DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION
UQLab: a framework for UncertaintyQuantification in Matlab
Stefano Marelli and Bruno Sudret
Chair of Risk, Safety & Uncertainty Quantification
MascotNum201424.04.2014
IntroductionUQLab in action
Summary and Outlook
Outline
1 Introduction
2 UQLab in action
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 1 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Outline
1 IntroductionComputer SimulationsA global frameworkThe UQLab project
2 UQLab in action
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 1 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Introduction
Computer simulations and uncertainty quantificationComputer simulations increasingly substitute expensive experimentalinvestigations
Massive increase in availability of computational resources andcomputational algorithmsLogarithmic decrease of cost/flop in High Performance ComputinginfrastructuresComputer 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
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 2 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Introduction
Computer simulations and uncertainty quantificationComputer simulations increasingly substitute expensive experimentalinvestigations
Massive increase in availability of computational resources andcomputational algorithmsLogarithmic decrease of cost/flop in High Performance ComputinginfrastructuresComputer 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
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 2 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Introduction
Computer simulations and uncertainty quantificationComputer simulations increasingly substitute expensive experimentalinvestigations
Massive increase in availability of computational resources andcomputational algorithmsLogarithmic decrease of cost/flop in High Performance ComputinginfrastructuresComputer 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
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 2 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Sources of uncertainty
Aleatory uncertaintyUncertainty 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 uncertaintyLack of knowledge about the parameters of a system, e.g. measurementuncertainty, lack of dataIn principle reducible by acquiring additional information
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 3 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Sources of uncertainty
Aleatory uncertaintyUncertainty 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 uncertaintyLack of knowledge about the parameters of a system, e.g. measurementuncertainty, lack of dataIn principle reducible by acquiring additional information
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 3 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Outline
1 IntroductionComputer SimulationsA global frameworkThe UQLab project
2 UQLab in action
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 3 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Global framework for managing uncertainties
PhysicalModel
Model(s) of the system
Assessment criteria
Probabilistic InputModel
Quantification of
sources of uncertainty
Analysis
Uncertainty propagation
Random variables Computational model Moments
Probability 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. http://www.ibk.ethz.ch/su/publications/Reports/HDRSudret.pdf
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 4 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Global framework for managing uncertainties
PhysicalModel
Model(s) of the system
Assessment criteria
Probabilistic InputModel
Quantification of
sources of uncertainty
Analysis
Uncertainty propagation
Random variables Computational model Moments
Probability 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. http://www.ibk.ethz.ch/su/publications/Reports/HDRSudret.pdf
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 4 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Global framework for managing uncertainties
PhysicalModel
Model(s) of the system
Assessment criteria
Probabilistic InputModel
Quantification of
sources of uncertainty
Analysis
Uncertainty propagation
Random variables Computational model Moments
Probability 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. http://www.ibk.ethz.ch/su/publications/Reports/HDRSudret.pdf
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 4 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Global framework for managing uncertainties
PhysicalModel
Model(s) of the system
Assessment criteria
Probabilistic InputModel
Quantification of
sources of uncertainty
Analysis
Uncertainty propagation
Random variables Computational model Moments
Probability 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. http://www.ibk.ethz.ch/su/publications/Reports/HDRSudret.pdf
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 4 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The physical model
Computational models of physical and engineering systemsSolution of differential equations (e.g. FEM, FD, PS, etc. )Multi-physics simulations (e.g. Comsol, etc. )
Functional approximations, surrogate modelsInterpolation methods (Kriging)Regression methods (Polynomial Chaos, Support vector regression)
Measurements/databasesExperimental data from literatureNew 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)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 5 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The physical model
Computational models of physical and engineering systemsSolution of differential equations (e.g. FEM, FD, PS, etc. )Multi-physics simulations (e.g. Comsol, etc. )
Functional approximations, surrogate modelsInterpolation methods (Kriging)Regression methods (Polynomial Chaos, Support vector regression)
Measurements/databasesExperimental data from literatureNew 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)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 5 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The probabilistic input model
Experimental data availableDescriptive Statistics: moments, histograms,kernel smoothingStatistical inference: fitting marginals, copula
Only prior/expert knowledgeMaximum entropy principle: maximizeinformation under constraintsPrior knowledge: e.g. physical constraints onsystem variables, literature
Scarce data + expert informationBayesian inference methods to combine expertjudgment and experimental information
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
Clayton copula sampling
u1
u1
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 6 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The statistical analysisMany possibilities
Full response characterization(distribution analysis)Reliability analysis (rare events simulation)Analysis of the momentsSensitivity analysis/model reductionStochastic/parametric inversionModel calibrationDesign optimization
Examples
Monte Carlo SimulationApproximation methods(FORM/SORM)
Surrogate modelling (PolynomialChaos, Kriging)Sobol’ sensitivity indices
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 7 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The statistical analysisMany possibilities
Full response characterization(distribution analysis)Reliability analysis (rare events simulation)Analysis of the momentsSensitivity analysis/model reductionStochastic/parametric inversionModel calibrationDesign optimization
Examples
Monte Carlo SimulationApproximation methods(FORM/SORM)
Surrogate modelling (PolynomialChaos, Kriging)Sobol’ sensitivity indices
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 7 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Summary
Uncertainty Quantification (UQ) propagates theuncertainty in model parameters to the model responseEvery UQ problem can be decomposed in input, modeland analysisThe framework introduced can be used as a guideline insetting up and solving any UQ problem
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 8 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Outline
1 IntroductionComputer SimulationsA global frameworkThe UQLab project
2 UQLab in action
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 8 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The current situation in UQ software
Available software for Uncertainty QuantificationNon general: few combine all the available UQ techniquesToo complex for some communities: requires advanced ITknowledgePoorly extendable: is either closed source or very difficult toextend (documentation, complex build system, etc. )Lack of HPC interface: even if supporting it, often difficult to useIntrusive: black-box approach is not often encouragedNon-portable: lack of support for linux/windows/mac
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 9 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The UQLab Software Framework
UQLab: Uncertainty Quantification Lab
Focus on:GeneralityEase of useNon-intrusivenessHPCExtendibilityCollaboration
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 10 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
The UQLab Software Framework
UQLab: Uncertainty Quantification Lab
Focus on:GeneralityEase of useNon-intrusivenessHPCExtendibilityCollaboration
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 10 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Extendibility: an effort to collaborative development
Multi-level collaborative development
End Users: Field Engineers, Industrial Users, Students, etc.No extension of the existing codebase.Scientific Developers: Scientists, extending the scientific baseline of theframework (implementation of new algorithms)
Modular Content Management System
Builtin: Native or contributed, after review and optimization by coredevelopers.Contrib: Users/partner contributed extensions: must include workingdemos and detailed documentation. May be enabled/disabled.External: Locally handled by the developer. No restrictions.
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 11 / 33
IntroductionUQLab in action
Summary and Outlook
Computer SimulationsA global frameworkThe UQLab project
Extendibility: an effort to collaborative development
Multi-level collaborative development
End Users: Field Engineers, Industrial Users, Students, etc.No extension of the existing codebase.Scientific Developers: Scientists, extending the scientific baseline of theframework (implementation of new algorithms)
Modular Content Management System
Builtin: Native or contributed, after review and optimization by coredevelopers.Contrib: Users/partner contributed extensions: must include workingdemos and detailed documentation. May be enabled/disabled.External: Locally handled by the developer. No restrictions.
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 11 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Outline
1 Introduction
2 UQLab in actionCurrent status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 11 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Current Features of UQLabRepresentation of
The physical modelFunctions, strings, function handlesPolynomial Chaos Expansions:
- Orthonormal polynomials- Full and sparse (Smolyak) quadrature-based projection- Standard and adaptive basis selection regression (OLS, Lars)
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- Plugin to DiceKriging (R)
The probabilistic input model (copula formalism)Standard marginals (support for custom)Elliptic copulaeGeneralized isoprobabilistic transformSampling strategies (MC, LHS, quasi-random sequences)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 12 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Current Features of UQLab (cont’d)
UQLab featuresStatistical analysis
Reliability analysis- Simple Monte-Carlo reliability analysis with advanced sampling- Approximation methods: FORM and SORM (a la FERUM) with
revisited algorithms- Importance Sampling (FORM-based, or user specified)- FERUMLink plugin to FERUM
Global sensitivity analysis- Screening: Cotter measure, Morris method- Variance decomposition: Sobol’ indices- PCE-based Sobol’ indices- Plugin to R-based ”Sensitivity” package
High Performance Computing (HPC)HPC Dispatcher
Simple interface to common HPC facilitiesParallelization of core algorithms (e.g. model evaluations)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 13 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Outline
1 Introduction
2 UQLab in actionCurrent status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 13 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Metamodelling: polynomial chaos expansion I
DefinitionsGiven a probabilistic Hilbert space H endowed with a scalar product
〈g(x), h(x)〉 = E [g(x)h(x)] =∫
x∈Dx
g(x)h(x)fxdx
where fx is a short form for the joint pdf of the random vector X:
fx(x) :∫
x∈Dx
fx(x)dx = 1
Orthonormal polynomial basisLet Ψ be an orthonormal polynomial basis of the space H for the randomvector X such that:
〈Ψα(x),Ψβ(x)〉 = δαβ
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 14 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Metamodelling: polynomial chaos expansion I
DefinitionsGiven a probabilistic Hilbert space H endowed with a scalar product
〈g(x), h(x)〉 = E [g(x)h(x)] =∫
x∈Dx
g(x)h(x)fxdx
where fx is a short form for the joint pdf of the random vector X:
fx(x) :∫
x∈Dx
fx(x)dx = 1
Orthonormal polynomial basisLet Ψ be an orthonormal polynomial basis of the space H for the randomvector X such that:
〈Ψα(x),Ψβ(x)〉 = δαβ
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 14 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Metamodelling: polynomial chaos expansion II
Polynomial Chaos ExpansionGiven an input vector X ∈ RM and a model Y =M(X), then
M(X) ≈MPC (X) =∑α∈A
yαΨα(X)
is the truncated polynomial chaos expansion of M of degree p, i.e. Acontains all polynomials of maximal degree p.
Calculating the yα
Projection or regression methodsRelatively small Experimental Design (full model evaluations)Advanced adaptive techniques available (e.g., LARS)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 15 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Metamodelling: polynomial chaos expansion II
Polynomial Chaos ExpansionGiven an input vector X ∈ RM and a model Y =M(X), then
M(X) ≈MPC (X) =∑α∈A
yαΨα(X)
is the truncated polynomial chaos expansion of M of degree p, i.e. Acontains all polynomials of maximal degree p.
Calculating the yα
Projection or regression methodsRelatively small Experimental Design (full model evaluations)Advanced adaptive techniques available (e.g., LARS)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 15 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Metamodelling: polynomial chaos expansion III
AdvantagesNon-intrusive (black box exp. design)Because the Ψα are orthogonal, it is uniqueSparse for most physical modelsInteresting properties of the coefficients (postprocessing):
µY = y0σ2
Y =∑
α∈Aα 6=0
y2α
Surrogate model evaluation only takes a few matrix multiplicationsA posteriori error estimates → accuracy driven adaptivity
DisadvantagesBasis size can quickly explodeAccuracy depends on the experimental design
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 16 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Metamodelling: polynomial chaos expansion III
AdvantagesNon-intrusive (black box exp. design)Because the Ψα are orthogonal, it is uniqueSparse for most physical modelsInteresting properties of the coefficients (postprocessing):
µY = y0σ2
Y =∑
α∈Aα 6=0
y2α
Surrogate model evaluation only takes a few matrix multiplicationsA posteriori error estimates → accuracy driven adaptivity
DisadvantagesBasis size can quickly explodeAccuracy depends on the experimental design
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 16 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 = 0E(x, ω) is the (spatially variable) Young’s modulus of the rodf (x) is the uniform axial loadF is a pinpoint load at x = L
The diffusion coefficient E(x, ω) is a lognormal random field with exponentialcorrelation function:
E(x, ω) = eλE +ζE g(x,ω)
Cov[g(x)g(x′)
]= e−|x−x′|/l
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 17 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 variablesThe diffusion equation is solved numerically with a Matlab FEMcode for each realization of the random vector Ξ = {ξ1...ξM}, with1000 FEM elementsPCE problem with 62 input variables, 1000 output variables(displacement at each node of the FE mesh)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 18 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 variablesThe diffusion equation is solved numerically with a Matlab FEMcode for each realization of the random vector Ξ = {ξ1...ξM}, with1000 FEM elementsPCE problem with 62 input variables, 1000 output variables(displacement at each node of the FE mesh)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 18 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 = 1:62Marg(ii ). Type = 'Gaussian ';Marg(ii ). Parameters =[0 1];endmyInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'FEM1DDiffusion ';myModel = uq_create_model ( Model );
Meta.Type='uq_metamodel ';Meta. MetaType ='PCE ';Meta. Coeff . Degree =1:5;Meta. ExpDesign . NSamples =500;Meta. ExpDesign . Sampling ='LHS ';Meta. Input = myInput ;Meta. FullModel = myModel ;PCModel = uq_create_model (Meta );uq_calculate_metamodel ;
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 19 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 = 1:62Marg(ii ). Type = 'Gaussian ';Marg(ii ). Parameters =[0 1];endmyInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'FEM1DDiffusion ';myModel = uq_create_model ( Model );
Meta.Type='uq_metamodel ';Meta. MetaType ='PCE ';Meta. Coeff . Degree =1:5;Meta. ExpDesign . NSamples =500;Meta. ExpDesign . Sampling ='LHS ';Meta. Input = myInput ;Meta. FullModel = myModel ;PCModel = uq_create_model (Meta );uq_calculate_metamodel ;
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 19 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 (X);
% % V a l i d a t i o nXval = uq_getSample (100);uq_select_model ( myModel );Yval = uq_evalModel (Xval );
1 independent PCE for each FEMelement!!
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 20 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 (X);
% % V a l i d a t i o nXval = uq_getSample (100);uq_select_model ( myModel );Yval = uq_evalModel (Xval );
1 independent PCE for each FEMelement!!
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 20 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
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 (X);
% % V a l i d a t i o nXval = uq_getSample (100);uq_select_model ( myModel );Yval = uq_evalModel (Xval );
1 independent PCE for each FEMelement!!
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 20 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 1: Remarks
This was a brute-force showcase exampleMany alternatives exist, e.g., Principal Component AnalysisCan be handled on a normal computer (ran on this laptop in approx38 minutes)Very easy to deploy after calculation
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 21 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Outline
1 Introduction
2 UQLab in actionCurrent status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 21 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Sensitivity analysis: Sobol’ indices
Variance decompositionThe total model variance D can be decomposed in a sum ofpartial-variances D1,2..M
D = Var(M(x)) =M∑
i=1Di +
∑1≤i≤j≤M
Dij + ...+ D12...M
Definition: Sobol’ and Total Sobol’ indices of order s
Sj1...js = Dj1...js/D
STi =∑Ji
Dj1...js/D, Ji = {{j1, ..., js} ⊃ {i}}
Sobol’, 1993: Sensitivity estimates for nonlinear mathematical models. Math. Modeling & Comp.Exp. 1, 407-414.
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 22 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Sensitivity analysis: Sobol’ indices
Variance decompositionThe total model variance D can be decomposed in a sum ofpartial-variances D1,2..M
D = Var(M(x)) =M∑
i=1Di +
∑1≤i≤j≤M
Dij + ...+ D12...M
Definition: Sobol’ and Total Sobol’ indices of order s
Sj1...js = Dj1...js/D
STi =∑Ji
Dj1...js/D, Ji = {{j1, ..., js} ⊃ {i}}
Sobol’, 1993: Sensitivity estimates for nonlinear mathematical models. Math. Modeling & Comp.Exp. 1, 407-414.
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 22 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Sobol’ Indices and PCE
Re-grouping PCE coefficients
MPC (X) =∑α∈A
yαΨα (X) =
y0 +M∑
i=1
∑α∈Ai
yαΨα (xi) +∑
α∈A1,2,...,M
yαΨα(x1, x2, ..., xM )
where Ai ⊂ A is a set of α such that Ψα is an orthonormal polynomialthat depends on the s variables xi1 , ..., xis
PCE based Sobol’ indices
SUi1,...,is = 1σ2,PC
Y
∑α∈Ii1,...,is
y2α =
∑α∈Ii1,...,is
y2α
/ ∑1≤|α|≤P
y2α
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 23 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Sobol’ Indices and PCE
Re-grouping PCE coefficients
MPC (X) =∑α∈A
yαΨα (X) =
y0 +M∑
i=1
∑α∈Ai
yαΨα (xi) +∑
α∈A1,2,...,M
yαΨα(x1, x2, ..., xM )
where Ai ⊂ A is a set of α such that Ψα is an orthonormal polynomialthat depends on the s variables xi1 , ..., xis
PCE based Sobol’ indices
SUi1,...,is = 1σ2,PC
Y
∑α∈Ii1,...,is
y2α =
∑α∈Ii1,...,is
y2α
/ ∑1≤|α|≤P
y2α
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 23 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: simple function with many dimensions
M = 86: realistic engineering applicationX ∼ U([1, 2])M , with X20 ∈ U([1, 3])Strongly non-linearContinuous and smooth functionEasy to predict sensitivity patternsWell-located peaks in the sensitivity
High-dimensional test function
y =3 −5M
M∑k=1
kxk +1M
M∑k=1
kx3k + ln
(1
3M
M∑k=1
k(
x2k + x4
k))
+ x1x22 − x5x3 + x2x4 + x50 + x50x2
54
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 24 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: Reference results
Model responseTotal Sobol’ indicesSecond Order Sobol’indices
Analytical function
y =3 −5M
M∑k=1
kxk +1M
M∑k=1
kx3k + ln
(1
3M
M∑k=1
k(
x2k + x4
k))
+ x1x22 − x5x3 + x2x4 + x50 + x50x2
54
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 25 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: Reference results
Model responseTotal Sobol’ indicesSecond Order Sobol’indices
Analytical function
y =3 −5M
M∑k=1
kxk +1M
M∑k=1
kx3k + ln
(1
3M
M∑k=1
k(
x2k + x4
k))
+ x1x22 − x5x3 + x2x4 + x50 + x50x2
54
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 25 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: Reference results
Model responseTotal Sobol’ indicesSecond Order Sobol’indices
Analytical function
y =3 −5M
M∑k=1
kxk +1M
M∑k=1
kx3k + ln
(1
3M
M∑k=1
k(
x2k + x4
k))
+ x1x22 − x5x3 + x2x4 + x50 + x50x2
54
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 25 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: Polynomial Chaos-based Sobol’ indicesMultidimensional function
Parameter distributions:Name Type a b{X1..,X19,X21..,X86} Uniform 1 2X20 Uniform 1 3
uqlabfor ii = 1:86Marg(ii ). Type = 'Uniform ';Marg(ii ). Parameters =[1 2];endMarg (20). Parameters =[2 3];myInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'uq_multidim86 ';myModel = uq_create_model ( Model );
Meta.Type='uq_metamodel ';Meta. MetaType ='PCE ';Meta. Coeff . Degree =3:5;Meta. ExpDesign . NSamples =900;Meta. ExpDesign . Sampling ='Sobol ';Meta. Input = myInput ;Meta. FullModel = myModel ;PCModel = uq_create_model (Meta );uq_calculate_metamodel ;
Analysis .Type='uq_sensitivity ';Analysis . Method ='Sobol ';uq_create_analysis ( Analysis );uq_runAnalysis ;
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 26 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: Polynomial Chaos-based Sobol’ indicesMultidimensional function
Parameter distributions:Name Type a b{X1..,X19,X21..,X86} Uniform 1 2X20 Uniform 1 3
uqlabfor ii = 1:86Marg(ii ). Type = 'Uniform ';Marg(ii ). Parameters =[1 2];endMarg (20). Parameters =[2 3];myInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'uq_multidim86 ';myModel = uq_create_model ( Model );
Meta.Type='uq_metamodel ';Meta. MetaType ='PCE ';Meta. Coeff . Degree =3:5;Meta. ExpDesign . NSamples =900;Meta. ExpDesign . Sampling ='Sobol ';Meta. Input = myInput ;Meta. FullModel = myModel ;PCModel = uq_create_model (Meta );uq_calculate_metamodel ;
Analysis .Type='uq_sensitivity ';Analysis . Method ='Sobol ';uq_create_analysis ( Analysis );uq_runAnalysis ;
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 26 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: Sensitivity results
Total Sobol’ indices
PCE-based Sobol’ indices(1200 model runs)
MCS-based Sobol’ indices(1,76 Mio model runs)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 27 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 2: Sensitivity results
Second order Sobol’ indices
PCE-based second order Sobol’ indices(1200 model runs)
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 27 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Outline
1 Introduction
2 UQLab in actionCurrent status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 27 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Reliability analysis
Definition:Structural reliability analysis is the process of calculating theprobability of failure of a structure with respect to some failure criterionon one or more quantities of interest.
Setting up a structural reliability problemStep 1: Define the model and the quantities of interestStep 2: Define the input parameters and their uncertaintiesStep 3: Define a failure criterion and calculate the failure probability
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 28 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Reliability analysis
Definition:Structural reliability analysis is the process of calculating theprobability of failure of a structure with respect to some failure criterionon one or more quantities of interest.
Setting up a structural reliability problemStep 1: Define the model and the quantities of interestStep 2: Define the input parameters and their uncertaintiesStep 3: Define a failure criterion and calculate the failure probability
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 28 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 3: Problem set-up and UQLab solution
Truss structure:
Parameter distributions:Name Type µ σ/µ
E1, E2 (Pa) Lognorm 2.1× 1011 10%A1 (m2) Lognorm 2.0× 10−3 10%A2 (m2) Lognorm 1.0× 10−3 10%P1 - P6 (N) Gumbel 5.0× 104 15%
Pf (V1 > 0.13m) = 1.18 × 10−4
code:uqlabMarg (1). Name = 'E1 ';Marg (1). Type = 'Lognormal ';Marg (1). Moments =[2.1 e11 2.1 e10 ];Marg (2). Name = 'E2 ';...myInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'uq_truss ';myModel = uq_create_model ( Model );
Analysis .Type='uq_reliability ';Analysis . limit_state = 0.13;Analysis . Method = 'IS ';Analysis . MaxSamples = 1e4;uq_create_analysis ( Analysis );
uq_runAnalysis ;
Pf (V1 > 0.13m) = 1.17 × 10−4
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 29 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 3: Problem set-up and UQLab solution
Truss structure:
Parameter distributions:Name Type µ σ/µ
E1, E2 (Pa) Lognorm 2.1× 1011 10%A1 (m2) Lognorm 2.0× 10−3 10%A2 (m2) Lognorm 1.0× 10−3 10%P1 - P6 (N) Gumbel 5.0× 104 15%
Pf (V1 > 0.13m) = 1.18 × 10−4
code:uqlabMarg (1). Name = 'E1 ';Marg (1). Type = 'Lognormal ';Marg (1). Moments =[2.1 e11 2.1 e10 ];Marg (2). Name = 'E2 ';...myInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'uq_truss ';myModel = uq_create_model ( Model );
Analysis .Type='uq_reliability ';Analysis . limit_state = 0.13;Analysis . Method = 'IS ';Analysis . MaxSamples = 1e4;uq_create_analysis ( Analysis );
uq_runAnalysis ;
Pf (V1 > 0.13m) = 1.17 × 10−4
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 29 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 3: Problem set-up and UQLab solution
Truss structure:
Parameter distributions:Name Type µ σ/µ
E1, E2 (Pa) Lognorm 2.1× 1011 10%A1 (m2) Lognorm 2.0× 10−3 10%A2 (m2) Lognorm 1.0× 10−3 10%P1 - P6 (N) Gumbel 5.0× 104 15%
Pf (V1 > 0.13m) = 1.18 × 10−4
code:uqlabMarg (1). Name = 'E1 ';Marg (1). Type = 'Lognormal ';Marg (1). Moments =[2.1 e11 2.1 e10 ];Marg (2). Name = 'E2 ';...myInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'uq_truss ';myModel = uq_create_model ( Model );
Analysis .Type='uq_reliability ';Analysis . limit_state = 0.13;Analysis . Method = 'IS ';Analysis . MaxSamples = 1e4;uq_create_analysis ( Analysis );
uq_runAnalysis ;
Pf (V1 > 0.13m) = 1.17 × 10−4
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 29 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 3: Problem set-up and UQLab solution
Truss structure:
Parameter distributions:Name Type µ σ/µ
E1, E2 (Pa) Lognorm 2.1× 1011 10%A1 (m2) Lognorm 2.0× 10−3 10%A2 (m2) Lognorm 1.0× 10−3 10%P1 - P6 (N) Gumbel 5.0× 104 15%
Pf (V1 > 0.13m) = 1.18 × 10−4
code:uqlabMarg (1). Name = 'E1 ';Marg (1). Type = 'Lognormal ';Marg (1). Moments =[2.1 e11 2.1 e10 ];Marg (2). Name = 'E2 ';...myInput = uq_create_input (Marg );
Model .Type='mfile ';Model . Function = 'uq_truss ';myModel = uq_create_model ( Model );
Analysis .Type='uq_reliability ';Analysis . limit_state = 0.13;Analysis . Method = 'IS ';Analysis . MaxSamples = 1e4;uq_create_analysis ( Analysis );
uq_runAnalysis ;
Pf (V1 > 0.13m) = 1.17 × 10−4
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 29 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 3: Adding HPC, PCE and Sobol’ indices...
Sensitivity analysis:HPC Scalability: code:
HPCopts .nCPU = 4;HPCopts . Profile ='credentials .txt ';uq_create_dispatcher ( HPCopts );
Metaopts .Type = 'uq_metamodel ';Metaopts . MetaType = 'PCE ';Metaopts . ExpDesign . NSamples = 200;Metaopts . Input = myInput ;Metaopts . FullModel = myModel ;PCmodel = uq_create_model ( Metaopts );uq_calculate_metamodel ;
Analysis .Type='uq_sensitivity ';Analysis . Method = 'Sobol ';uq_create_analysis ( Analysis );uq_runAnalysis ;
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 30 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 3: Adding HPC, PCE and Sobol’ indices...
Sensitivity analysis:
HPC Scalability:
code:HPCopts .nCPU = 4;HPCopts . Profile ='credentials .txt ';uq_create_dispatcher ( HPCopts );
Metaopts .Type = 'uq_metamodel ';Metaopts . MetaType = 'PCE ';Metaopts . ExpDesign . NSamples = 200;Metaopts . Input = myInput ;Metaopts . FullModel = myModel ;PCmodel = uq_create_model ( Metaopts );uq_calculate_metamodel ;
Analysis .Type='uq_sensitivity ';Analysis . Method = 'Sobol ';uq_create_analysis ( Analysis );uq_runAnalysis ;
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 30 / 33
IntroductionUQLab in action
Summary and Outlook
Current status of UQLabExample 1: PCE of a stochastic diffusion problemExample 2: Sensitivity analysis in high dimensionExample 3: Reliability analysis of a truss structure
Example 3: Adding HPC, PCE and Sobol’ indices...
Sensitivity analysis:
HPC Scalability:
code:HPCopts .nCPU = 4;HPCopts . Profile ='credentials .txt ';uq_create_dispatcher ( HPCopts );
Metaopts .Type = 'uq_metamodel ';Metaopts . MetaType = 'PCE ';Metaopts . ExpDesign . NSamples = 200;Metaopts . Input = myInput ;Metaopts . FullModel = myModel ;PCmodel = uq_create_model ( Metaopts );uq_calculate_metamodel ;
Analysis .Type='uq_sensitivity ';Analysis . Method = 'Sobol ';uq_create_analysis ( Analysis );uq_runAnalysis ;
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 30 / 33
IntroductionUQLab in action
Summary and Outlook
Outline
1 Introduction
2 UQLab in action
3 Summary and Outlook
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 30 / 33
IntroductionUQLab in action
Summary and Outlook
Planned features
What’s nextReliability based design optimizationAdvanced/alternative metamodelling tools (support vector machines,vector polynomial chaos, etc.)Bayesian methods (e.g. MCMC) for inversion and model calibrationPlugins to many available commercial modelling platforms (e.g.Abaqus)Higher degree of parallelization at many levelsUser friendly graphical user interface
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 31 / 33
IntroductionUQLab in action
Summary and Outlook
Summary
The UQLab projectThe UQLab development has started and is well under wayMany new features are being implemented, but many are ready foruseThe introduction of HPC in many steps of UQ problems is possible
OutlookComplete the documentation at all levelsSet a release scheduleWe are in closed alpha testing, soon to move to closed beta testingWe are collecting feedback from collaborations and teaching
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 32 / 33
IntroductionUQLab in action
Summary and Outlook
Summary
The UQLab projectThe UQLab development has started and is well under wayMany new features are being implemented, but many are ready foruseThe introduction of HPC in many steps of UQ problems is possible
OutlookComplete the documentation at all levelsSet a release scheduleWe are in closed alpha testing, soon to move to closed beta testingWe are collecting feedback from collaborations and teaching
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 32 / 33
IntroductionUQLab in action
Summary and Outlook
Thank you for your attention!!
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 32 / 33
IntroductionUQLab in action
Summary and Outlook
Some referencesUQLab at the Chair of Risk, Safety and Uncertainty Quantification at ETH Zurich:
http://www.ibk.ethz.ch/su/research/uqlabBlatman, G. and Sudret, B. (2011). Adaptive sparse polynomial chaos expansion based on Least
Angle Regression. J. Comput. Phys., 230, 2345-2367.Blatman, G. and Sudret, B. (2013). Sparse polynomial chaos expansions of vector-valued
response quantities. Proc. 11th Int. Conf. Struct. Safety and Reliability (ICOSSAR 2013),New York, USA.
Bourinet, J.-M., Mattrand, C., and Dubourg, V. (2009). A review of recent features andimprovements added to FERUM software. Proc. 10th Int. Conf. Struct. Safety andReliability (ICOSSAR 2009), Osaka, Japan.
Lebrun, R. and Dutfoy, A. (2009). A generalization of the Nataf transformation to distributionswith elliptical copula. Probabilistic Engineering Mechanics, Volume 24, Issue 2, April 2009,Pages 172-178.
Lemaire, M. (2010). Structural reliability (Vol. 84). John Wiley & Sons.Roustant, O., Ginsbourger, D., and Deville, Y. (2009). The DiceKriging package: kriging-based
metamodeling and optimization for computer experiments. Book of abstract of the R UserConference.
Sobol’, I. (1993). Sensitivity estimates for nonlinear mathematical models. Math. Modeling &Comp. Exp. 1, 407-414.
Sudret, B. (2007). Uncertainty propagation and sensitivity analysis in mechanical models -Contributions to structural reliability and stochastic spectral methods. Habilitation a dirigerdes recherches, Universite Blaise-Pascal, Clermont-Ferrand, France, 2007.
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 33 / 33
Statistical Analysis methodsDam breach modelling
Appendix
Backup Slides
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 34 / 33
Statistical Analysis methodsDam breach modelling
Statistical Analysis: Distribution analysisQuestion?Given a computer model M and a probabilisticmodel of its input parameters X ∼ fX , what arethe characteristics of the output distribution of Y =M(X)?
range / shape (uni-/multi-modal?)quantiles (median, inter-quartile, 99%-quantile, etc.)
MethodsMonte Carlo simulation + kernel smoothing (if large sample set available)Surrogate-based methods: polynomial chaos expansions, Kriging
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 35 / 33
Statistical Analysis methodsDam breach modelling
Model calibration
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 36 / 33
Statistical Analysis methodsDam breach modelling
Model calibration
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 36 / 33
Statistical Analysis methodsDam breach modelling
Model calibrationQuestion?Given a computer model M and a set of experimental results, what arethe best-fitting input parameters?
account for measurement uncertainty“best-fitting” parameters and residual model erroraccuracy of the fitting through the epistemic uncertainty of thefitted parameters
Stochastic inverse problemsintrinsic variability (aleatory uncertainty) in the input parameterscomputed from sets of similar experiments
MethodsBayesian calibration using prior information on the rangeMarkov chain Monte Carlo simulationStochastic inverse methods
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 37 / 33
Statistical Analysis methodsDam breach modelling
Analysis of the Macchione model
6 uncertainparameters:
G ∈ [5.7 11.4]α ∈ [1 4]β ∈ [ 1
18π13π]
Ys0 ∈ [0.2 0.99]s ∈ [2.2 2.8]wcs ∈ [0.4 0.8]
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 38 / 33
Statistical Analysis methodsDam breach modelling
Sensitivity analysis
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 39 / 33
Statistical Analysis methodsDam breach modelling
Accuracy
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 40 / 33
Statistical Analysis methodsDam breach modelling
Accuracy
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 41 / 33
Statistical Analysis methodsDam breach modelling
Convergence
Marelli - Sudret (RSUQ, ETH Zurich) UQLab: a framework for Uncertainty Quantification 24.04.2014 42 / 33