Young Researchers' Minisymposium

YR Ma1 Numerical Methods for PDEs with uncertainties
Eveline Rosseel (KU Leuven), Elisabeth Ullmann (TU Freiberg)

The numerical simulation of many physical processes in different areas of science, engineering and industry requires input data which are often subject to a considerable degree of uncertainty. Examples include applications in groundwater contaminant transport or solid mechanics, where material parameters, e.g., the permeability, elastic modulus, and process conditions, e.g., the amounts of precipitation, are not exactly known due to measurement errors or incomplete models. Quantitative statements on the effect of these data uncertainties are highly desirable for the evaluation of simulation results.

Mathematically, such processes can be formulated in terms of partial differential equations (PDEs) with uncertain or stochastic coefficients. The efficient discretization of these PDEs, along with robust solvers for the resulting (non)linear systems of equations are key building blocks for uncertainty quantification in simulations - once the stochastic solution is available, the actual post-processing, i.e., the computation of quantities of interest, can be performed, e.g., the computation of travel times of contaminants or failure probabilities of beams.

Many popular discretization schemes for PDE with random data result in parameterized, deterministic PDEs. Depending on the statistical properties of the random input data, the parameter space can be very high-dimensional - a problem which is often referred to as 'curse of dimensionality'. In any case the total number of degrees of freedom for a discretized PDE with random data is huge due to the coupling of physical and stochastic degrees of freedom in a simulation - even if the parameter space involves only a small number of statistically independent directions. This minisymposium highlights linear algebra techniques and approximation schemes which are being developed to tackle such challenging problems.

For problems posed in low to moderate-dimensional parameter spaces, we focus on very efficient alternatives to Monte-Carlo methods, namely, stochastic Galerkin methods and stochastic collocation schemes. One goal of this minisymposium is to present state-of-the-art iterative solvers for such discretizations. In stochastic Galerkin methods a coupled system of deterministic PDEs arises. Here, successful solution approaches are based on extending multigrid ideas to the combined physical-stochastic discretization and exploiting the structure of the Galerkin matrix. Stochastic collocation methods - including the popular sparse grid schemes - yield a decoupled sequence of deterministic PDEs, where the coefficient functions are sampled at distinct locations in the parameter space. In this case, the challenge lies in exploiting the systems' similarities to reuse information and minimize the costs for solving the entire sequence.

However, for problems in very high-dimensional parameter spaces, data sparse representations of all unknowns in the simulation are required to combat the curse of dimensionality. Hence a second aim of this minisymposium is to present some of the latest developments in this area. We address tensor approximations and sparse tensor techniques which exploit the structure of the high-dimensional stochastic approximation spaces as well as the regularity of the solution to the parameterized PDE in order to identify low-dimensional, tractable representations of the solution. In turn, novel linear algebra techniques are needed for tensor-based methods leaving ample opportunities for researchers from both analysis and numerical linear algebra to collaborate on this subject and to deliver efficient algorithms for uncertainty quantification.

Finally, we wish to present recent work on PDEs where the uncertainties are modelled in terms of fuzzy numbers and fuzzy fields. This approach is particularly useful in applications where information on the statistical properties of the uncertain coefficients, e.g., a covariance function or probability density function, is not available. Instead, the uncertainty is described with the help of membership functions assigning certain weights to possible outcomes of the coefficients. This approach, however, requires the solution of nontrivial optimization problems. By applying discretization techniques similar to stochastic Galerkin and collocation methods the efficiency of solvers for fuzzy PDEs can be enhanced.

This minisymposium aims to bring together young researchers from numerical linear algebra and analysis working on efficient algorithmic tools for uncertainty quantification in simulations.

Roman Andreev (ETH Zurich): Parametric eigenvalue problems: high-dimensional analyticity and sparse tensor collocation

Samuel Corveleyn (KU Leuven): Solving fuzzy PDEs by a polynomial chaos expansion

Andrew Gordon (U Manchester): Solving stochastic collocation systems with algebraic multigrid

Eveline Rosseel (KU Leuven): Multigrid algorithms for stochastic finite element computations

Christine Tobler (ETH Zurich): Applying low-rank tensor methods to parametric linear systems and stochastic PDEs

Elisabeth Ullmann (TU Bergakademie Freiberg): PDEs with uncertainties: discretization schemes and computational challenges