uqlab: a framework for uncertainty quantification …...chair of risk, safety & uncertainty...

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

Summary and Outlook

Computer SimulationsA global frameworkThe UQLab project

Outline

1 IntroductionComputer SimulationsA global frameworkThe UQLab project

2 UQLab in action

Summary and Outlook

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

Summary and Outlook

Introduction

Summary and Outlook

Introduction

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

knowledge, measurement- and model errors

Summary and Outlook

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

Summary and Outlook

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

Summary and Outlook

Outline

2 UQLab in action

Summary and Outlook

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

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

Summary and Outlook

PhysicalModel

Assessment criteria

Quantification of

Analysis

Response PDF

Summary and Outlook

PhysicalModel

Assessment criteria

Quantification of

Analysis

Response PDF

Summary and Outlook

PhysicalModel

Assessment criteria

Quantification of

Analysis

Response PDF

Summary and Outlook

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)

Summary and Outlook

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)

Summary and Outlook

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

Clayton copula sampling

Summary and Outlook

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

Summary and Outlook

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

Summary and Outlook

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

Summary and Outlook

Outline

2 UQLab in action

Summary and Outlook

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

Summary and Outlook

The UQLab Software Framework

UQLab: Uncertainty Quantification Lab

Focus on:GeneralityEase of useNon-intrusivenessHPCExtendibilityCollaboration

Summary and Outlook

The UQLab Software Framework

UQLab: Uncertainty Quantification Lab

Focus on:GeneralityEase of useNon-intrusivenessHPCExtendibilityCollaboration

Summary and Outlook

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.

Summary and Outlook

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.

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

Summary and Outlook

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)

Summary and Outlook

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)

Summary and Outlook

Outline

1 Introduction

Summary and Outlook

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

Summary and Outlook

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

Summary and Outlook

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)

Summary and Outlook

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)

Summary and Outlook

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

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

Summary and Outlook

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

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

Summary and Outlook

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

Summary and Outlook

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∑

√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)

Summary and Outlook

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∑

√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)

Summary and Outlook

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 ;

Summary and Outlook

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 ;

Summary and Outlook

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!!

Summary and Outlook

Example 1: Results

Summary and Outlook

Example 1: Results

Summary and Outlook

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

Summary and Outlook

Outline

1 Introduction

Summary and Outlook

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.

Summary and Outlook

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.

Summary and Outlook

Sobol’ Indices and PCE

Re-grouping PCE coefficients

MPC (X) =∑α∈A

yαΨα (X) =

y0 +M∑

∑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

∑α∈Ii1,...,is

y2α =

∑α∈Ii1,...,is

/ ∑1≤|α|≤P

Summary and Outlook

Sobol’ Indices and PCE

Re-grouping PCE coefficients

MPC (X) =∑α∈A

yαΨα (X) =

y0 +M∑

∑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

∑α∈Ii1,...,is

y2α =

∑α∈Ii1,...,is

/ ∑1≤|α|≤P

Summary and Outlook

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

M∑k=1

x2k + x4

+ x1x22 − x5x3 + x2x4 + x50 + x50x2

Summary and Outlook

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

M∑k=1

x2k + x4

+ x1x22 − x5x3 + x2x4 + x50 + x50x2

Summary and Outlook

Analytical function

y =3 −5M

M∑k=1

kxk +1M

M∑k=1

kx3k + ln

M∑k=1

x2k + x4

+ x1x22 − x5x3 + x2x4 + x50 + x50x2

Summary and Outlook

Analytical function

y =3 −5M

M∑k=1

kxk +1M

M∑k=1

kx3k + ln

M∑k=1

x2k + x4

+ x1x22 − x5x3 + x2x4 + x50 + x50x2

Summary and Outlook

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 ;

Summary and Outlook

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 ;

Summary and Outlook

Example 2: Sensitivity results

Total Sobol’ indices

PCE-based Sobol’ indices(1200 model runs)

MCS-based Sobol’ indices(1,76 Mio model runs)

Summary and Outlook

Example 2: Sensitivity results

Second order Sobol’ indices

PCE-based second order Sobol’ indices(1200 model runs)

Summary and Outlook

Outline

1 Introduction

Summary and Outlook

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

Summary and Outlook

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

Summary and Outlook

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

Summary and Outlook

Truss structure:

Pf (V1 > 0.13m) = 1.18 × 10−4

uq_runAnalysis ;

Pf (V1 > 0.13m) = 1.17 × 10−4

Summary and Outlook

Truss structure:

Pf (V1 > 0.13m) = 1.18 × 10−4

uq_runAnalysis ;

Pf (V1 > 0.13m) = 1.17 × 10−4

Summary and Outlook

Truss structure:

Pf (V1 > 0.13m) = 1.18 × 10−4

uq_runAnalysis ;

Pf (V1 > 0.13m) = 1.17 × 10−4

Summary and Outlook

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 ;

Summary and Outlook

Sensitivity analysis:

HPC Scalability:

code:HPCopts .nCPU = 4;HPCopts . Profile ='credentials .txt ';uq_create_dispatcher ( HPCopts );

Summary and Outlook

Sensitivity analysis:

HPC Scalability:

code:HPCopts .nCPU = 4;HPCopts . Profile ='credentials .txt ';uq_create_dispatcher ( HPCopts );

Summary and Outlook

Outline

1 Introduction

2 UQLab 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

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

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

Summary and Outlook

Thank you for your attention!!

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.

Statistical Analysis methodsDam breach modelling

Appendix

Backup Slides

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

Model calibration

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

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]

Sensitivity analysis

Accuracy

Convergence

uqlab: a framework for uncertainty quantification …...chair of risk, safety & uncertainty...

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