Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification
File(s)
Published version
Author(s)
Kantarakias, Kyriakos D
Papadakis, George
Type
Journal Article
Abstract
We consider the Least Squares (LSQ) regression method for Uncertainty Quantification
(UQ) using generalised polynomial chaos (gPC) and augment the linear system with the
gradient of the Quantity of Interest (QoI) with respect to the stochastic variables. The
gradient is computed very efficiently for all variables from the adjoint system of equations.
To minimise the condition number of the augmented LSQ system, an effective sampling
strategy of the stochastic space is required. We compare two strategies. In the first,
we apply pivoted QR decomposition to the standard LSQ matrix and evaluate both the
QoI and its gradient at the sample points identified. In the second strategy, we apply
QR decomposition directly to the augmented matrix. We find that the first strategy is
more efficient in terms of accuracy vs number of evaluations. We call the new approach
sensitivity-enhanced generalised polynomial chaos, or se-gPC, and apply it to several
test cases including an aerodynamic case with 40 stochastic parameters. The method
can produce accurate estimations of the statistical moments using a small number of
sampling points. The computational cost scales as ∼ mp−1, instead of ∼ mp of the standard
LSQ formulation, where m is the number of stochastic variables and p the chaos order.
The solution of the adjoint system of equations is implemented in many computational
mechanics packages, thus the infrastructure exists for the application of the method to a
wide variety of engineering problems.
(UQ) using generalised polynomial chaos (gPC) and augment the linear system with the
gradient of the Quantity of Interest (QoI) with respect to the stochastic variables. The
gradient is computed very efficiently for all variables from the adjoint system of equations.
To minimise the condition number of the augmented LSQ system, an effective sampling
strategy of the stochastic space is required. We compare two strategies. In the first,
we apply pivoted QR decomposition to the standard LSQ matrix and evaluate both the
QoI and its gradient at the sample points identified. In the second strategy, we apply
QR decomposition directly to the augmented matrix. We find that the first strategy is
more efficient in terms of accuracy vs number of evaluations. We call the new approach
sensitivity-enhanced generalised polynomial chaos, or se-gPC, and apply it to several
test cases including an aerodynamic case with 40 stochastic parameters. The method
can produce accurate estimations of the statistical moments using a small number of
sampling points. The computational cost scales as ∼ mp−1, instead of ∼ mp of the standard
LSQ formulation, where m is the number of stochastic variables and p the chaos order.
The solution of the adjoint system of equations is implemented in many computational
mechanics packages, thus the infrastructure exists for the application of the method to a
wide variety of engineering problems.
Date Issued
2023-10-15
Date Acceptance
2023-07-12
Citation
Journal of Computational Physics, 2023, 491
ISSN
0021-9991
Publisher
Elsevier
Journal / Book Title
Journal of Computational Physics
Volume
491
Copyright Statement
© 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://
creativecommons.org/licenses/by/4.0/).
creativecommons.org/licenses/by/4.0/).
License URL
Identifier
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Subjects
ADJOINT FORMULATION
AERODYNAMIC SHAPE OPTIMIZATION
Computer Science
Computer Science, Interdisciplinary Applications
DERIVATIVE INFORMATION
DESIGN
EXPANSIONS
Generalised polynomial chaos
Optimal sampling of stochastic space
Physical Sciences
Physics
Physics, Mathematical
REGRESSION
Science & Technology
Technology
Uncertainty quantification
Publication Status
Published
Article Number
112377
Date Publish Online
2023-07-18