Numerical Solution of Systems with Stochastic Uncertainties: A General Purpose Framework for Stochastic Finite Elements

This work develops numerical techniques for the simulation of systems with stochastic parameters, modelled by stochastic partial differential equations (SPDEs). After treating the theory of linear and nonlinear elliptic SPDEs, discretisation techniques are presented. The spatial discretisation is performed by existing simulation software and the stochastic discretisation is carried out by directly integrating statistics or by expansions in tensor products of finite element shape functions times stochastic functions. Monte Carlo and Smolyak integration techniques are employed for the direct integration of statistics, whereas the discretisation by series expansions is realised either by orthogonal projections or by Galerkin methods, which yield large systems of coupled block equations. For the solution of linear SPDEs, efficient representations of the linear block equations are developed and used in iterative solvers. Due to the size of the equations, a parallel solver is supplied. The solution of nonlinear SPDEs is performed by approximate and by quasi-Newton methods. An adaptive refinement of the stochastic ansatz-spaces is implemented based on the solution of dual problems. The numerical techniques described in this thesis are implemented in a general purpose software for stochastic finite elements that allows to introduce stochastic uncertainties into existing simulation codes and that permits to propagate the input uncertainties to the system response.Inhalt der Arbeit ist die numerische Simulation von Systemen mit stochastischen Parametern, die durch stochastische partielle Differentialgleichungen (SPDGLn) beschrieben werden. Es werden die Theorie linearer und nichtlinearer elliptischer SPDGLn sowie Diskretisierungsverfahren beschrieben. Für die räumliche Diskretisierung wird eine existierende Simulationssoftware verwendet, während die stochastische Diskretisierung durch die direkte numerische Integration von Statistiken unter Verwendung von Monte Carlo- und Smolyak-Quadraturverfahren oder durch Reihenentwicklungen in Tensorprodukten finiter Elemente und stochastischer Ansatzfunktionen erfolgt. Die Reihenentwicklung wird dabei durch orthogonale Projektionen oder durch Galerkinverfahren gewonnen. Bei der Anwendung stochastischer Galerkinvervahren entstehen große Systeme gekoppelter Blockgleichungssysteme, welche hier durch iterative Verfahren gelöst werden. Zur Lösung linearer SPDGln werden effiziente Darstellungen der Gleichungssysteme und iterative Löser entwickelt. Aufgrund der Größe der entstehenden Gleichungssysteme wird ein paralleler Löser bereitgestellt. Die Lösung nichtlinearer SPDGLn geschieht durch approximative und Quasi-Newtonverfahren. Ein duales Verfahren ermöglicht die adaptive Verfeinerung der Lösung. Diese Verfahren werden in einer Allzwecksoftware für stochastische finite Elemente implementiert, die es erlaubt, existierende Simulationscodes um stochastische Unsicherheiten zu erweitern

A Review of Recent Developments in the Numerical Solution of Stochastic Partial Differential Equations (Stochastic Finite Elements)

The present review discusses recent developments in numerical techniques for the solution of systems with stochastic uncertainties. Such systems are modelled by stochastic partial differential equations (SPDEs), and techniques for their discretisation by stochastic finite elements (SFEM) are reviewed. Also, short overviews of related fields are given, e.g. of mathematical properties of random fields and SPDEs and of techniques for high-dimensional integration. After a summary of aspects of stochastic analysis, models and representations of random variables are presented. Then mathematical theories for SPDEs with stochastic operator are reviewed. Discretisation-techniques for random fields and for SPDEs are summarised and solvers for the resulting discretisations are reviewed, where the main focus lies on series expansions in the stochastic dimensions with an emphasis on Galerkin-schemes

FAST AND OPTIMAL SOLUTION ALGORITHMS FOR PARAMETERIZED PARTIAL DIFFERENTIAL EQUATIONS

This dissertation presents efficient and optimal numerical algorithms for the solution of parameterized partial differential equations (PDEs) in the context of stochastic Galerkin discretization. The stochastic Galerkin method often leads to a large coupled system of algebraic equations, whose solution is computationally expensive to compute using traditional solvers. For efficient computation of such solutions, we present low-rank iterative solvers, which compute low-rank approximations to the solutions of those systems while not losing much accuracy. We first introduce a low-rank iterative solver for linear systems obtained from the stochastic Galerkin discretization of linear elliptic parameterized PDEs. Then we present a low-rank nonlinear iterative solver for efficiently computing approximate solutions of nonlinear parameterized PDEs, the incompressible Navier–Stokes equations. Along with the computational issue, the stochastic Galerkin method suffers from an optimality issue. The method, in general, does not minimize the solution error in any measure. To address this issue, we present an optimal projection method, a least-squares Petrov–Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted l2-norm of the solution error over all solutions in a given finite-dimensional subspace. The method can be adapted to minimize the solution error in different weighted l2-norms by simply choosing a specific weighting function within the least-squares formulation

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length

Fuzzy-stochastic FEM-based homogenization framework for materials with polymorphic uncertainties in the microstructure

Uncertainties in the macroscopic response of heterogeneous materials result from two sources: the natural variability in the microstructure's geometry and the lack of sufficient knowledge regarding the microstructure. The first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function. The second type of uncertainty is denoted epistemic uncertainty. This kind of uncertainty cannot be described using probabilistic methods. Models considering both sources of uncertainties are called polymorphic. In the case of polymorphic uncertainties, some combination of stochastic methods and fuzzy arithmetic should be used. Thus, in the current work, we examine a fuzzy‐stochastic finite element method–based homogenization framework for materials with random inclusion sizes. We analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element. The stochastic finite element method is used to obtain the material response in the case of random inclusion radii. Due to unavoidable noise in experimental data, an insufficient number of samples, and limited accuracy of the fitting procedure, the radii distribution density cannot be obtained exactly; thus, it is described in terms of fuzzy location and scale parameters. The influence of fuzzy input on the homogenized stress measures is analyzed

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length