Generalized Polynomial Chaos Expansion for Fast and Accurate Uncertainty Quantification in Geomechanical Modelling
Abstract
:1. Introduction
2. Mathematical Framework
2.1. Geomechanical Governing Equations and Model Parameterization
2.2. gPCE Surrogate Model
2.3. Sobol’ Indices in the gPCE Framework
2.4. gPCEES and gPCEMDA
3. Numerical Results
Algorithm 1 gPCEES and gPCEMDA for the forward model $\mathcal{S}$. 

3.1. Model SetUp and Random Parameters
3.2. gPCE Surrogate Model
3.3. Sensitivity Analysis
3.4. Bayesian Update
3.4.1. ES and gPCEES
3.4.2. gPCEMDA
4. Discussion and Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
UQ  Uncertainty Quantification 
SA  Sensitivity Analysis 
ES  Ensemble Smoother 
FE  Finite Element 
gPCE  generalized Polynomial Chaos Expansion 
MC  Monte Carlo 
MDA  Multiple Data Assimilation 
PSI  Permanent Scatterer Interferometry 
VN  VermeerNeher 
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$\mathit{N}=1$ (${\mathit{n}}_{\mathit{q}}$ = 16)  $\mathit{N}=2$ (${\mathit{n}}_{\mathit{q}}$ = 81)  $\mathit{N}=3$ (${\mathit{n}}_{\mathit{q}}$ = 137)  

t [yr]  ${\mathit{\mu}}_{\mathit{u}}$ [m]  ${\mathit{\sigma}}_{\mathit{u}}^{2}$ [m${}^{2}$]  Q${}^{2}$  ${\mathit{\mu}}_{\mathit{u}}$ [m]  ${\mathit{\sigma}}_{\mathit{u}}^{2}$ [m${}^{2}$]  Q${}^{2}$  ${\mathit{\mu}}_{\mathit{u}}$ [m]  ${\mathit{\sigma}}_{\mathit{u}}^{2}$ [m${}^{2}$]  Q${}^{2}$  
1  −2.1× 10${}^{3}$  2.0× 10${}^{6}$  0.820  −2.3× 10${}^{3}$  3.1× 10${}^{6}$  0.931  −2.3× 10${}^{3}$  3.0× 10${}^{6}$  0.823  
5  −1.7× 10${}^{1}$  1.4× 10${}^{2}$  0.860  −1.7× 10${}^{1}$  1.5× 10${}^{2}$  0.995  −1.7× 10${}^{1}$  1.5× 10${}^{2}$  0.999  
10  −3.6× 10${}^{1}$  2.6× 10${}^{2}$  0.904  −3.7× 10${}^{1}$  2.8× 10${}^{2}$  0.999  −3.7× 10${}^{1}$  2.8× 10${}^{2}$  1.000 
ES  gPCEES  

${\mathit{n}}_{\mathit{MC}}=\mathbf{25}$  ${\mathit{n}}_{\mathit{MC}}=\mathbf{50}$  ${\mathit{n}}_{\mathit{MC}}=\mathbf{100}$  ${\mathit{n}}_{\mathit{MC}}=\mathbf{25}$  ${\mathit{n}}_{\mathit{MC}}=\mathbf{50}$  ${\mathit{n}}_{\mathit{MC}}=\mathbf{100}$  
$A{E}^{p}\left({\mathrm{u}}_{z}\right)$ [m]  0.1667  0.1642  0.1656  0.1667  0.1642  0.1656  
$A{E}^{u}\left({\mathrm{u}}_{z}\right)$ [m]  0.0190  0.0104  0.0099  0.0189  0.0117  0.0101  
$AE{S}^{p}\left({\mathrm{u}}_{z}\right)$ [m]  0.0834  0.0836  0.0803  0.0833  0.0836  0.0803  
$AE{S}^{u}\left({\mathrm{u}}_{z}\right)$ [m]  0.0149  0.0010  0.0016  0.0149  0.0010  0.0017  
$A{E}^{p}\left({\lambda}^{*}\right)$  0.0028  0.0029  0.0029  0.0028  0.0029  0.0029  
$A{E}^{u}\left({\lambda}^{*}\right)$  0.0021  0.0014  0.0005  0.0021  0.0024  0.0004  
$AE{S}^{p}\left({\lambda}^{*}\right)$  0.0027  0.0025  0.0024  0.0027  0.0025  0.0024  
$AE{S}^{u}\left({\lambda}^{*}\right)$  0.0020  0.0002  0.0002  0.0020  0.0001  0.0003  
$A{E}^{p}\left(R\right)$  0.1109  0.1065  0.1069  0.1109  0.1065  0.1069  
$A{E}^{u}\left(R\right)$  0.0567  0.0186  0.0192  0.0568  0.0641  0.0184  
$AE{S}^{p}\left(R\right)$  0.0787  0.0724  0.0729  0.0787  0.0724  0.0729  
$AE{S}^{u}\left(R\right)$  0.0538  0.0186  0.0188  0.0538  0.0196  0.0182 
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Zoccarato, C.; Gazzola, L.; Ferronato, M.; Teatini, P. Generalized Polynomial Chaos Expansion for Fast and Accurate Uncertainty Quantification in Geomechanical Modelling. Algorithms 2020, 13, 156. https://doi.org/10.3390/a13070156
Zoccarato C, Gazzola L, Ferronato M, Teatini P. Generalized Polynomial Chaos Expansion for Fast and Accurate Uncertainty Quantification in Geomechanical Modelling. Algorithms. 2020; 13(7):156. https://doi.org/10.3390/a13070156
Chicago/Turabian StyleZoccarato, Claudia, Laura Gazzola, Massimiliano Ferronato, and Pietro Teatini. 2020. "Generalized Polynomial Chaos Expansion for Fast and Accurate Uncertainty Quantification in Geomechanical Modelling" Algorithms 13, no. 7: 156. https://doi.org/10.3390/a13070156