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Andrea Barth (Universität Stuttgart)

On adaptive methods for uncertainty quantification

The accurate and efficient estimation of moments of (functionals of) solutions to stochastic problems is of high interest in the field of uncertainty quantification. Random multiscale features or when adding nonlinearities to the underlying stochastic model may lead to large local effects in the solutions and inefficiencies in the moment estimation. Standard multilevel Monte Carlo methods (MLMC) are robust, but not able to efficiently account for these local effects, resulting in high computational cost. The standard continuous level Monte Carlo method (CLMC) can account for local solution features, but with its level distribution sampled by (pseudo) random numbers, it is troubled by a high variance. Thus, we consider the quasi continuous level Monte Carlo method (QCLMC), which combines adaptivity to local solution features via samplewise a-posteriori error estimates with quasi random numbers to sample its underlying continuous level distribution. Therefore, QCLMC has the potential of a high cost reduction and improved performance in comparison to MLMC and standard CLMC, which is demonstrated via applications to random elliptic equations with discontinuous coefficients and to a random inviscid Burgers’ equation.

Alexey Chernov (Universität Oldenburg)

Parametric regularity for the stationary incompressible Navier-Stokes equation

The stationary incompressible Navier-Stokes equation is an important model in fluid mechanics. Here we are interested in the case when the coefficients and/or the forcing, and thereby the solution, may depend on parameters. The main goal is to determine the regularity of the solution with respect to the parameters. Such regularity results are necessary for the design and convergence analysis of efficient numerical approximations in the parameter domain.

The distinguishing feature of our setup is that rather general smooth but not necessarily analytic parametrizations are allowed. This makes the holomorphy arguments inapplicable. Instead we utilize the novel falling factorial argument, developed recently in [1, 2] for the related nonlinear parametric PDE models. We demonstrate the argument in the simplified setting and discuss possible applications.

Annika Lang (Chalmers University of Technology)

Stochastic partial differential equations on surfaces and evolving random surfaces: a computational approach

Looking around us, many surfaces including the Earth are no plain Euclidean domains but special cases of Riemannian manifolds. One way of describing uncertain physical phenomena on these surfaces is via stochastic partial differential equations. In this talk, I will introduce how to compute approximations of solutions to such equations and give convergence results to characterize the quality of the approximations. Furthermore, I will show how these solutions on surfaces are a first step towards the computation of time-evolving stochastic manifolds.