the estimator's efficacy, achieving significant computational savings while maintaining accuracy.
Kinetic equations are used to model a wide range of phenomena important for real-world applications. Their applications span astrophysics, nuclear physics, engineering, and social sciences. Due to their high-dimensional phase space, modelling and quantifying uncertainties, relevant for applications, poses a significant challenge even for modern computing infrastructure. In recent years, dynamical low-rank approximation (DLRA) has gained popularity for making fine grid simulations of high-dimensional problems feasible by evolving the solution of a time-dependent PDE as a low-rank factorization. This reduces the computational and memory requirements significantly. In this work, we propose a low-rank multilevel Monte Carlo estimator for kinetic equations based on a probabilistic rank-adaptive DLRA time integrator. The level hierarchy of the low-rank multilevel estimator is constructed through spatial refinement and by ensuring that the low-rank error remains below the spatial discretization error. We demonstrate the efficacy of the estimator through several numerical experiments from radiation transport, radiation therapy, and shallow water flow.