Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
5
3
2015
EDITORIAL PREFACE: NUMERICAL METHODS FOR UNCERTAINTY QUANTIFICATION
vii-viii
Alexey
Chernov
Institute for Mathematics, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
Fabio
Nobile
Mathematics Institute of Computational Science and Engineering, EPF Lausanne, 1015 Lausanne, Switzerland
UNCERTAINTY QUANTIFICATION FOR MAXWELL'S EQUATIONS USING STOCHASTIC COLLOCATION AND MODEL ORDER REDUCTION
195-208
Peter
Benner
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
Judith
Schneider
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
Modeling and simulation are indispensable for the design process of new semiconductor structures. Difficulties arise from shrinking structure sizes, increasing working frequencies, and uncertainties of materials and geometries. Therefore, we consider the time-harmonic Maxwell's equations for the simulation of a coplanar waveguide with uncertain material parameters. To analyze the uncertainty of the system, we use stochastic collocation with Stroud and sparse grid points. The results are compared to a Monte Carlo simulation. Both methods rely on repetitive runs of a deterministic solver. To accelerate this, we compute a reduced model by means of proper orthogonal decomposition to reduce the computational cost. The Monte Carlo simulation and the stochastic collocation are both applied to the full and the reduced model. All results are compared concerning accuracy and computation time.
SECOND-ORDER SENSITIVITY ANALYSIS OF PARAMETER ESTIMATION PROBLEMS
209-231
Ekaterina
Kostina
Institute for Apllied Mathematics, Heidelberg University, Im Neuenheimer Feld 293, 69120 Heidelberg, Germany
Max
Nattermann
Institute for Apllied Mathematics, Heidelberg University, Im Neuenheimer Feld 293, 69120 Heidelberg, Germany
The use of model-based simulation to gain knowledge of unknown phenomena and processes behavior is a challenging task in many natural sciences. In order to get a full description of an underlying process, an important issue is to estimate unknown parameters from real but erroneous observations. Thus the whole system is affected by uncertainties and a sensitivity analysis is necessary. Usually one applies first-order sensitivity analysis and resulting linearized confidence regions to determine the statistical accuracy of the solution to parameter estimation problems. But especially in significantly nonlinear cases linearized regions may not be an adequate representation. In this paper, we suggest quadratic regions based on the second-order sensitivity analysis. The new region definition is based on a map that transforms the input uncertainties onto the parameter space. Furthermore, the approximation accuracy of the quadratic confidence regions is exemplary illustrated at two examples.
A HERMITE SPECTRAL METHOD FOR A FOKKER-PLANCK OPTIMAL CONTROL PROBLEM IN AN UNBOUNDED DOMAIN
233-254
Masoumeh
Mohammadi
Institut fur Mathematik, Universitat Wurzburg, Campus Hubland Nord, Emil-Fischer-Str. 30, 97074 Wurzburg, Germany
Alfio
Borzi
Institut fur Mathematik, Universitat Wurzburg, Campus Hubland Nord, Emil-Fischer-Str. 30, 97074 Wurzburg, Germany
A Hermite spectral discretization method to approximate the solution of a Fokker-Planck optimal control problem in an unbounded domain is presented. It is proved that the solution of the corresponding discretized optimality system is spectrally accurate and the numerical scheme preserves the required conservativity property of the forward solution. The theoretical results are verified with numerical experiments.
STOCHASTIC GALERKIN METHODS AND MODEL ORDER REDUCTION FOR LINEAR DYNAMICAL SYSTEMS
255-273
Roland
Pulch
Institute of Mathematics and Computer Science, University of Greifswald,
Walther-Rathenau-Straße 47, D-17489 Greifswald, Germany
E. Jan W.
ter Maten
Centre for Analysis, Scientific computing and Applications (CASA), Dept. Mathematics & Computer Science, Technische Universiteit Eindhoven, P.O.Box 513, NL-5600 MB Eindhoven, The Netherlands; Bergische Universitat Wuppertal, D-42119 Wuppertal, Germany
Linear dynamical systems are considered in the form of ordinary differential equations or differential algebraic equations. We change their physical parameters into random variables to represent uncertainties. A stochastic Galerkin method yields a larger linear dynamical system satisfied by an approximation of the random processes. If the original systems own a high dimensionality, then a model order reduction is required to decrease the complexity. We investigate two approaches: the system of the stochastic Galerkin scheme is reduced and, vice versa, the original systems are reduced followed by an application of the stochastic Galerkin method. The properties are analyzed in case of reductions based on moment matching with the Arnoldi algorithm. We present numerical computations for two test examples.
AN OPTIMAL SAMPLING RULE FOR NONINTRUSIVE POLYNOMIAL CHAOS EXPANSIONS OF EXPENSIVE MODELS
275-295
Michael
Sinsbeck
Institute for Modeling Hydraulic and Environmental Systems (LS3)/SimTech, University of Stuttgart, Stuttgart, Germany
Wolfgang
Nowak
Institute for Modeling Hydraulic and Environmental Systems (LS3)/SimTech, University of Stuttgart, Stuttgart, Germany
In this work we present the optimized stochastic collocation method (OSC). OSC is a new sampling rule that can be applied to polynomial chaos expansions (PCE) for uncertainty quantification. Given a model function, the goal of PCE is to find the polynomial from a given polynomial space that is closest to the model function with respect to the L2-norm induced by a given probability measure. Many PCE methods approximate the involved projection integral by discretization with a finite set of integration points. Our key idea is to choose these integration points through numerical optimization based on an operator norm derived from the discretized projection operator. OSC is a generalization of Gaussian quadrature: both methods coincide for one-dimensional integration and under appropriate problem settings in multidimensional problems. As opposed to many established integration rules, OSC does not generally lead to tensor grids in multidimensional problems. With OSC, the user can specify the number of integration points independently of the problem dimension and PCE expansion order. This allows one to reduce the number of model evaluations and still achieve a high accuracy. The input parameters can follow any kind of probability distribution, as long as the statistical moments up to a certain order are available. Even statistically dependent parameters can be handled in a straightforward and natural fashion. Moreover, OSC allows reusing integration points, if results from earlier model evaluations are available. Gauss-Kronrod and Stroud integration rules can be reproduced with OSC for the respective special cases.