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- Elisabeth Ullmann
- SIAM J. Scientific Computing
- 2010

The discretization of linear partial differential equations with random data by means of the stochastic Galerkin finite element method results in general in a large coupled linear of system of equations. Using the stochastic diffusion equation as a model problem, we introduce and study a symmetric positive definite Kronecker product preconditioner for the… (More)

- Oliver G. Ernst, Elisabeth Ullmann
- SIAM J. Matrix Analysis Applications
- 2010

We investigate the structural, spectral and sparsity properties of Stochastic Galerkin matrices as arise in the discretization of linear differential equations with random coefficient functions. These matrices are characterized as the Galerkin representation of polynomial multiplication operators. In particular, it is shown that the global Galerkin matrix… (More)

A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We… (More)

- Aretha L. Teckentrup, Robert Scheichl, Michael B. Giles, Elisabeth Ullmann
- Numerische Mathematik
- 2013

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried… (More)

- Elisabeth Ullmann, Howard C. Elman, Oliver G. Ernst
- SIAM J. Scientific Computing
- 2012

We consider the numerical solution of a steady-state diffusion problem where the diffusion coefficient is the exponent of a random field. The standard stochastic Galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it. We consider a reformulated version of this problem as a… (More)

- Oliver G. Ernst, Catherine Elizabeth Powell, David J. Silvester, Elisabeth Ullmann
- SIAM J. Scientific Computing
- 2009

Abstract. We introduce a stochastic Galerkin mixed formulation of the steady-state diffusion equation and focus on the efficient iterative solution of the saddle-point systems obtained by combining standard finite element discretisations with two distinct types of stochastic basis functions. So-called mean-based preconditioners, based on fast solvers for… (More)

We present an overview of the stochastic finite element method with an emphasis on the computational tasks involved in its implementation.

This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control problem constrained by a diffusion equation with random coefficient together with box constraints for the control. For each… (More)

Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in… (More)

- Thomas F. Münzel, Andreas Daiber, +13 authors Swenja Kröller-Schön
- European heart journal
- 2017

Aims
Epidemiological studies indicate that traffic noise increases the incidence of coronary artery disease, hypertension and stroke. The underlying mechanisms remain largely unknown. Field studies with nighttime noise exposure demonstrate that aircraft noise leads to vascular dysfunction, which is markedly improved by vitamin C, suggesting a key role of… (More)