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

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

Numerical Methods for Random Parameter Optimal Control and the Optimal Control of Stochastic Differential Equations

This thesis considers the investigation and development of numerical methods for optimal control problems that are influenced by stochastic phenomena of various type. The first part treats tasks characterized by random parameters, while in the subsequent second part time-dependent stochastic processes are the basis of the dynamics describing the analyzed systems. In each case the investigations aim to transform the original problem into one that can be tackled by existing (direct) methods of deterministic optimal control - here we prefer Bock's direct multiple shooting approach. In the context of this transformation, in the first part approaches from stochastic programming as well as robust and probabilistic optimization are used. Regarding a specific application from mathematical economics, which considers pricing conspicuous consumption products in periods of recession, new numerical procedures are developed and analyzed with due regard to those techniques - in particular, a scenario tree approach, approximations of robust worst-case settings, and financial tools as the Value at Risk and Conditional Value at Risk. Furthermore, necessary reformulations of the resulting optimal control problems, in particular for Value at Risk and Conditional Value at Risk, as well as the discussion and interpretation of results determined depending on an uncertain recession duration, an uncertain recession strength, and control delays are in focus. The gained economic insight can be seen as an important step in the direction of a better understanding of real-world pricing strategies. In the second part of the thesis, based on the Wiener chaos expansion of a stochastic process and on Malliavin calculus, a system of coupled ordinary differential equations is developed that completely characterizes the stochastic differential equation describing the dynamics of the process. As in general this system includes infinitely many equations, a rigorous error estimation depending on the order of the chaos decomposition is proven in order to guarantee the numerical applicability. To transfer the generic procedure of the chaos expansion to stochastic optimal control problems, a method to preserve the feedback character of the occurring control process is shown. This allows the derivation of a novel direct method to solve finite-horizon stochastic optimal control problems. The appropriability and accuracy of this methodology are demonstrated by treating several problem instances numerically. Finally, the economic application of the first part is revisited under the viewpoint of dealing with a time-dependent recession strength, i.e., a stochastic process. In particular, those applications illustrate that the existing methods of deterministic optimal control can be extended to problems including stochastic differential equations

Reactive Flow and Transport Through Complex Systems

The meeting focused on mathematical aspects of reactive flow, diffusion and transport through complex systems. The research interest of the participants varied from physical modeling using PDEs, mathematical modeling using upscaling and homogenization, numerical analysis of PDEs describing reactive transport, PDEs from fluid mechanics, computational methods for random media and computational multiscale methods

Stochastic finite element modelling of elementary random media.

Following a stochastic approach, this thesis presents a numerical framework for elastostatics of random media. Firstly, after a mathematically rigorous investigation of the popular white noise model in an engineering context, the smooth spatial stochastic dependence between material properties is identified as a fundamental feature of practical random media. Based on the recognition of the probabilistic essence of practical random media and driven by engineering simulation requirements, a comprehensive random medium model, namely elementary random media (ERM), is consequently defined and its macro-scale properties including stationarity, smoothness and principles for material measurements are systematically explored. Moreover, an explicit representation scheme, namely the Fourier-Karhunen-Loeve (F-K-L) representation, is developed for the general elastic tensor of ERM by combining the spectral representation theory of wide-sense stationary stochastic fields and the standard dimensionality reduction technology of principal component analysis. Then, based on the concept of ERM and the F-K-L representation for its random elastic tensor, the stochastic partial differential equations regarding elastostatics of random media are formulated and further discretized, in a similar fashion as for the standard finite element method, to obtain a stochastic system of linear algebraic equations. For the solution of the resulting stochastic linear algebraic system, two different numerical techniques, i.e. the joint diagonalization solution strategy and the directed Monte Carlo simulation strategy, are developed. Original contributions include the theoretical analysis of practical random medium modelling, establishment of the ERM model and its F-K-L representation, and development of the numerical solvers for the stochastic linear algebraic system. In particular, for computational challenges arising from the proposed framework, two novel numerical algorithms are developed: (a) a quadrature algorithm for multidimensional oscillatory functions, which reduces the computational cost of the F-K-L representation by up to several orders of magnitude; and (b) a Jacobi-like joint diagonalization solution method for relatively small mesh structures, which can effectively solve the associated stochastic linear algebraic system with a large number of random variables

Singular SPDEs and fluctuations of particle systems

This thesis studies the large scale behaviour of biological processes in a random en- vironment. We start by considering a system of branching random walks in which the branching rates are determined by a random spatial catalyst. In an appropriate setting we show that this process converges to a superBrownian motion in a space white noise potential. We study the asymptotic properties of this superprocess and prove that it survives with positive probability. We then consider scaling limits of a spatial Λ–Fleming–Viot model, relating it both to the process we just introduced and to a stochastic Fisher-KPP equation. Finally, we study the longtime behaviour of the Kardar–Parisi–Zhang equation on finite volume, proving asymptotic synchroniza- tion and a one force, one solution principle. Our analyses rely on techniques from singular stochastic partial differential equations for the parabolic Anderson model and the KPZ equation, and on the theory of superprocesses

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal