Communications in Mathematical Sciences

Volume 22 (2024)

Number 1

Description of random level sets by polynomial chaos expansions

Pages: 95 – 112

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n1.a4

Authors

Markus Bambach (Advanced Manufacturing Lab, ETH Zürich, Switzerland)

Stephan Gerster (Institut für Mathematik, University of Mainz, Germany)

Michael Herty (Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany)

Aleksey Sikstel (Department of Mathematics, TU Darmstadt, Germany)

Abstract

We present a novel approach to determine the evolution of level sets under uncertainties in their velocity fields. This leads to a stochastic description of level sets. To compute the quantiles of random level sets, we use the stochastic Galerkin method for a hyperbolic reformulation of the equations for the propagation of level sets. A novel intrusive Galerkin formulation is presented and proven to be hyperbolic. It induces a corresponding finite-volume scheme that is specifically tailored to uncertain velocities.

Keywords

level sets, uncertainty quantification, Hamilton–Jacobi equations, hyperbolic conservation laws, stochastic Galerkin, finite-volume method

2010 Mathematics Subject Classification

35F21, 37L45, 60D05, 60H15

Received 15 October 2021

Received revised 18 September 2022

Accepted 8 May 2023

Published 7 December 2023