Learning UQLab by example

 

We made a significant effort to provide UQLab users with a number of examples that can be used to gradually learn all the features of the software.

 

Each example is especially prepared to provide users with enough experience and background to develop their own applications in no time. The examples are divided into different categories, as explained below.

Contents

  • Reliability-based design optimization

  • Bayesian inversion

  • UQLink

  • Support vector machines

  • Probabilistic input

  • Basic modeling

  • Polynomial chaos expansions

  • Kriging

  • Polynomial chaos-Kriging

  • Low-rank approximations

  • Sensitivity analysis

  • Reliability analysis

 

 NEW! 

Reliability-based design optimization

Reliability-based design optimization (RBDO) is a powerful tool for the design of structures under uncertainty. UQLab offers an intuitive way to set-up and solve RBDO problems using either state-of-the-art algorithms or custom solution schemes, which combine various reliability, optimization and surrogate modeling techniques.

Learn how to set up and solve a simple problem using state-of-the-art RBDO approaches.

Learn how to set up your own RBDO solution scheme by combining reliability and optimization techniques available in UQLab.

Fig_Example_RBDO_01.jpg
Fig_Example_RBDO_03.jpg

Learn how to use surrogate models to alleviate the computational burden of RBDO.

 
Bayesian inversion

Bayesian inversion is a powerful tool for probabilistic model calibration and validation. UQLab offers users an intuitive way to set up Bayesian inverse problems with customizable likelihood functions and discrepancy model options; and solve the problems with state-of-the art Markov Chain
Monte Carlo (MCMC) algorithms.

Learn how to set up and solve a Bayesian calibration problem using a simple model of beam deflection.

Posterior marginals of unknown residual variance
Posterior marginals of known residual variance

Learn how to solve the classical Bayesian linear regression problem, with known and unknown residual variance.

Learn how to calibrate a conceptual watershed model (HYMOD) using historical data.

Learn how to calibrate a predator-prey model of lynx and hare populations with multiple model discrepancy options.

Calibration of two deformation models: Mid-span deflection and elongation

Learn how to set up and calibrate simultaneously multiple forward models of a simple beam.

Calibration with a user-defined likelihood function

Set up and use a custom user-defined likelihood function to model dependence between data points.

Compute the MAP estimator of the model parameters of a linear regression model.

Conduct a Bayesian inversion analysis with a surrogate model to reduce the overall computational time.

 
UQLink

UQLab allows one to easily define computational models involving a third-party software through a built-in universal "code wrapper".

After the wrapper has been configured through input/output text files and ad-hoc text marking and parsing, the corresponding computational model can be seamlessly integrated with any techniques available in other UQLab modules to create complex analyses in no time.

Connect UQLab with a simply supported beam model in C code

Link UQLab to a C code implementing
a simply supported beam then carry out a reliability analysis using AK-MCS.

Connect UQLab with a ten-bar truss model in Abaqus

Link UQLab to an Abaqus model of
a ten-bar truss then carry out
sensitivity and reliability analyses.

Connect UQLab with a two-story structure model in OpenSees

Link UQLab to OpenSees for
a pushover analysis of a two-story
one-bay structure.

 
Support vector machines
Support vector machines (SVM) belong to a class of machine learning techniques developed since the mid-'90s. In the context of uncertainty quantification, support vector machines for classification (SVC) can be used to build a classifier given an experimental design. They can be used for reliability analysis (a.k.a. rare events estimation). Support vector machines for regression (SVR) can be used as a metamodeling tool to approximate a black-box or expensive-to-evaluate computational model.
 
Classification (SVC)
UQLab offers a straightforward parametrization of a SVC model to be fitted on the data set at hand (e.g. a design of computer experiments): linear and quadratic penalization schemes, separable and elliptic kernels based on classical SVM kernels (linear, polynomial, sigmoid) and others (Gaussian, exponential, Matérn, user-defined). Different techniques including the span leave-one-out and the cross-validation error estimation methods as well as various optimization algorithms are available to estimate the hyperparameters.
Classification of Fisher's iris data set

Learn how to select different kernel families through an introductory SVC example based on the Fisher’s iris data set.

Classification of simple function

Learn how to use the span estimate of the leave-one-out error or K-fold cross-validation as well as various optimization strategies.

Trade-off between false negatives and overall accuracy of an SVC model

Create an SVC model from existing data (a breast cancer data set).

Regression (SVR)

UQLab offers a straightforward parametrization of an SVR model to be fitted on the data set at hand (e.g., an experimental design):
L1- and L2-penalization schemes, separable and elliptic kernels based on classical SVM kernels (linear, polynomial, sigmoid) and others (Gaussian, exponential, Matérn, user-defined). Different techniques, including the span leave-one-out and the cross-validation error estimation methods, as well as various optimization algorithms, are available to estimate the hyperparameters.

 
SVR prediction of the X.SinX function

Learn how to select different kernel families through an introductory SVR example.

SVR prediction of the X.SinX function using different estimation methods

Learn how to use the span estimate of the leave-one-out error or K-fold cross-validation as well as various optimization strategies.

SVR prediction of a model with multiple outputs

Apply SVR to a computational model with multiple outputs.

Validation error of a SVR model of an existing data

Create an SVR model from existing data (the Boston housing data set).

 
Probabilistic Input

Whether the uncertainty sources are simple independent uniform variables or complex combinations of fancy marginals glued together by a copula, UQLab provides simple methods to define, sample or transform your input distributions. If the theoretical marginals and/or copulas are unknown and only data are available, uqlab provides the possibility to infer the unknown distributions from the data in a simple way.

 UPDATED! 

Samples generated by different sampling strategies

Learn how to specify a random vector and draw samples using various sampling strategies.

Marginal and joint distributions of uncorrelated and correlated samples

Learn how to specify marginal distributions for the elements of a random vector and a Gaussian copula.

Learn how to specify marginal distributions for the elements of a random vector and a Pair copula.

Learn how to specify marginal distributions for the elements of a random vector and a (C- or D-) vine copula.

Learn how to group uncertain inputs into mutually independent subsets (blocks) via multiple copulas.

Learn how to infer the marginal distributions of an input from data.

Learn how to infer the copula of an input from data, and how to perform independence tests.

 

Basic modeling
 

UQLab allows one to simply define new computational models, either based on existing m-code files, function handles, or simple text strings.

After a computational model has been configured, it can be seamlessly integrated with any other technique to create complex analyses in no time.

Histogram of the Ishigami function output

Learn different ways to define a computational model.

Histogram of the Ishigami function output for different parameter values

Learn how to pass parameters to a computational model.

Model with multiple outputs

Learn how to handle models that return vector outputs instead of scalars.

 

Polynomial chaos expansions

Modern computational models are often expensive-to-evaluate. In the context of uncertainty quantification, where repeated runs are required,
only a limited budget of simulations is usually affordable. We consider a “reasonable budget” to be in the order of 50-500 runs.

 

With this limited number of runs, a polynomial surrogate may be built, which mimics the behavior of the true model at a close-to-zero computational cost. Polynomial chaos expansions are a particularly powerful technique which makes use of so-called orthogonal polynomials with respect to the input probability distributions.

Non-zero coefficients of PCE with different calculation strategies

Learn how to deploy various strategies to compute PCE coefficients.

Validation error of PCE models with different experimental designs

Try out various methods to generate an experimental design (DOE) and compute PCE coefficients with least-square minimization.

Convergence of statistical moments estimation with PCE metamodel

Apply sparse PCE for the estimation of the mid-span deflection of a simply supported beam subject to a uniform random load.

PCE prediction of a model with multiple outputs

Apply sparse PCE to surrogate a model with multiple outputs.

Validation error of a PCE model of an existing data

Build a sparse PCE surrogate model from existing data.

Non-zero coefficients of PCE with an arbitrary distribution

Build a sparse PCE using polynomials orthogonal to arbitrary distributions

bPCE.png

Apply the bootstrap method to estimate the local errors of PCE metamodel predictions.

 
Kriging

Kriging (Gaussian process modeling) considers the computational model as a realization of a Gaussian process indexed by the parameters
in the input space.

 

UQLab offers a straightforward parametrization of the Gaussian process to be fitted to the experimental design points: constant, linear, polynomial, or arbitrary trends, with separable and elliptic kernels based on different one-dimensional families (Gaussian, exponential, Matérn, or user-defined).

The hyperparameters can be estimated either using the Maximum-Likelihood or the Cross-Validation method using various optimization techniques (local and global). UQLab supports both interpolation and regression (Kriging with noisy model response) mode.

 UPDATED! 

Kriging prediction of the X.SinX function

Learn how to select different correlation families through an introductory Kriging example.

Kriging prediction of the X.sinX function using CV estimation

Learn how to use maximum likelihood or leave-one-out cross-validation as well as various optimization strategies.

Kriging prediction of the X.SinX function using different trend function

Learn how to specify various trend types.

Contour plot of the Branin-Hoo function

Create a  Kriging surrogate for a function with multiple input dimensions (the Branin-Hoo function).

Validation error of a Kriging model of an existing data (truss)

Create a Kriging surrogate from existing data (the truss structure data set).

Validation error of a Kriging model of an existing data (Boston housing)

Create a Kriging surrogate from existing data (the Boston housing data set).

Kriging prediction of a model with multiple outputs

Apply Kriging to a computational model with multiple outputs.

KrigingRegression2.png
KrigingRegression1.png

Learn how to create a Gaussian process regression (GPR) model for model with noisy response.

KrigingRegressionHousing.png

Apply GPR to create a regression model for the Boston housing data set.

 
Polynomial Chaos-Kriging

Polynomial Chaos-Kriging (PC-Kriging) combines features from polynomial chaos expansions and Kriging in a single efficient surrogate.

A PC-Kriging model is effectively a universal Kriging model with a sophisticated trend based on orthogonal polynomials. By adopting advanced sparse polynomial chaos expansion techniques based on compressive sensing, PC-Kriging can effectively reproduce the global approximation behavior typical of PCE, while at the same time retaining the interpolatory characteristics of Kriging.

The PC-Kriging implementation in UQLab leverages on the PCE and Kriging modules, to provide a fully-configurable tool that is seamlessly integrated with the whole metamodeling tools offered by the platform.

PC-Kriging prediction of the X.SinX function

Learn how to create a PC-Kriging surrogate of a simple one-dimensional function.

Histogram of the Ishigami function with PC-Kriging

Learn how to create a PC-Kriging metamodel of a multi-dimensional function (the Ishigami function).

PC-Kriging prediction of a model with multiple outputs

Apply PC-Kriging to a computational model with multiple outputs.

Validation of an PC-Kriging model of an existing data

Create a PC-Kriging metamodel from existing data (the truss data set).

 
Low-rank approximations

Canonical low-rank polynomial approximations (LRA), also known as separated representations, have recently been introduced in the field of uncertainty quantification as a promising tool for effectively dealing with high-dimensional model inputs. The key idea is to approximate a response quantity of interest with a sum of a small number of appropriate rank-one tensors, which are products of univariate polynomial functions.

An important property of LRA is that the number of unknowns increases only linearly with the input dimension, hence making them a powerful tool to tackle high dimensional problems.

The LRA module in UQLab capitalizes on the PCE module to create low-rank representations, hence offering similar flexibility in terms of supported polynomial types.

Validation error of an LRA metamodel of the simply supported beam model

Create a canonical low-rank tensor approximation of a simple engineering model.

Validation error of an LRA metamodel of the Ishigami function

Create a canonical low-rank tensor approximation of a highly non-linear function.

Validation error of an LRA metamodel of the Sobol' function

Create a canonical low-rank tensor approximation of a complex function.

LRA prediction of a model with multiple outputs

Learn how to create a canonical low-rank tensor approximation of a model with multiple outputs.

Validation error of an LRA model for existing data

Learn how to create a canonical low-rank tensor approximation from existing data (the truss data set).

 
Sensitivity analysis

Sensitivity analysis is a powerful tool to identify which variables contribute the most to the variability of the response of a computational model.

 

UQLab offers a wide selection of sensitivity analysis tools, ranging from sample-based methods (e.g., input/output correlation), linearization methods (e.g., perturbation analysis), screening methods (e.g., Morris' elementary effects), moment-independent method (e.g., Borgonovo indices), to the more advanced ANOVA- (ANalysis Of Variance) measures for independent (e.g., Sobol' indices) and dependent inputs parameters (e.g., ANCOVA indices). 

The algorithms take advantage of the latest available sampling-based algorithms as well as some metamodel-specific techniques to extract the most accurate information based on relatively small sample size (e.g., polynomial chaos expansion- and low-rank-approximation-based Sobol' indices).

Sensitivity analysis of the Borehole function using different methods

Apply various sensitivity analysis techniques to a benchmark problem (borehole function).

Estimation of Sobol' sensitivity indices with different methods

Learn how to obtain the Sobol' indices using either the sampling-based or the PCE/LRA-based methods.

Total and second-order Sobol' sensitivity indices of a 100-dimensional model

Discover the amazing efficiency of
PCE-based Sobol' analysis on
a 100-dimensional example.

Total Sobol' sensitivity indices of a model with multiple outputs

Apply sensitivity analysis techniques to model with multiple outputs.

Sensitivity analysis of the short-column function with dependent inputs

Apply sensitivity analysis techniques to model with dependent inputs.

 
Reliability analysis

The reliability analysis (rare events estimation) module provides a set of tools to determine the probability that some criterion on the model response is satisfied, e.g., the probability of exceedance of some prescribed admissible threshold. 

 

The reliability analysis module in UQLab comprises a number of classical techniques (e.g., FORM and SORM approximation methods, classical Monte-Carlo Sampling (MCS), Importance Sampling, and Subset Simulation) as well as the more recent surrogate modeling-based methods (e.g., Adaptive Kriging Monte-Carlo simulation).

 

All the available algorithms take advantage of the high interoperability with the other modules of UQLab, hence making their deployment time-efficient.

IS results for the R-S problem

Learn how to apply all of the available reliability analysis algorithms on a classic linear example: the R-S case.

Subset simulation for the 2D hat function

See how different simulation-based methods perform on a highly non-linear limit state function.

Reliability of a damped oscillator model

Apply several different algorithms in a high-dimensional, strongly non-linear test case.

Reliability of a parallel system

Perform reliability analysis on a parallel system with the FORM method.

Computation of the outcrossing rate in time-variant reliability analysis

Compute the outcrossing rate in a non-stationary time-variant reliability problem using the PHI2 method.