Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
10
2
2020
MODEL REDUCTION FOR LARGE-SCALE EARTHQUAKE SIMULATION IN AN UNCERTAIN 3D MEDIUM
101-127
10.1615/Int.J.UncertaintyQuantification.2020031165
Pierre
Sochala
Bureau de Recherches Géologiques et Minières, 45060 Orléans, France
F.
De Martin
Bureau de Recherches Géologiques et Minières, 45060 Orléans, France
O.
Le Maître
Centre de Mathématiques Appliquées, CNRS & Inria, Ecole Polytechnique, 91120 Palaiseau,
France
seismic wave propagation
uncertainty quantification
empirical orthogonal functions
polynomial chaos
interval probability
global sensitivity analysis
In this paper, we are interested in the seismic wave propagation into an uncertain medium. To this end, we performed an ensemble of 400 large-scale simulations that requires 4 million core-hours of CPU time. In addition to the large computational load of these simulations, solving the uncertainty propagation problem requires dedicated procedures to handle the complexities inherent to large dataset size and the low number of samples. We focus on the peak ground motion at the free surface of the 3D domain, and our analysis utilizes a surrogate model combining two key ingredients for complexity mitigation: (i) a dimension reduction technique using empirical orthogonal basis functions, and (ii) a functional approximation of the uncertain reduced coordinates by polynomial chaos expansions. We carefully validate the resulting surrogate model by estimating its predictive error using bootstrap, truncation, and cross-validation procedures. The surrogate model allows us to compute various statistical information of the uncertain prediction, including marginal and joint probability distributions, interval probability maps, and 2D fields of global sensitivity indices.
A STOCHASTIC COLLOCATION METHOD FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS AND RANDOM INPUTS
129-143
10.1615/Int.J.UncertaintyQuantification.2020031754
Ling
Guo
Department of Mathematics, Shanghai Normal University No. 100, Guilin Road Shanghai,200234 China
Lijun
Yi
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
Volterra integro-differential equations
hp-version
continuous and discontinuous Galerkin methods
time-stepping method
stochastic collocation method
exponential rates of convergence
In this paper we propose a stochastic collocation method to solve the Volterra integro-differential equations with weakly singular kernels, random coefficients, and forcing terms. The input data are assumed to depend on a finite number of random variables. The method consists of the hp-versions of the continuous Galerkin and discontinuous Galerkin time-stepping schemes in time and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, which naturally leads to the solution of uncoupled deterministic problems. We establish a priori error estimates that are completely explicit with respect to all the discretization parameters. In particular, we show that exponential rates of convergence can be achieved in both the temporal direction and the probability space for solutions with start-up singularities by using geometrically refined time-steps and linearly increasing polynomial degrees. Numerical experiments are provided to illustrate the theoretical results.
SURROGATE MODELING OF INDOOR DOWN-LINK HUMAN EXPOSURE BASED ON SPARSE POLYNOMIAL CHAOS EXPANSION
145-163
10.1615/Int.J.UncertaintyQuantification.2020031452
Zicheng
Liu
Chaire C2M, LTCI, Télécom Paris, Palaiseau, France
Dominique
Lesselier
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes,
Gif-sur-Yvette, 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
specific absorption rate
surrogate model
polynomial chaos expansion
least angle regression
orthogonal matching pursuit
cross-model validation
double cross validation
Sobol' indices
global sensitivity analysis
data preprocessing
Human exposure induced by wireless communication systems increasingly draws the public attention. Here, an indoor down-link scenario is concerned and the exposure level is statistically analyzed. The electromagnetic field emitted by a WiFi box is measured and electromagnetic dosimetry features are evaluated from the whole-body specific absorption rate as computed with a finite-difference time-domain (a.k.a. FDTD) code. Due to computational cost, a statistical analysis is performed based on a surrogate model, which is constructed by means of so-called sparse polynomial chaos expansion, where the inner cross validation (ICV) is used to select the optimal hyperparameters during the model construction and assess the model performance. However, the model assessment based on ICV tends to be overly optimistic with small data sets. The method of cross-model validation is used and outer cross validation is carried out for the model assessment. The effects of the data preprocessing are investigated as well. On the basis of the surrogate model, the global sensitivity of the exposure to input parameters is analyzed from Sobol' indices.
ENHANCED ADAPTIVE SURROGATE MODELS WITH APPLICATIONS IN UNCERTAINTY QUANTIFICATION FOR NANOPLASMONICS
165-193
10.1615/Int.J.UncertaintyQuantification.2020031727
Niklas
Georg
Institut für Dynamik und Schwingungen, Technische Universität Braunschweig,
Braunschweig, Germany; Centre for Computational Engineering, Technische Universität Darmstadt, Darmstadt,
Germany; Institut für Teilchenbeschleunigung und Elektromagnetische Felder (TEMF), Technische
Universität Darmstadt, Darmstadt, Germany
Dimitrios
Loukrezis
Centre for Computational Engineering, Technische Universität Darmstadt, Darmstadt,
Germany; Institut für Teilchenbeschleunigung und Elektromagnetische Felder (TEMF), Technische
Universität Darmstadt, Darmstadt, Germany
Ulrich
Römer
Institut für Dynamik und Schwingungen, Technische Universität Braunschweig, Schleinitzstraße
20, 38106 Braunschweig, Germany
Sebastian
Schöps
Centre for Computational Engineering, Technische Universität Darmstadt, Darmstadt,
Germany; Institut für Teilchenbeschleunigung und Elektromagnetische Felder (TEMF), Technische
Universität Darmstadt, Darmstadt, Germany
adaptivity
adjoint error indicator
conformal maps
hierarchical interpolation
stochastic sparse grid collocation
Maxwell's source problem
plasmonics
We propose an efficient surrogate modeling technique for uncertainty quantification. The method is based on a well-known dimension-adaptive collocation scheme. We improve the scheme by enhancing sparse polynomial surrogates with conformal maps and adjoint error correction. The methodology is applied to Maxwell's source problem with random input data. This setting comprises many applications of current interest from computational nanoplasmonics, such as grating couplers or optical waveguides. Using a nontrivial benchmark model, we show the benefits and drawbacks of using enhanced surrogate models through various numerical studies. The proposed strategy allows us to conduct a thorough uncertainty analysis, taking into account a moderately large number of random parameters.