Navigating Uncertainty: Stochastic Methods for Nonlinear Systems and the Women Leading the Way
Alina Chertock (Prof. Dr.)

Many important scientific problems involve several sources of uncertainties, such as
model parameters and initial and boundary conditions. Quantifying these uncertainties
is essential for many applications since it helps to conduct sensitivity analysis and
provides guidance for improving the models. The design of reliable numerical methods
for models with uncertainties has seen a lot of activity lately. One of the most popular
methods is Monte Carlo-type simulations, which are generally good but inefficient due
to the large number of realizations required. In addition to Monte Carlo methods, a
widely used approach for solving partial differential equations with uncertainties is the
generalized polynomial chaos (gPC), where stochastic processes are represented in
terms of orthogonal polynomials series of random variables. It is well-known that gPC-
based methods, which are spectral-type methods, exhibit fast convergence when the
solution depends smoothly on random parameters. However, their application to
nonlinear systems of conservation/balance laws still encounters some significant
difficulties. The latter is related to the presence of discontinuities that may develop in
numerical solutions in finite time, triggering the appearance of aliasing errors and
Gibbs-type phenomena. This talk will provide an overview of numerical methods for
models with uncertainties and explore strategies to address the challenges
encountered when applying these methods to nonlinear hyperbolic systems of
conservation and balance laws.