Description of random level sets by polynomial chaos expansions

TL;DR

This paper introduces a new stochastic Galerkin method for modeling the evolution of random level sets under uncertain velocity fields, providing a hyperbolic reformulation and finite-volume scheme for quantile computation.

Contribution

It presents a novel intrusive Galerkin formulation for hyperbolic level-set equations with uncertainties, tailored for uncertain velocities.

Findings

The method is proven hyperbolic.

A finite-volume scheme is developed for uncertain velocities.

Quantiles of random level sets can be computed efficiently.

Abstract

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