Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
9
5
2019
ROBUST UNCERTAINTY QUANTIFICATION USING RESPONSE SURFACE APPROXIMATIONS OF DISCONTINUOUS FUNCTIONS
415-437
10.1615/Int.J.UncertaintyQuantification.2019026974
Timothy
Wildey
Optimization and Uncertainty Quantification Department, Center for Computing Research,
Sandia National Laboratories, Albuquerque, NM 87185
A. A.
Gorodetsky
University of Michigan, Department of Aerospace Engineering, Ann Arbor, MI 48109
A.
Belme
Sorbonne Universités, UPMC Univ Paris 06, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France;
CNRS, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France
John N.
Shadid
Computational Mathematics Department, Center for Computing Research, Sandia National
Laboratories, Albuquerque, NM 87185, and Department of Mathematics and Statistics,
University of New Mexico
response surface approximation
discontinuity
machine learning
gradient-enhanced
epistemic uncertainty
robust error bounds
This paper considers response surface approximations for discontinuous quantities of interest. Our objective is not
to adaptively characterize the interface defining the discontinuity. Instead, we utilize an epistemic description of the uncertainty in the location of a discontinuity to produce robust bounds on sample-based estimates of probabilistic quantities of interest. We demonstrate that two common machine learning strategies for classification, one based on nearest neighbors (Voronoi cells) and one based on support vector machines, provide reasonable descriptions of the region where the discontinuity may reside. In higher dimensional spaces, we demonstrate that support vector machines are more accurate for discontinuities defined by smooth interfaces. We also show how gradient information, often available via adjoint-based approaches, can be used to define indicators to effectively detect a discontinuity and to decompose the samples into clusters using an unsupervised learning technique. Numerical results demonstrate the
epistemic bounds on probabilistic quantities of interest for simplistic models and for a compressible fluid model with a
shock-induced discontinuity.
BAYESIAN OPTIMAL EXPERIMENTAL DESIGN INVOLVING MULTIPLE SETUPS FOR DYNAMIC STRUCTURAL TESTING
439-452
10.1615/Int.J.UncertaintyQuantification.2019025897
Sahil
Bansal
Indian Institute of Technology Delhi
optimal experimental design
Bayesian statistical framework
multiple setups
model updating
genetic algorithm
In an experimental design for dynamic structural testing, it is a common practice to obtain data from a structure using multiple setups, with each setup covering a different part of the structure. This is generally due to availability of limited number of sensors with synchronous data acquisition. This paper considers the problem of optimal placement of actuators and sensors with the aim of maximizing the data information in an experimental design with multiple setups so that the structural dynamic behavior can be fully characterized. The uncertainty in the model parameters is computed by a Bayesian statistical framework and expected utility is used to quantify the uncertainty of the set of identified model parameters. The problem of optimal experimental design with multiple setups is formulated as an optimization problem in which the actuators' and sensors' configuration, which maximizes the expected utility, is selected as the optimal one. The proposed approach can be used to compare and evaluate the quality of the parameter estimates that can be achieved with different numbers of setups in an experimental design and a different number of actuators and sensors in each setup. The effectiveness of the proposed approach is illustrated by optimal experimental design for a simple 12-degree-of-freedom (dof) chainlike spring-mass model of a structure and a 37-dof truss structure.
ROBUSTNESS OF THE SOBOL' INDICES TO DISTRIBUTIONAL UNCERTAINTY
453-469
10.1615/Int.J.UncertaintyQuantification.2019030553
Joseph
Hart
Department of Mathematics, North Carolina State University, Raleigh, NC
Pierre A.
Gremaud
Department of Mathematics, North Carolina State University, Raleigh, North Carolina
27695-8205
global sensitivity analysis
Sobol' indices
uncertain distributions
deep uncertainty
Global sensitivity analysis (GSA) is used to quantify the influence of uncertain variables in a mathematical model. Prior to performing GSA, the user must specify (or implicitly assume) a probability distribution to model the uncertainty, and possibly statistical dependencies, of the variables. Determining this distribution is challenging in practice as the user has limited and imprecise knowledge of the uncertain variables. This paper analyzes the robustness of the Sobol' indices, a commonly used tool in GSA, to changes in the distribution of the uncertain variables. A method for assessing such robustness is developed that requires minimal user specification and no additional evaluations of the model. Theoretical and computational aspects of the method are considered and illustrated through examples.
KERNEL-BASED STOCHASTIC COLLOCATION FOR THE RANDOM TWO-PHASE NAVIER-STOKES EQUATIONS
471-492
10.1615/Int.J.UncertaintyQuantification.2019029228
M.
Griebel
Institute for Numerical Simulation, Bonn University, Endenicher Allee 19b, D-53115 Bonn,
Germany; Fraunhofer Institute for Algorithms and Scientific Computing SCAI, Schloss Birlinghoven,
D-53754 Sankt Augustin, Germany
C.
Rieger
Institute for Numerical Simulation, Bonn University, Endenicher Allee 19b, D-53115 Bonn,
Germany; Department of Mathematics, RWTH Aachen University, Schinkelstr. 2, D-52062 Aachen,
Germany
Peter
Zaspel
Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051
Basel, Switzerland
stochastic collocation
incompressible two-phase Navier-Stokes
uncertainty quantification
In this work, we apply stochastic collocation methods with radial kernel basis functions for an uncertainty quantification of the random incompressible two-phase Navier-Stokes equations. Our approach is nonintrusive and we use the existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase Navier-Stokes equation for each given realization. We are able to empirically show that the resulting kernel-based stochastic collocation is highly competitive in this setting and even outperforms some other standard methods.
SHAPLEY EFFECTS FOR SENSITIVITY ANALYSIS WITH CORRELATED INPUTS: COMPARISONS WITH SOBOL' INDICES, NUMERICAL ESTIMATION AND APPLICATIONS
493-514
10.1615/Int.J.UncertaintyQuantification.2019028372
Bertrand
Iooss
EDF Lab Chatou, 6 Quai Watier, 78401 Chatou, France; Institut de Mathématiques de
Toulouse, 31062, Toulouse, France
Clementine
Prieur
Université Grenoble Alpes, CNRS, LJK, F-38000 Grenoble, France; Inria Project/Team
AIRSEA, France
dependence
metamodel
sensitivity analysis
Shapley
Sobol'
variance contribution
The global sensitivity analysis of a numerical model aims to quantify, by means of sensitivity indices estimates, the contributions of each uncertain input variable to the model output uncertainty. The so-called Sobol' indices, which are based on functional variance analysis, present a difficult interpretation in the presence of statistical dependence between inputs. The Shapley effects were recently introduced to overcome this problem as they allocate the mutual contribution (due to correlation and interaction) of a group of inputs to each individual input within the group. In this paper, using several new analytical results, we study the effects of linear correlation between some Gaussian input variables on Shapley effects, and compare these effects to classical first-order and total Sobol' indices. This illustrates the interest, in terms of sensitivity analysis setting and interpretation, of the Shapley effects in the case of dependent inputs. For the practical issue of computationally demanding computer models, we show that the substitution of the original model by a metamodel (here, kriging) makes it possible to estimate these indices with precision at a reasonable computational cost.