Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
10
1
2020
SIMPLEX STOCHASTIC COLLOCATION FOR PIECEWISE SMOOTH FUNCTIONS WITH KINKS
1-24
10.1615/Int.J.UncertaintyQuantification.2019030208
Barbara
Fuchs
Fraunhofer SCAI, Sankt Augustin, Germany
Jochen
Garcke
Fraunhofer SCAI, Sankt Augustin, Germany;
Fraunhofer Center for Machine Learning, Sankt Augustin, Germany;
Institute for Numerical Simulation, University of Bonn, Germany
uncertainty quantification
stochastic collocation
energy and the environment
stochastic discontinuity
adaptivity
Delaunay triangulation
gas networks
Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for nonsmooth functions with kinks. For example, such kinks can arise in the uncertainty quantification of quantities of interest for gas networks. This is due to the regulation of the gas flow, pressure, or temperature. But, one can exploit that, for each sample in the parameter space it is known if a regulator was active or not, which can be obtained from the result of the corresponding numerical solution. This information can be exploited in a stochastic collocation method. We approximate the function separately on each smooth region by polynomial interpolation and obtain an approximation to the kink. Note that we do not need information about the exact location of kinks, but only an indicator assigning each sample point to its smooth region. We obtain a global order of convergence of (p + 1)/d, where p is the degree of the employed polynomials and d the dimension of the parameter space.
SENSITIVITY ANALYSIS FOR STOCHASTIC SIMULATORS USING DIFFERENTIAL ENTROPY
25-33
10.1615/Int.J.UncertaintyQuantification.2020031610
Soumaya
Azzi
LTCI, Télécom ParisTech, Chair C2M, 46 Rue Barrault, 75013 Paris, France
Bruno
Sudret
Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Stefano-Franscini-Platz 5,
8093 Zurich, Switzerland
Joe
Wiart
LTCI, Télécom ParisTech, Chair C2M, 46 Rue Barrault, 75013 Paris, France
sensitivity analysis
differential entropy
uncertainty quantification
stochastic processes
This paper is dedicated to the sensitivity analysis of stochastic simulators. Stochastic simulators inherently contain some sources of randomness; in this case the output of the simulator in a given point is a random variable. In this paper, the stochastic simulator is represented as a stochastic process and the sensitivity analysis is performed on the differential entropy of the stochastic process. The method's performance is illustrated on a toy example, then on an electromagnetic dosimetry example.
OPTIMAL UNCERTAINTY QUANTIFICATION OF A RISK MEASUREMENT FROM A THERMAL-HYDRAULIC CODE USING CANONICAL MOMENTS
35-53
10.1615/Int.J.UncertaintyQuantification.2020030800
Jerome
Stenger
EDF R&D, 6 quai Watier, 78401 Chatou, France; Université Paul Sabatier, 118 route de Narbonne, 31400 Toulouse, France
Fabrice
Gamboa
Université Paul Sabatier, 118 route de Narbonne, 31400 Toulouse, France; Artificial and Natural Intelligence Toulouse Institute (ANITI), France
M.
Keller
EDF R&D, 6 quai Watier, 78401 Chatou, France
Bertrand
Iooss
EDF R&D, 6 quai Watier, 78401 Chatou, France; Université Paul Sabatier, 118 route de Narbonne, 31400 Toulouse, France
uncertainty quantification
canonical moments
computational statistics
stochastic optimization
In uncertainty quantification studies, a major topic of interest lies in assessing the uncertainties tainting the results of a computer simulation. In this work we seek to gain robustness on the quantification of a risk measurement by accounting for all sources of uncertainties tainting the inputs of a computer code. To that end, we evaluate the maximum quantile over a class of bounded distributions satisfying moments constraint. Two options are available when dealing with such complex optimization problems: one can either optimize under constraints, or preferably, one should reformulate the objective function. We identify a well suited parameterization to compute the maximal quantile based on the theory of canonical moments. It allows an effective, free of constraints, optimization. This methodology is applied to an industrial computer code related to nuclear safety.
EXTENDING CLASSICAL SURROGATE MODELING TO HIGH DIMENSIONS THROUGH SUPERVISED DIMENSIONALITY REDUCTION: A DATA-DRIVEN APPROACH
55-82
10.1615/Int.J.UncertaintyQuantification.2020031935
Christos
Lataniotis
Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Stefano-Franscini-Platz 5,
8093 Zurich, Switzerland
Stefano
Marelli
Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Stefano-Franscini-Platz 5,
8093 Zurich, Switzerland
Bruno
Sudret
Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Stefano-Franscini-Platz 5,
8093 Zurich, Switzerland
dimensionality reduction
surrogate modeling
Kriging
polynomial chaos expansions
kernel principal component analysis
Thanks to their versatility, ease of deployment, and high performance, surrogate models have become staple tools in the arsenal of uncertainty quantification (UQ). From local interpolants to global spectral decompositions, surrogates are characterized by their ability to efficiently emulate complex computational models based on a small set of model runs used for training. An inherent limitation of many surrogate models is their susceptibility to the curse of dimensionality, which traditionally limits their applicability to a maximum of O(102) input dimensions. We present a novel approach at high-dimensional surrogate modeling that is model-, dimensionality reduction-, and surrogate model-agnostic (black box), and can enable the solution of high-dimensional [i.e., up to O(104)] problems. After introducing the general algorithm, we demonstrate its performance by combining Kriging and polynomial chaos expansion surrogates and kernel principal component analysis. In particular, we compare the generalization performance that the resulting surrogates achieve to the classical sequential application of dimensionality reduction followed by surrogate modeling on several benchmark applications, comprising an analytical function and two engineering applications of increasing dimensionality and complexity.
UNCERTAINTY QUANTIFICATION OF DETONATION THROUGH ADAPTED POLYNOMIAL CHAOS
83-100
10.1615/Int.J.UncertaintyQuantification.2020030630
Xiao
Liang
School of Mathematics and System Science, Shandong University of Science and Technology,
Qingdao, Shandong, China and Sony Astani Department of Civil and Environmental
Engineering, University of Southern California, Los Angeles, CA
R.
Wang
Institute of Applied Physics and Computational Mathematics, Beijing, China
Roger
Ghanem
Sony Astani Department of Aerospace and Mechanical Engineering, University of Southern California, 210 KAP Hall, Los Angeles, California 90089, USA
uncertainty quantification
basis adaptation
quadratic adaptation
detonation diffraction
model reduction
Mathematical models used to describe detonation consist usually of coupled nonlinear partial differential equations, with phenomena occurring at a multitude of scales. While numerical solutions of these problems require significant computational resources, the evolution of the physics along multiple spatial and temporal scales makes the associated predictions sensitive to fluctuations that are beyond normal experimental control. Modeling, characterizing, and propagating uncertainties in predictions of detonation dynamics exacerbates both the mathematical, algorithmic, and computational challenges. These challenges are addressed in the present paper by using basis adaptation in the context of polynomial chaos expansions. The multivariate Rosenblatt transformation is used to first map all the random variables to independent Gaussian variables, following which a rotation is affected on these Gaussians that is adapted to any specified quantity of interest. Thus, accurate estimates of statistical moments and even probability density functions are obtained at specified Lagrangian reference points.