Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
10
3
2020
GOAL-ORIENTED MODEL ADAPTIVITY IN STOCHASTIC ELASTODYNAMICS: SIMULTANEOUS CONTROL OF DISCRETIZATION, SURROGATE MODEL AND SAMPLING ERRORS
195-223
10.1615/Int.J.UncertaintyQuantification.2020031735
Pedro
Bonilla-Villalba
Institute of Materials, Mechanics and Advanced Manufacturing, Cardiff University, 5 the
Parade, Cardiff, UK
S.
Claus
ONERA, 6 Chemin de la Vauve aux Granges, 91120 Palaiseau, France
A.
Kundu
Institute of Materials, Mechanics and Advanced Manufacturing, Cardiff University, 5 the
Parade, Cardiff, UK
Pierre
Kerfriden
Institute of Materials, Mechanics and Advanced Manufacturing, Cardiff University, 5 the
Parade, Cardiff, UK; Centre des Matériaux, MINES ParisTech/PSL University, 63-65 rue Henri Auguste
Desbruères, Corbeil-essonnes, France
adaptivity
stochastic control and estimation
stochastic finite element method
solid mechanics
The presented adaptive modeling approach aims to jointly control the level of refinement for each of the building blocks
employed in a typical chain of finite element approximations for stochastically parametrized systems, namely: (i) finite
error approximation of the spatial fields, (ii) surrogate modeling to interpolate quantities of interest(s) in the parameter domain, and (iii) Monte Carlo sampling of associated probability distribution(s). The control strategy seeks accurate calculation of any statistical measure of the distributions at minimum cost, given an acceptable margin of error as the only tunable parameter. At each stage of the greedy-based algorithm for spatial discretization, the mesh is selectively refined in the subdomains with highest contribution to the error in the desired measure. The strictly incremental
complexity of the surrogate model is controlled by enforcing preponderant discretization error integrated across the
parameter domain. Finally, the number of Monte Carlo samples is chosen such that either (a) the overall precision of the chain of approximations can be ascertained with sufficient confidence or (b) the fact that the computational model requires further mesh refinement is statistically established. The efficiency of the proposed approach is discussed for a frequency-domain vibration structural dynamics problem with forward uncertainty propagation. Results show that locally adapted finite element solutions converge faster than those obtained using uniformly refined grids.
HYPERDIFFERENTIAL SENSITIVITY ANALYSIS OF UNCERTAIN PARAMETERS IN PDE-CONSTRAINED OPTIMIZATION
225-248
10.1615/Int.J.UncertaintyQuantification.2020032480
Joseph
Hart
Optimization and Uncertainty Quantification, Sandia National Laboratories, P.O. Box 5800,
Albuquerque, New Mexico 87123-1320, USA
Bart van Bloemen
Waanders
Optimization and Uncertainty Quantification, Sandia National Laboratories, P.O. Box 5800,
Albuquerque, New Mexico 87123-1320, USA
Roland
Herzog
Technical University Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany
sensitivity analysis
PDE-constrained optimization
randomized linear algebra
low rank approximations
Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyperdifferential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyperdifferential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multiphysics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.
REPLICATION-BASED EMULATION OF THE RESPONSE DISTRIBUTION OF STOCHASTIC SIMULATORS USING GENERALIZED LAMBDA DISTRIBUTIONS
249-275
10.1615/Int.J.UncertaintyQuantification.2020033029
Xujia
Zhu
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
stochastic simulators
surrogate modeling
generalized lambda distributions
sparse polynomial chaos expansions
Due to limited computational power, performing uncertainty quantification analyses with complex computational models can be a challenging task. This is exacerbated in the context of stochastic simulators, the response of which to a given set of input parameters, rather than being a deterministic value, is a random variable with unknown probability density function (PDF). Of interest in this paper is the construction of a surrogate that can accurately predict this response PDF for any input parameters. We suggest using a flexible distribution family−the generalized lambda distribution−to approximate the response PDF. The associated distribution parameters are cast as functions of input parameters and represented by sparse polynomial chaos expansions. To build such a surrogate model, we propose an approach based on a local inference of the response PDF at each point of the experimental design based on replicated model evaluations. Two versions of this framework are proposed and compared on analytical examples and case studies.