Abstract

As engineered systems face an ever-increasing array of threats, the ability to perform real-time predictions and decision-making is central for coping with potential menaces. For thorough real-time predictions, uncertainty measures need to be incorporated, albeit conventional uncertainty quantification (UQ) efforts are often time-consuming and thus go against the essence of real-time decision-making. A fast, metamodel-based Bayesian updating scheme within a decision-making framework under uncertainty framework is proposed. The latter combines data harnessing, assimilation, and parameter updating in an online setting. For fast parameter updating in the online phase, the adaptation property of polynomial chaos (PC) metamodels is exploited. In the offline phase, a global Gaussian approximation of the response surface is obtained through a first-order PC surrogate, from which, a rotated basis of the underlying Gaussian Hilbert space is identified. In the subsequent online phase, a cheap, locally adapted solution embedded in a lower dimensional subspace is leveraged within a Bayesian updating setting to perform fast data assimilation. Incorporating uncertainty into real-time decision-making paves the way for higher confidence in online predictions, increased system resilience, and adaptivity to surroundings. The aforementioned fast parameter updating scheme based on adapted PC metamodels is evaluated on a set of problems with varying complexity.

References

References
1.
Bui-Thanh
,
T.
,
Willcox
,
K.
, and
Ghattas
,
O.
,
2008
, “
Model Reduction for Large-Scale Systems With High-Dimensional Parametric Input Space
,”
SIAM J. Sci. Comput.
,
30
(
6
), pp.
3270
3288
.10.1137/070694855
2.
Carlberg
,
K.
,
Bou-Mosleh
,
C.
, and
Farhat
,
C.
,
2011
, “
Efficient Non-Linear Model Reduction Via Least-Squares Petrov-Galerkin Projection and Compressive Tensor Approximations
,”
Int. J. Numer. Methods Eng.
,
86
(
2
), pp.
155
181
.10.1002/nme.3050
3.
Chaturantabut
,
S.
, and
Sorensen
,
D.
,
2010
, “
Nonlinear Model Reduction Via Discrete Empirical Interpolation
,”
SIAM J. Sci. Comput.
,
32
(
5
), pp.
2737
2764
.10.1137/090766498
4.
Rowley
,
C.
,
2005
, “
Model Reduction for Fluids Using Balanced Proper Orthogonal Decomposition
,”
Int. J. Bifucation Chaos Appl. Sci. Eng.
,
15
, pp.
997
1013
.10.1142/S0218127405012429
5.
Noor
,
A.
, and
Peters
,
J.
,
1980
, “
Reduced Basis Technique for Nonlinear Analysis of Structures
,”
AIAA J.
,
18
(
4
), pp.
455
462
.10.2514/3.50778
6.
Benner
,
P.
,
Gugercin
,
S.
, and
Willcox
,
K.
,
2015
, “
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
,”
SIAM Rev.
,
57
(
4
), pp.
483
531
.10.1137/130932715
7.
Constantine
,
P.
,
2015
,
Active Subspaces—Emerging Ideas for Dimension Reduction in Parameter Studies
,
SIAM
, Philadelphia, PA.
8.
Berguin
,
S.
, and
Marvis
,
D.
,
2014
, “
Dimensionality Reduction Using Principal Component Analysis With Gradient Information
,” AIAA Paper No.
0112
.10.2514/1.J053372
9.
Tipireddy
,
R.
, and
Ghanem
,
R.
,
2014
, “
Basis Adaptation in Homogeneous Chaos Spaces
,”
J. Comput. Phys.
,
259
, pp.
304
317
.10.1016/j.jcp.2013.12.009
10.
Stoyanov
,
M.
, and
Webster
,
C.
,
2015
, “
A Gradient-Based Sampling Approach for Dimension Reduction of Partial Differential Equations With Stochastic Coefficients
,”
Int. J. Uncertainty Quantif.
,
5
(
1
), pp.
49
72
.10.1615/Int.J.UncertaintyQuantification.2014010945
11.
Lieberman
,
C.
,
Willcox
,
K.
, and
Ghattas
,
O.
,
2010
, “
Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems
,”
SIAM J. Sci. Comput.
,
32
(
5
), pp.
2523
2542
.10.1137/090775622
12.
Lecerf
,
M.
,
Allaire
,
D.
, and
Willcox
,
K.
,
2015
, “
Methodology for Dynamic Data-Driven Online Flight Capability Estimation
,”
AIAA J.
,
53
(
10
), pp.
3073
3087
.10.2514/1.J053893
13.
Singh
,
V.
, and
Willcox
,
K. E.
,
2017
, “
Methodology for Path Planning With Dynamic Data-Driven Flight Capability Estimation
,”
AIAA J.
,
55
(
8
), pp.
2727
2738
.10.2514/1.J055551
14.
Kalman
,
R.
,
1960
, “
A New Approach to Linear Filtering and Prediction Problems
,”
ASME J. Basic Eng.
,
82
(
1
), pp.
35
45
.10.1115/1.3662552
15.
Smith
,
G.
,
Schmidt
,
S.
, and
McGee
,
L.
,
1962
,
Application of Statistical Filter Theory to the Optimal Estimation of Position and Velocity on Board a Circumlunar Vehicle
, NASA Ames Research Center, Moffett Field, California, CA.
16.
Evensen
,
G.
,
2003
, “
The Ensemble Kalman Filter: Theoretical Formulation and Practical Implementation
,”
Ocean Dyn.
,
53
(
4
), pp.
343
367
.10.1007/s10236-003-0036-9
17.
Gordon
,
N.
,
Salmond
,
D.
, and
Smith
,
A.
,
1993
, “
Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation
,”
IEE Proc. F-Radar Signal Process.
,
140
, pp.
107
113
.10.1049/ip-f-2.1993.0015
18.
Luts
,
J.
,
Broderick
,
T.
, and
Wand
,
M. P.
,
2014
, “
Real-Time Semiparametric Regression
,”
J. Comput. Graphical Stat.
,
23
(
3
), pp.
589
615
.10.1080/10618600.2013.810150
19.
Giordano
,
R.
,
Broderick
,
T.
,
Meager
,
R.
,
Huggins
,
J.
, and
Jordan
,
M.
,
2016
, “
Fast Robustness Quantification With Variational Bayes
,”
ICML Workshop on #Data4Good: Machine Learning in Social Good Applications
, New York.
20.
Wang
,
J.
, and
Zabaras
,
N.
,
2005
, “
Using Bayesian Statistics in the Estimation of Heat Source in Radiation
,”
Int. J. Heat Mass Transfer
,
48
(
1
), pp.
15
29
.10.1016/j.ijheatmasstransfer.2004.08.009
21.
Balakrishnan
,
S.
,
Roy
,
A.
,
Ierapetritou
,
M.
,
Flach
,
G.
, and
Georgopoulos
,
P.
,
2003
, “
Uncertainty Reduction and Characterization for Complex Environmental Fate and Transport Models: An Empirical Bayesian Framework Incorporating the Stochastic Response Surface Method
,”
Water Resour. Res.
,
39
(
12
), pp.
1
13
.10.1029/2002WR001810
22.
Higdon
,
D.
,
Lee
,
H.
, and
Holloman
,
C.
,
2003
, “
Markov Chain Monte Carlo-Based Approaches for Inference in Computationally Intensive Inverse Problems
,”
Bayesian Stat.
,
7
, pp.
181
197
.
23.
Broderick
,
T.
,
Dudik
,
M.
,
Tkacik
,
G.
,
Schapire
,
R. E.
, and
Bialek
,
W.
,
2008
, “
Faster Solutions of the Inverse Pairwise Ising Problem
,” arXiv preprint arXiv:0712.2437.
24.
Aster
,
R.
,
Borchers
,
B.
, and
Thurber
,
C.
,
2004
,
Parameter Estimation and Inverse Problems
,
Academic Press
, Cambridge, MA.
25.
Kaipio
,
J.
, and
Somersalo
,
E.
,
2005
,
Statistical and Computational Inverse Problems
,
Springer
, Berlin.
26.
Gouveia
,
W.
, and
Scales
,
J.
,
1997
, “
Resolution of Seismic Waveform Inversion: Bayes Versus Occam
,”
Inverse Probl.
,
13
(
2
), pp.
323
349
.10.1088/0266-5611/13/2/009
27.
Jackson
,
C.
,
Sen
,
M.
, and
Stoffa
,
P.
,
2004
, “
An Efficient Stochastic Bayesian Approach to Optimal Parameter and Uncertainty Estimation for Climate Model Predictions
,”
J. Clim.
,
17
(
14
), pp.
2828
2841
.10.1175/1520-0442(2004)017<2828:AESBAT>2.0.CO;2
28.
Ghanem
,
R. G.
, and
Spanos
,
P. D.
,
1991
,
Stochastic Finite Element: A Spectral Approach
, Springer-Verlag, New York.
29.
Xiu
,
D.
, and
Karniadakis
,
G. E.
,
2002
, “
The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations
,”
SIAM J. Sci. Comput.
,
24
(
2
), pp.
619
644
.10.1137/S1064827501387826
30.
Ghanem
,
R.
,
1998
, “
Scales of Fluctuation and the Propagation of Uncertainty in Random Porous Media
,”
Water Resour. Res.
,
34
(
9
), pp.
2123
2136
.10.1029/98WR01573
31.
Reagan
,
M. T.
,
Najm
,
H. N.
,
Ghanem
,
R. G.
, and
Knio
,
O. M.
,
2003
, “
Uncertainty Quantification in Reacting-Flow Simulations Through Non-Intrusive Spectral Projection
,”
Combust. Flame
,
132
(
3
), pp.
545
555
.10.1016/S0010-2180(02)00503-5
32.
Ghanem
,
R.
, and
Red-Horse
,
J.
,
1999
, “
Propagation of Probabilistic Uncertainty in Complex Physical Systems Using a Stochastic Finite Element Approach
,”
Phys. D
,
133
(
1–4
), pp.
137
144
.10.1016/S0167-2789(99)00102-5
33.
Najm
,
H. N.
,
2009
, “
Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics
,”
Annu. Rev. Fluid Mech.
,
41
(
1
), pp.
35
52
.10.1146/annurev.fluid.010908.165248
34.
Marzouk
,
Y. M.
,
Najm
,
H. N.
, and
Rahn
,
L. A.
,
2007
, “
Stochastic Spectral Methods for Efficient Bayesian Solution of Inverse Problems
,”
J. Comput. Phys.
,
224
(
2
), pp.
560
586
.10.1016/j.jcp.2006.10.010
35.
Marzouk
,
Y. M.
, and
Najm
,
H. N.
,
2009
, “
Dimensionality Reduction and Polynomial Chaos Acceleration of Bayesian Inference in Inverse Problems
,”
J. Comput. Phys.
,
228
(
6
), pp.
1862
1902
.10.1016/j.jcp.2008.11.024
36.
Sudret
,
B.
,
2008
, “
Global Sensitivity Analysis Using Polynomial Chaos Expansions
,”
Reliab. Eng. Syst. Saf.
,
93
(
7
), pp.
964
979
.10.1016/j.ress.2007.04.002
37.
Crestaux
,
T.
,
Le Maı̂tre
,
O.
, and
Martinez
,
J.-M.
,
2009
, “
Polynomial Chaos Expansion for Sensitivity Analysis
,”
Reliab. Eng. Syst. Saf.
,
94
(
7
), pp.
1161
1172
.10.1016/j.ress.2008.10.008
38.
Ghanem
,
R.
,
1999
, “
Higher-Order Sensitivity of Heat Conduction Problems to Random Data Using the Spectral Stochastic Finite Element Method
,”
ASME. J. Heat Transfer
,
121
(
2
), pp.
290
299
.10.1115/1.2825979
39.
Alexanderian
,
A.
,
Winokur
,
J.
,
Sraj
,
I.
,
Srinivasan
,
A.
,
Iskandarani
,
M.
,
Thacker
,
W. C.
, and
Knio
,
O. M.
,
2012
, “
Global Sensitivity Analysis in an Ocean General Circulation Model: A Sparse Spectral Projection Approach
,”
Comput. Geosci.
,
16
(
3
), pp.
757
778
.10.1007/s10596-012-9286-2
40.
Tsilifis
,
P.
,
Ghanem
,
R.
, and
Hajali
,
P.
,
2017
, “
Efficient Bayesian Experimentation Using an Expected Information Gain Lower Bound
,”
SIAM/ASA J. Uncertainty Quantif.
,
5
(
1
), pp.
30
62
.10.1137/15M1043303
41.
Huan
,
X.
, and
Marzouk
,
Y.
,
2014
, “
Gradient-Based Stochastic Optimization Methods in Bayesian Experimental Design
,”
Int. J. Uncertainty Quantif.
,
4
(
6
), pp.
479
510
.10.1615/Int.J.UncertaintyQuantification.2014006730
42.
Ghauch
,
Z. G.
,
Aitharaju
,
V.
,
Rodgers
,
W. R.
,
Pasupuleti
,
P.
,
Dereims
,
A.
, and
Ghanem
,
R. G.
,
2019
, “
Integrated Stochastic Analysis of Fiber Composites Manufacturing Using Adapted Polynomial Chaos Expansions
,”
Compos. Part A
,
118
, pp.
179
193
.10.1016/j.compositesa.2018.12.029
43.
Thimmisetty
,
C.
,
Tsilifis
,
P.
, and
Ghanem
,
R.
,
2017
, “
Homogeneous Chaos Basis Adaptation for Design Optimization Under Uncertainty: Application to the Oil Well Placement Problem
,”
Artif. Intell. Eng. Des., Anal. Manuf.
,
31
(
3
), pp.
265
276
.10.1017/S0890060417000166
44.
Tsilifis
,
P.
, and
Ghanem
,
R. G.
,
2017
, “
Reduced Wiener Chaos Representation of Random Fields Via Basis Adaptation and Projection
,”
J. Comput. Phys.
,
341
, pp.
102
120
.10.1016/j.jcp.2017.04.009
45.
Tsilifis
,
P.
, and
Ghanem
,
R. G.
,
2018
, “
Bayesian Adaptation of Chaos Representations Using Variational Inference and Sampling on Geodesics
,”
Proc. R. Soc. A
,
474
(
2217
), p.
20180285
.10.1098/rspa.2018.0285
46.
Li
,
J.
, and
Marzouk
,
Y.
,
2014
, “
Adaptive Construction of Surrogates for the Bayesian Solution of Inverse Problems
,”
SIAM J. Sci. Comput.
,
36
(
3
), pp.
A1163
A1186
.10.1137/130938189
47.
Saad
,
G.
, and
Ghanem
,
R.
,
2009
, “
Characterization of Reservoir Simulation Models Using a Polynomial Chaos-Based Ensemble Kalman Filter
,”
Water Resour. Res.
,
45
(
4
), pp.
1
19
.10.1029/2008WR007148
48.
Eldred
,
M.
, and
Burkardt
,
J.
,
2009
, “
Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification
,” AIAA. Paper No.
2009-976
.10.2514/6.2009-976
49.
Sues
,
R.
,
Aminpour
,
M.
, and
Shin
,
Y.
,
2001
, “
Reliability-Based Multidisciplinary Optimization for Aerospace Systems
,”
AIAA
Paper No. 2001-1521.
50.
An
,
J.
, and
Owen
,
A.
,
2001
, “
Quasi-Regression
,”
J. Complexity
,
17
(
4
), pp.
588
607
.10.1006/jcom.2001.0588
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