advancements in the uqlab framework for uncertainty...

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

investigations

Introduction

investigations

Uncertainty quantification aims at making the best use ofcomputer models by dealing rigorously with variability, lack of

knowledge, measurement- and model errors

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

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

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

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

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

PhysicalModel

UncertaintyAnalysis

Response PDF

PhysicalModel

UncertaintyAnalysis

Response PDF

PhysicalModel

UncertaintyAnalysis

Response PDF

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)

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)

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

Clayton copula sampling

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’

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’

The UQLab project What is UQLab

Outline

1 Introduction

2 The UQLab projectWhat is UQLabCurrent statusSome Statistics

3 Case studies

4 Outlook

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

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)

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

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

The UQLab project Some Statistics

Some Statistics

Since release of the public beta (July 1st, 2015):• 340+ users from 42 countries worldwide...• ...and counting!

Some StatisticsSince release of the public beta (July 1st, 2015):

• 340+ users from 42 countries worldwide...• ...and counting!

Some StatisticsSince release of the public beta (July 1st, 2015):

• 340+ users from 42 countries worldwide...• ...and counting!

Case studies

Outline

1 Introduction

2 The UQLab project

3 Case studiesStochastic PDEKriging-based rare event estimationSubsurface contaminant diffusion

4 Outlook

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

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

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)

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)

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 );

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 );

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

Example 1: Results

Remarks:

Example 1: Results

Remarks:

Example 1: Results

Remarks:

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)

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)

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!

{X1, X2} Normal 0 1

PMCSf = 3.85× 10−4

PAK-MCSf = (3.76± 0.38)× 10−4

{X1, X2} Normal 0 1

PMCSf = 3.85× 10−4

PAK-MCSf = (3.76± 0.38)× 10−4

{X1, X2} Normal 0 1

PMCSf = 3.85× 10−4

PAK-MCSf = (3.76± 0.38)× 10−4

{X1, X2} Normal 0 1

PMCSf = 3.85× 10−4

PAK-MCSf = (3.76± 0.38)× 10−4

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)

Probabilistic model of porosity / conductivity

0 0.05 0.1 0.15 0.2

L1bL2aL2bL2c

• In each layer, bounds on porosity:

φi ∼ U [φimin , φimax]

• Deterministic mapping to the conductivity

Probabilistic model of porosity / conductivity

10−12

10−10

10−8

10−6

10−4

10−2

L1bL2aL2bL2c

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)

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)

Other parameters

Parameter Notation RangePorosity φi, i = 1, . . . , 15 [φimin , φ

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

PCE-based Sobol’ sensitivity indices

φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4

KAC3ab

Total Sobol’ IndicesSTot

φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4K

AC3aba

Sobol’ Indices Order 1

Parameter∑

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

PCE-based Sobol’ sensitivity indices

φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4

KAC3ab

Total Sobol’ IndicesSTot

φD4 φC3ab φL1b φL1a φC1 ∇H2 φL2a φD1 AD4K

AC3aba

Sobol’ Indices Order 1

Parameter∑

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

Outlook

Outline

1 Introduction

2 The UQLab project

3 Case studies

4 OutlookTentative release schedule

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!

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!!

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!!

Thank you very much for yourattention!

UQLabThe Framework for Uncertainty Quantification

www.uqlab.com

Chair of Risk, Safety & Uncertainty Quantificationhttp://www.rsuq.ethz.ch

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