Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
8
5
2018
DATA ASSIMILATION FOR NAVIER-STOKES USING THE LEAST-SQUARES FINITE-ELEMENT METHOD
383-403
10.1615/Int.J.UncertaintyQuantification.2018021021
Alexander
Schwarz
Institut für Mechanik, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen,
Germany
Richard P.
Dwight
Aerodynamics Group, Faculty of Aerospace, TU Delft, P.O. Box 5058, 2600GB Delft, The
Netherlands
least-squares finite-element method
data assimilation
inverse problems
fluid dynamics
Navier-Stokes
Bayesian inference
uncertainty quantification
We investigate theoretically and numerically the use of the least-squares finite-element method (LSFEM) to approach
data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is
based on a stress-velocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces
H(div)×H1×L2 for the variables respectively. In general, S-V-P formulations are promising when the stresses are of special interest, e.g., for non-Newtonian, multiphase or turbulent flows. Resolution of the system is via minimization of a least-squares functional representing the magnitude of the residual of the equations. A simple and immediate approach to extend this solver to data assimilation is to add a data-discrepancy term to the functional. Whereas most data assimilation techniques require a large number of evaluations of the forward simulation and are therefore very
expensive, the approach proposed in this work uniquely has the same cost as a single forward run. However, the question arises: what is the statistical model implied by this choice? We answer this within the Bayesian framework, establishing the latent background covariance model and the likelihood. Further we demonstrate that−in the linear case−the method is equivalent to application of the Kalman filter, and derive the posterior covariance. We practically demonstrate the capabilities of our method on a backward-facing step case. Our LSFEM formulation (without data) is shown to have good approximation quality, even on relatively coarse meshes−in particular with respect to mass conservation and reattachment location. Adding limited velocity measurements from experiment, we show that the method is able to correct for discretization error on very coarse meshes, as well as correct for the influence of unknown and uncertain boundary conditions.
CUBIC INTUITIONISTIC FUZZY AGGREGATION OPERATORS
405-427
10.1615/Int.J.UncertaintyQuantification.2018020471
Gagandeep
Kaur
School of Mathematics, Thapar Institute of Engineering and Technology (Deemed University)
Patiala 147004, Punjab, India
Harish
Garg
School of Mathematics, Thapar Institute of Engineering and Technology (Deemed University)
Patiala 147004, Punjab, India
cubic intuitionistic fuzzy set
membership degrees
aggregation operators
multicriteria decision-making
The objective of this manuscript is to present some series of aggregation operators under the cubic intuitionistic fuzzy
set (CIFS) and their suitable properties. Firstly an operational law, score function, and accuracy function between
the cubic intuitionistic fuzzy numbers (CIFNs) under the P-order and R-order are defined and hence based on them, some weighted averaging and geometric aggregation operators, namely, cubic intuitionistic fuzzy weighted, ordered weighted, hybrid averaging, and geometric aggregation operators are proposed. A decision-making method based on these operators is proposed for ranking the different sets of the alternative under CIFS domain. Finally, an illustrative example is given to demonstrate the proposed approach.
UNCERTAINTY QUANTIFICATION FOR INCIDENT HELIUM FLUX IN PLASMA-EXPOSED TUNGSTEN
429-446
10.1615/Int.J.UncertaintyQuantification.2018025120
Ozgur
Cekmer
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831
Khachik
Sargsyan
Sandia National Laboratories, Livermore, CA, USA
Sophie
Blondel
Department of Nuclear Engineering, University of Tennessee, Knoxville, TN 37996
Habib N.
Najm
Sandia National Laboratories
P.O. Box 969, MS 9051, Livermore, CA 94551, USA
David E.
Bernholdt
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831
Brian D.
Wirth
Department of Nuclear Engineering, University of Tennessee, Knoxville, TN 37996; Nuclear Science and Engineering Directorate, Oak Ridge National Laboratory, Oak Ridge, TN 37831
nuclear fusion
plasma-facing components
uncertainty quantification
Bayesian inference
model error
In this work, the surface response of a tungsten plasma-facing component was simulated by a cluster-dynamics code,
Xolotl, with a focus on quantifying the impact of uncertainty in one of the input parameters to Xolotl, namely, the incident helium flux. The simulated conditions involve a tungsten surface exposed to 100 eV helium ion implantations with a flux of either 4 × 1022 or 4 × 1025 He m-2s-1. Two sources were used to describe the implanted helium depth distribution in tungsten, either molecular dynamics (MD) or a binary collision approximation code, the stopping and range of ions in matter (SRIM). The aim of this work is to evaluate and examine uncertain predictions on the
helium retention based on these two different modeling methodologies that either neglect electronic energy loss or the crystalline structure of the solid, respectively. An embedded model-form error approach was pursued here in order to arrive at predictions that account for variability due to the two different data sources, and the impact of this model-form uncertainty in incident helium flux on Xolotl output was presented for the two implantation fluxes.
BEYOND BLACK-BOXES IN BAYESIAN INVERSE PROBLEMS AND MODEL VALIDATION: APPLICATIONS IN SOLID MECHANICS OF ELASTOGRAPHY
447-482
10.1615/Int.J.UncertaintyQuantification.2018025837
L.
Bruder
Mechanics and High Performance Computing Group, Technical University of Munich,
Parkring 35, 85748 Garching, Germany
Phaedon-Stelios
Koutsourelakis
Continuum Mechanics Group, Technical University of Munich, Boltzmannstrasse 15, 85748
Garching, Germany
uncertainty quantification
variational inference
inverse problems
model error
Stochastic optimization
Bayesian modeling
The present paper is motivated by one of the most fundamental challenges in inverse problems, that of quantifying model discrepancies and errors. While significant strides have been made in calibrating model parameters, the overwhelming majority of pertinent methods is based on the assumption of a perfect model. Motivated by problems in solid mechanics which, as all problems in continuum thermodynamics, are described by conservation laws and phenomenological constitutive closures, we argue that in order to quantify model uncertainty in a physically meaningful manner, one should break open the black-box forward model. In particular, we propose formulating an undirected probabilistic model that explicitly accounts for the governing equations and their validity. This recasts the solution of both forward and inverse problems as probabilistic inference tasks where the problem's state variables should not only be compatible with the data but also with the governing equations as well. Even though the probability densities involved do not contain any black-box terms, they live in much higher-dimensional spaces. In combination with the intractability of the normalization constant of the undirected model employed, this poses significant challenges which we propose to address
with a linearly scaling, double layer of stochastic variational inference. We demonstrate the capabilities and efficacy of the proposed model in synthetic forward and inverse problems (with and without model error) in elastography.