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Advancements in the UQLab Framework forUncertainty Quantification
S. Marelli and B. Sudret
Chair of Risk, Safety and Uncertainty Quantification
SIAM UQ2016, Lausanne, 06.04.2016
Introduction Computer Simulations
Introduction
Computer simulations and uncertainty quantification• Computer simulations increasingly substitute expensive experimental
investigations
• Massive increase in availability of computational resources andcomputational algorithms
• Logarithmic decrease of cost/flop in High Performance Computinginfrastructures
• Computer models only provide a simplified representation of reality andare prone to intrinsic model errors and uncertainty
“Essentially, all models are wrong, but some are useful”,George E.P. Box, 1987
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 2 / 25
Introduction Computer Simulations
Introduction
Computer simulations and uncertainty quantification• Computer simulations increasingly substitute expensive experimental
investigations
• Massive increase in availability of computational resources andcomputational algorithms
• Logarithmic decrease of cost/flop in High Performance Computinginfrastructures
• Computer models only provide a simplified representation of reality andare prone to intrinsic model errors and uncertainty
“Essentially, all models are wrong, but some are useful”,George E.P. Box, 1987
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 2 / 25
Introduction Computer Simulations
Introduction
Computer simulations and uncertainty quantification• Computer simulations increasingly substitute expensive experimental
investigations
• Massive increase in availability of computational resources andcomputational algorithms
• Logarithmic decrease of cost/flop in High Performance Computinginfrastructures
• Computer models only provide a simplified representation of reality andare prone to intrinsic model errors and uncertainty
“Essentially, all models are wrong, but some are useful”,George E.P. Box, 1987
Uncertainty quantification aims at making the best use ofcomputer models by dealing rigorously with variability, lack of
knowledge, measurement- and model errors
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 2 / 25
Introduction Computer Simulations
Sources of uncertainty
Aleatory uncertainty• Uncertainty in the occurrence of events,
e.g. earthquakes, floods, tsunami, etc.• Natural variability of physical quantities:
e.g. radioactive decay, flood waveproperties, earthquake spectra etc.
• Not reducible
Epistemic uncertainty• Lack of knowledge about the parameters of a system, e.g. measurement
uncertainty, lack of data• In principle reducible by acquiring additional information
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 3 / 25
Introduction Computer Simulations
Sources of uncertainty
Aleatory uncertainty• Uncertainty in the occurrence of events,
e.g. earthquakes, floods, tsunami, etc.• Natural variability of physical quantities:
e.g. radioactive decay, flood waveproperties, earthquake spectra etc.
• Not reducible
Epistemic uncertainty• Lack of knowledge about the parameters of a system, e.g. measurement
uncertainty, lack of data• In principle reducible by acquiring additional information
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 3 / 25
Introduction Computer Simulations
Outline
1 IntroductionComputer SimulationsA global framework
2 The UQLab projectWhat is UQLabCurrent statusSome Statistics
3 Case studiesStochastic PDEKriging-based rare event estimationSubsurface contaminant diffusion
4 OutlookTentative release schedule
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 3 / 25
Introduction A global framework
Global framework for managing uncertainties
PhysicalModel
Model(s) of the systemAssessment criteria
Probabilistic InputModel
Quantification ofsources of uncertainty
UncertaintyAnalysis
Uncertainty propagation
Random variables Computational model MomentsProbability of failure
Response PDF
IterationSensitivity analysis
IterationSensitivity analysis
Sudret, B. (2007). Uncertainty propagation and sensitivity analysis in mechanical models - Contributions to structural reliability andstochastic spectral methods. Habilitation a diriger des recherches, Universite Blaise Pascal, Clermont-Ferrand
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 4 / 25
Introduction A global framework
Global framework for managing uncertainties
PhysicalModel
Model(s) of the systemAssessment criteria
Probabilistic InputModel
Quantification ofsources of uncertainty
UncertaintyAnalysis
Uncertainty propagation
Random variables Computational model MomentsProbability of failure
Response PDF
IterationSensitivity analysis
IterationSensitivity analysis
Sudret, B. (2007). Uncertainty propagation and sensitivity analysis in mechanical models - Contributions to structural reliability andstochastic spectral methods. Habilitation a diriger des recherches, Universite Blaise Pascal, Clermont-Ferrand
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 4 / 25
Introduction A global framework
Global framework for managing uncertainties
PhysicalModel
Model(s) of the systemAssessment criteria
Probabilistic InputModel
Quantification ofsources of uncertainty
UncertaintyAnalysis
Uncertainty propagation
Random variables Computational model MomentsProbability of failure
Response PDF
IterationSensitivity analysis
IterationSensitivity analysis
Sudret, B. (2007). Uncertainty propagation and sensitivity analysis in mechanical models - Contributions to structural reliability andstochastic spectral methods. Habilitation a diriger des recherches, Universite Blaise Pascal, Clermont-Ferrand
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 4 / 25
Introduction A global framework
Global framework for managing uncertainties
PhysicalModel
Model(s) of the systemAssessment criteria
Probabilistic InputModel
Quantification ofsources of uncertainty
UncertaintyAnalysis
Uncertainty propagation
Random variables Computational model MomentsProbability of failure
Response PDF
IterationSensitivity analysis
IterationSensitivity analysis
Sudret, B. (2007). Uncertainty propagation and sensitivity analysis in mechanical models - Contributions to structural reliability andstochastic spectral methods. Habilitation a diriger des recherches, Universite Blaise Pascal, Clermont-Ferrand
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 4 / 25
Introduction A global framework
The physical model
Computational models of physical and engineering systems• Solution of differential equations (e.g. FEM, FD, PS, etc. )• Multi-physics simulations (e.g. Comsol, etc. )
Functional approximations, surrogate models• Interpolation methods (Kriging)• Regression methods (Polynomial Chaos, Support vector regression)
Measurements/databases• Experimental data from literature• New in-situ measurements
A physical model Y =M(X) is the (possibly abstract) mapthat connects a set of entities X (the inputs) to a set of
quantities of interest Y (the responses)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 5 / 25
Introduction A global framework
The physical model
Computational models of physical and engineering systems• Solution of differential equations (e.g. FEM, FD, PS, etc. )• Multi-physics simulations (e.g. Comsol, etc. )
Functional approximations, surrogate models• Interpolation methods (Kriging)• Regression methods (Polynomial Chaos, Support vector regression)
Measurements/databases• Experimental data from literature• New in-situ measurements
A physical model Y =M(X) is the (possibly abstract) mapthat connects a set of entities X (the inputs) to a set of
quantities of interest Y (the responses)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 5 / 25
Introduction A global framework
The probabilistic input model
Experimental data available• Descriptive Statistics: moments, histograms,
kernel smoothing• Statistical inference: fitting marginals, copula
Only prior/expert knowledge• Maximum entropy principle: maximize
information under constraints• Prior knowledge: e.g. physical constraints on
system variables, literature
Scarce data + expert information• Bayesian inference methods to combine expert
judgment and experimental information0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
Clayton copula sampling
u1
u1
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 6 / 25
Introduction A global framework
The statistical analysisMany possibilities
• Full response characterization(distribution analysis)
• Reliability analysis (rare events simulation)• Analysis of the moments• Sensitivity analysis/model reduction• Stochastic/parametric inversion• Model calibration• Design optimization
Examples
• Monte Carlo Simulation• Approximation methods
(FORM/SORM)
• Sensitivity analysis: Morris’ andSobol’ indices
• Surrogate-model-based analyses:AK-MCS, PCE-based Sobol’
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 7 / 25
Introduction A global framework
The statistical analysisMany possibilities
• Full response characterization(distribution analysis)
• Reliability analysis (rare events simulation)• Analysis of the moments• Sensitivity analysis/model reduction• Stochastic/parametric inversion• Model calibration• Design optimization
Examples
• Monte Carlo Simulation• Approximation methods
(FORM/SORM)
• Sensitivity analysis: Morris’ andSobol’ indices
• Surrogate-model-based analyses:AK-MCS, PCE-based Sobol’
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 7 / 25
The UQLab project What is UQLab
Outline
1 Introduction
2 The UQLab projectWhat is UQLabCurrent statusSome Statistics
3 Case studies
4 Outlook
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 7 / 25
The UQLab project What is UQLab
The UQLab Software Framework
UQLab: Uncertainty Quantification Lab
Focus on:
• Generality• Ease of use• Documentation• Non-intrusiveness• Extendibility• Collaboration
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 8 / 25
The UQLab project Current status
Current Features of UQLab
Representation of
• The physical model• Matlab-based functions
- m-files- strings- function handles
• Support for model parameters• Simple API to connect to external solvers• Pre-computed surrogate models
• The probabilistic input model (copula formalism)• Standard marginals (support for user-defined)• Truncation of marginals (including user-defined)• Gaussian copula• Generalized isoprobabilistic transforms• Sampling strategies (MC, LHS, quasi-random sequences)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 9 / 25
The UQLab project Current status
Current Features of UQLab (cont’d)
Surrogate modelling
• Polynomial Chaos Expansions- Full and sparse (Smolyak) quadrature- Least-square analysis- Sparse expansions (LARS)- Polynomials orthogonal to arbitrary distributions [upcoming]
• Gaussian process modelling (Kriging)- Simple, ordinary and universal Kriging- Arbitrary trends (function handles)- Maximum Likelihood and Cross-Validation objective functions- Local, global and mixed hyperparameter optimization- Support for user-defined correlation families/functions
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 10 / 25
The UQLab project Current status
Current Features of UQLab (cont’d)
UQLab features
• Statistical analysis• Reliability analysis
- Simple Monte-Carlo reliability analysis with advanced sampling- Approximation methods: FORM and SORM with revisited algorithms- Importance Sampling (FORM-based, or user specified)
• Global sensitivity analysis- Screening: Correlation analysis, Standard regression coefficients
(SRA/SRRA), Cotter measure, Morris method- Variance decomposition: Sobol’ indices- PCE-based Sobol’ indices
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 11 / 25
The UQLab project Some Statistics
Some Statistics
Since release of the public beta (July 1st, 2015):• 340+ users from 42 countries worldwide...• ...and counting!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 12 / 25
The UQLab project Some Statistics
Some StatisticsSince release of the public beta (July 1st, 2015):
• 340+ users from 42 countries worldwide...• ...and counting!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 12 / 25
The UQLab project Some Statistics
Some StatisticsSince release of the public beta (July 1st, 2015):
• 340+ users from 42 countries worldwide...• ...and counting!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 12 / 25
Case studies
Outline
1 Introduction
2 The UQLab project
3 Case studiesStochastic PDEKriging-based rare event estimationSubsurface contaminant diffusion
4 Outlook
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 12 / 25
Case studies Stochastic PDE
Example 1: Stochastic diffusion problem Blatman & Sudret (2013)
• 1D diffusion problem on D = [0, L]:[E(x, ω)u′(x, ω)
]′ + f(x) = 0u(0) = 0
E(u′)(L) = F
• u(x, ω) is the displacement field of a unit cross-section tension rodclamped at x = 0
• E(x, ω) is the (spatially variable) Young’s modulus of the rod• f(x) is the uniform axial load• F is a pinpoint load at x = L
Diffusion coefficientThe diffusion coefficient E(x, ω) is a lognormal random field with exponentialcorrelation function:
E(x, ω) = eλE+ζEg(x,ω)
Cov[g(x)g(x′)
]= e−|x−x
′|/l
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 13 / 25
Case studies Stochastic PDE
Example 1: Stochastic diffusion problem Blatman & Sudret (2013)
• 1D diffusion problem on D = [0, L]:[E(x, ω)u′(x, ω)
]′ + f(x) = 0u(0) = 0
E(u′)(L) = F
• u(x, ω) is the displacement field of a unit cross-section tension rodclamped at x = 0
• E(x, ω) is the (spatially variable) Young’s modulus of the rod• f(x) is the uniform axial load• F is a pinpoint load at x = L
Diffusion coefficientThe diffusion coefficient E(x, ω) is a lognormal random field with exponentialcorrelation function:
E(x, ω) = eλE+ζEg(x,ω)
Cov[g(x)g(x′)
]= e−|x−x
′|/l
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 13 / 25
Case studies Stochastic PDE
Example 1: Set-up of the UQ problem
• The Gaussian field g(x) can be represented by its Karhunen-Loveexpansion truncated at M = 62 (1% error in the variance):
g(x, ω) =M∑k=1
√lkφk(x)ξk(ω)
• ξk(ω) are standard normal variables• The diffusion equation is solved numerically with a Matlab FEM
code for each realization of the random vector Ξ = {ξ1...ξM}, with1000 FEM elements
• PCE problem with 62 input variables, 1000 output variables(displacement at each node of the FE mesh)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 14 / 25
Case studies Stochastic PDE
Example 1: Set-up of the UQ problem
• The Gaussian field g(x) can be represented by its Karhunen-Loveexpansion truncated at M = 62 (1% error in the variance):
g(x, ω) =M∑k=1
√lkφk(x)ξk(ω)
• ξk(ω) are standard normal variables• The diffusion equation is solved numerically with a Matlab FEM
code for each realization of the random vector Ξ = {ξ1...ξM}, with1000 FEM elements
• PCE problem with 62 input variables, 1000 output variables(displacement at each node of the FE mesh)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 14 / 25
Case studies Stochastic PDE
Example 1: Solution with UQLab
Realizations of E(x, ω):
from Blatman & Sudret, 2013
Parameter distributions:Name Type µ σ
{X1, ..., X62} Normal 0 1
Code:uqlab
for ii = 1:62In. Marginals (ii ). Type='Gaussian ';In. Marginals (ii ). Parameters =[0 1];endmyInput = uq_createInput (In );
modelOpts . mFile ='FEM1DDiffusion ';myModel = uq_createModel ( modelOpts );
metaOpts .Type='Metamodel ';metaOpts . MetaType ='PCE ';metaOpts . Degree =1:5;metaOpts . TruncOpts . qNorm = 0.75;metaOpts . ExpDesign . NSamples =500;metaOpts . ExpDesign . Sampling ='LHS ';myPCE = uq_createModel ( metaOpts );
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 15 / 25
Case studies Stochastic PDE
Example 1: Solution with UQLab
Realizations of E(x, ω):
from Blatman & Sudret, 2013
Parameter distributions:Name Type µ σ
{X1, ..., X62} Normal 0 1
Code:uqlab
for ii = 1:62In. Marginals (ii ). Type='Gaussian ';In. Marginals (ii ). Parameters =[0 1];endmyInput = uq_createInput (In );
modelOpts . mFile ='FEM1DDiffusion ';myModel = uq_createModel ( modelOpts );
metaOpts .Type='Metamodel ';metaOpts . MetaType ='PCE ';metaOpts . Degree =1:5;metaOpts . TruncOpts . qNorm = 0.75;metaOpts . ExpDesign . NSamples =500;metaOpts . ExpDesign . Sampling ='LHS ';myPCE = uq_createModel ( metaOpts );
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 15 / 25
Case studies Stochastic PDE
Example 1: Results
Confidence bounds on u(x, ω):
Confidence bounds created with kernelsmoothing of 1e5 samples from PCE
Code:XPC = uq_getSample (1 e5 );YPC = uq_evalModel (myPCE ,XPC );
% % V a l i d a t i o nXval = uq_getSample (100);Yval = uq_evalModel (myModel ,Xval );
Remarks:
• Brute-force showcase example(1 PCE × FEM element)
• Many alternatives exist, e.g.PCA-compression
• Very easy to deploy aftercalculation
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 16 / 25
Case studies Stochastic PDE
Example 1: Results
Confidence bounds on u(x, ω):
Confidence bounds created with kernelsmoothing of 1e5 samples from PCE
Code:XPC = uq_getSample (1 e5 );YPC = uq_evalModel (myPCE ,XPC );
% % V a l i d a t i o nXval = uq_getSample (100);Yval = uq_evalModel (myModel ,Xval );
Remarks:
• Brute-force showcase example(1 PCE × FEM element)
• Many alternatives exist, e.g.PCA-compression
• Very easy to deploy aftercalculation
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 16 / 25
Case studies Stochastic PDE
Example 1: Results
Confidence bounds on u(x, ω):
Confidence bounds created with kernelsmoothing of 1e5 samples from PCE
Code:XPC = uq_getSample (1 e5 );YPC = uq_evalModel (myPCE ,XPC );
% % V a l i d a t i o nXval = uq_getSample (100);Yval = uq_evalModel (myModel ,Xval );
Remarks:
• Brute-force showcase example(1 PCE × FEM element)
• Many alternatives exist, e.g.PCA-compression
• Very easy to deploy aftercalculation
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 16 / 25
Case studies Stochastic PDE
Example 1: Results
Confidence bounds on u(x, ω):
Confidence bounds created with kernelsmoothing of 1e5 samples from PCE
Code:XPC = uq_getSample (1 e5 );YPC = uq_evalModel (myPCE ,XPC );
% % V a l i d a t i o nXval = uq_getSample (100);Yval = uq_evalModel (myModel ,Xval );
Remarks:
• Brute-force showcase example(1 PCE × FEM element)
• Many alternatives exist, e.g.PCA-compression
• Very easy to deploy aftercalculation
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 16 / 25
Case studies Kriging-based rare event estimation
Example 2: Kriging-based rare event estimation - I
Reliability analysisCompute pf = P (g(x) ≤ 0), where
g(x) = 20−(x1−x2)2−8(x1 +x2−4)3
• Standard reliability analysisbenchmark
• 2-dimensional space, easy tovisualize
• Analytical function, “true” resultsavailable
The 2D “Hat” function
Method: Adaptive Kriging model (AK-MCS: enrichment of theexperimental design close to the limit state surface g(x) = 0)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 17 / 25
Case studies Kriging-based rare event estimation
Example 2: Kriging-based rare event estimation - I
Reliability analysisCompute pf = P (g(x) ≤ 0), where
g(x) = 20−(x1−x2)2−8(x1 +x2−4)3
• Standard reliability analysisbenchmark
• 2-dimensional space, easy tovisualize
• Analytical function, “true” resultsavailable
The 2D “Hat” function
Method: Adaptive Kriging model (AK-MCS: enrichment of theexperimental design close to the limit state surface g(x) = 0)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 17 / 25
Case studies Kriging-based rare event estimation
Example 2: Solution with UQLabParameter distributions:Name Type µ σ
{X1, X2} Normal 0 1
PMCSf = 3.85× 10−4
PAK-MCSf = (3.76± 0.38)× 10−4
uqlabfor ii = 1:2In. Marginals (ii ). Type='Gaussian ';In. Marginals (ii ). Parameters =[0 1];endmyInput = uq_createInput (In );
modelOpts . mFile ='HatFunction ';myModel = uq_createModel ( modelOpts );
% Reference MCS r e s u l t sXref = uq_getSample (1 e8 );Yref = uq_evalModel (Xref );PfMCS = sum(Yref <=0)/1 e8;
AKOptions .Type = 'Reliability ';AKOptions . Method = 'AKMCS ';AKOptions . AKMCS . IExpDesign .N = 10;AKOptions . AKMCS . MaxAddedED = 20;myAKMCS = uq_createAnalysis ( AKOpts );uq_display ( myAKMCS );
Only 25 model evaluations...and 14 lines of code!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 18 / 25
Case studies Kriging-based rare event estimation
Example 2: Solution with UQLabParameter distributions:Name Type µ σ
{X1, X2} Normal 0 1
PMCSf = 3.85× 10−4
PAK-MCSf = (3.76± 0.38)× 10−4
uqlabfor ii = 1:2In. Marginals (ii ). Type='Gaussian ';In. Marginals (ii ). Parameters =[0 1];endmyInput = uq_createInput (In );
modelOpts . mFile ='HatFunction ';myModel = uq_createModel ( modelOpts );
% Reference MCS r e s u l t sXref = uq_getSample (1 e8 );Yref = uq_evalModel (Xref );PfMCS = sum(Yref <=0)/1 e8;
AKOptions .Type = 'Reliability ';AKOptions . Method = 'AKMCS ';AKOptions . AKMCS . IExpDesign .N = 10;AKOptions . AKMCS . MaxAddedED = 20;myAKMCS = uq_createAnalysis ( AKOpts );uq_display ( myAKMCS );
Only 25 model evaluations...and 14 lines of code!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 18 / 25
Case studies Kriging-based rare event estimation
Example 2: Solution with UQLabParameter distributions:Name Type µ σ
{X1, X2} Normal 0 1
PMCSf = 3.85× 10−4
PAK-MCSf = (3.76± 0.38)× 10−4
uqlabfor ii = 1:2In. Marginals (ii ). Type='Gaussian ';In. Marginals (ii ). Parameters =[0 1];endmyInput = uq_createInput (In );
modelOpts . mFile ='HatFunction ';myModel = uq_createModel ( modelOpts );
% Reference MCS r e s u l t sXref = uq_getSample (1 e8 );Yref = uq_evalModel (Xref );PfMCS = sum(Yref <=0)/1 e8;
AKOptions .Type = 'Reliability ';AKOptions . Method = 'AKMCS ';AKOptions . AKMCS . IExpDesign .N = 10;AKOptions . AKMCS . MaxAddedED = 20;myAKMCS = uq_createAnalysis ( AKOpts );uq_display ( myAKMCS );
Only 25 model evaluations...and 14 lines of code!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 18 / 25
Case studies Kriging-based rare event estimation
Example 2: Solution with UQLabParameter distributions:Name Type µ σ
{X1, X2} Normal 0 1
PMCSf = 3.85× 10−4
PAK-MCSf = (3.76± 0.38)× 10−4
uqlabfor ii = 1:2In. Marginals (ii ). Type='Gaussian ';In. Marginals (ii ). Parameters =[0 1];endmyInput = uq_createInput (In );
modelOpts . mFile ='HatFunction ';myModel = uq_createModel ( modelOpts );
% Reference MCS r e s u l t sXref = uq_getSample (1 e8 );Yref = uq_evalModel (Xref );PfMCS = sum(Yref <=0)/1 e8;
AKOptions .Type = 'Reliability ';AKOptions . Method = 'AKMCS ';AKOptions . AKMCS . IExpDesign .N = 10;AKOptions . AKMCS . MaxAddedED = 20;myAKMCS = uq_createAnalysis ( AKOpts );uq_display ( myAKMCS );
Only 25 model evaluations...and 14 lines of code!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 18 / 25
Case studies Kriging-based rare event estimation
Example 2: Solution with UQLabParameter distributions:Name Type µ σ
{X1, X2} Normal 0 1
PMCSf = 3.85× 10−4
PAK-MCSf = (3.76± 0.38)× 10−4
uqlabfor ii = 1:2In. Marginals (ii ). Type='Gaussian ';In. Marginals (ii ). Parameters =[0 1];endmyInput = uq_createInput (In );
modelOpts . mFile ='HatFunction ';myModel = uq_createModel ( modelOpts );
% Reference MCS r e s u l t sXref = uq_getSample (1 e8 );Yref = uq_evalModel (Xref );PfMCS = sum(Yref <=0)/1 e8;
AKOptions .Type = 'Reliability ';AKOptions . Method = 'AKMCS ';AKOptions . AKMCS . IExpDesign .N = 10;AKOptions . AKMCS . MaxAddedED = 20;myAKMCS = uq_createAnalysis ( AKOpts );uq_display ( myAKMCS );
Only 25 model evaluations...and 14 lines of code!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 18 / 25
Case studies Subsurface contaminant diffusion
Example 3: Subsurface diffusion Joint work with University of Neuchatel
• Idealized model of theParis Basin
• Two-dimensional crosssection(25 km long / 1,040 mdepth) with 5× 5 mmesh (106 elements)
• 15 homogeneous layers
• Steady-state flow with Dirichlet boundary conditions:
∇ · (K · ∇H) = 0
Deman, Konakli, BS, Kerrou, Perrochet & Benabderrahmane, Using sparse polynomial chaos expansions for the global sensitivity analysis
of groundwater lifetime expectancy in a multi-layered hydrogeological model, Reliab. Eng. Sys. Safety, 147 (2015)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 19 / 25
Case studies Subsurface contaminant diffusion
Probabilistic model of porosity / conductivity
0 0.05 0.1 0.15 0.2
T
D1
D2
D3
D4
C1
C2
C3ab
L1a
L1bL2aL2bL2c
K1K2
K3
φ
T
D1
D2
D3
D4
C1
C2
C3ab
L1a
L1bL2aL2bL2c
K1K2
K3
• In each layer, bounds on porosity:
φi ∼ U [φimin , φimax]
• Deterministic mapping to the conductivity
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 20 / 25
Case studies Subsurface contaminant diffusion
Probabilistic model of porosity / conductivity
10−12
10−10
10−8
10−6
10−4
10−2
T
D1
D2
D3
D4
C1
C2
C3ab
L1a
L1bL2aL2bL2c
K1K2
K3
Kx[m/s]
Nominal conductivity (Kx) vs. porosityLayer Kx [m/s] φ [-]
K3 9.01E−09 0.0100K1-K2 4.53E−09 0.1150
L2c 1.10E−06 0.1389L2b 3.46E−07 0.1110L2a 1.62E−07 0.1139L1b 1.49E−05 0.1604L1a 1.17E−06 0.1549
C3ab 4.59E−08 0.0984C2 1.99E−13 0.1580C1 1.89E−06 0.0470D4 1.65E−05 0.0905D3 1.76E−06 0.1016D2 2.62E−07 0.0623D1 3.23E−06 0.0688T 1.95E−12 0.0810
• In each layer, bounds on porosity:φi ∼ U [φimin , φimax]
• Deterministic mapping to the conductivitylog10(Ki
x) = fi(φi) (layer-dependent)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 20 / 25
Case studies Subsurface contaminant diffusion
Mean life-time expectancy
DefinitionThe Mean Lifetime Expectancy MLE(x) is the time required for amolecule of water at point x to get out of the boundaries of the model
Map of mean lifetime expectancy (nominal case)
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 21 / 25
Case studies Subsurface contaminant diffusion
Other parameters
Parameter Notation RangePorosity φi, i = 1, . . . , 15 [φimin , φ
imax]
Anisotropy of hydraulic conductivitytensor
AiK , i = 1, . . . , 15 [0.01 , 1]
Euler angle of hydraulic conductivitytensor
θi, i = 1, . . . , 15 [−30 , 30](deg)
Longitudinal component of disper-sivity tensor
αiL, i = 1, . . . , 15 [5 , 25]
Anisotropy of dispersivity tensor Aiα, i = 1, . . . , 15 [5 , 25]
Hydraulic gradient (10−3m/m)Dogger sequence ∇HD [0.64 , 0.96]Oxfordian sequence ∇HO [2.40 , 3.60]Top of the model ∇Htop [2.72 , 4.08]
78 independent variables with uniform distributions
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 22 / 25
Case studies Subsurface contaminant diffusion
PCE-based Sobol’ sensitivity indices
0.01
0.2
0.4
0.6
0.8
φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4
KAC3ab
a
Total Sobol’ IndicesSTot
i
0.01
0.2
0.4
0.6
0.8
φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4K
AC3aba
Sobol’ Indices Order 1
S(1)
i
Parameter∑
jS
(1)j
φ 0.8664
AK 0.0088
θ 0.0029
αL 0.0076
Aα 0.0000
∇H 0.0057
• Uncertainties on the porosities (conductivities) drive the MLE uncertainty
Only 200 model runs allow one to detect the important parametersout of 78
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 23 / 25
Case studies Subsurface contaminant diffusion
PCE-based Sobol’ sensitivity indices
0.01
0.2
0.4
0.6
0.8
φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4
KAC3ab
a
Total Sobol’ IndicesSTot
i
0.01
0.2
0.4
0.6
0.8
φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4K
AC3aba
Sobol’ Indices Order 1
S(1)
i
Parameter∑
jS
(1)j
φ 0.8664
AK 0.0088
θ 0.0029
αL 0.0076
Aα 0.0000
∇H 0.0057
• Uncertainties on the porosities (conductivities) drive the MLE uncertainty
Only 200 model runs allow one to detect the important parametersout of 78
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 23 / 25
Outlook
Outline
1 Introduction
2 The UQLab project
3 Case studies
4 OutlookTentative release schedule
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 23 / 25
Outlook Tentative release schedule
Release schedule
Closed-Beta phase
• 2015/07/01 - V0.9 Beta release• 2016/03/01 - V0.92 Beta (Rare event estimation)
UQLab V1.0 [tentative!]
• 2016/07/01 [tentative] - V1.0!!• 2016/07/01 [tentative] - Open source release of the scientific code• end of 2016 - Lots of new features!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 24 / 25
Outlook Tentative release schedule
Upcoming features
• New surrogate modelling techniques:- Low Rank Approximations - review/documentation stage- PC-Kriging - review/documentation stage- Updates to the existing PCE and Kriging modules - complete
• New module: Bayesian inversion - in development• New module: Machine learning - in development• New module: Random fields simulation - in design• New Sensitivity analysis methods:
- DGSM indices (sampling+PCE-based) - in development- Borgonovo distribution-based indices - in design- LRA-based Sobol’ indices - in design
And much more!!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 25 / 25
Outlook Tentative release schedule
Upcoming features
• New surrogate modelling techniques:- Low Rank Approximations - review/documentation stage- PC-Kriging - review/documentation stage- Updates to the existing PCE and Kriging modules - complete
• New module: Bayesian inversion - in development• New module: Machine learning - in development• New module: Random fields simulation - in design• New Sensitivity analysis methods:
- DGSM indices (sampling+PCE-based) - in development- Borgonovo distribution-based indices - in design- LRA-based Sobol’ indices - in design
And much more!!
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 25 / 25
Outlook Tentative release schedule
Thank you very much for yourattention!
UQLabThe Framework for Uncertainty Quantification
www.uqlab.com
Chair of Risk, Safety & Uncertainty Quantificationhttp://www.rsuq.ethz.ch
S. Marelli and B. Sudret (RSUQ, ETH Zurich) Advancements in UQLab 06.04.2016 25 / 25
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