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

Reliabilitybased design optimization

Bayesian inversion

UQLink

Support vector machines

Probabilistic input

Basic modeling

Polynomial chaos expansions

Kriging

Polynomial chaosKriging

Lowrank approximations

Sensitivity analysis

Reliability analysis
NEW!
Reliabilitybased design optimization
Reliabilitybased design optimization (RBDO) is a powerful tool for the design of structures under uncertainty. UQLab offers an intuitive way to setup and solve RBDO problems using either stateoftheart 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 stateoftheart RBDO approaches.
Learn how to set up your own RBDO solution scheme by combining reliability and optimization techniques available in UQLab.
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 stateofthe 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.
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 predatorprey model of lynx and hare populations with multiple model discrepancy options.
Learn how to set up and calibrate simultaneously multiple forward models of a simple beam.
Set up and use a custom userdefined 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 thirdparty software through a builtin universal "code wrapper".
After the wrapper has been configured through input/output text files and adhoc 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.
Link UQLab to a C code implementing
a simply supported beam then carry out a reliability analysis using AKMCS.
Link UQLab to an Abaqus model of
a tenbar truss then carry out
sensitivity and reliability analyses.
Link UQLab to OpenSees for
a pushover analysis of a twostory
onebay 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 blackbox or expensivetoevaluate 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, userdefined). Different techniques including the span leaveoneout and the crossvalidation error estimation methods as well as various optimization algorithms are available to estimate the hyperparameters.
Learn how to select different kernel families through an introductory SVC example based on the Fisher’s iris data set.
Learn how to use the span estimate of the leaveoneout error or Kfold crossvalidation as well as various optimization strategies.
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 L2penalization schemes, separable and elliptic kernels based on classical SVM kernels (linear, polynomial, sigmoid) and others (Gaussian, exponential, Matérn, userdefined). Different techniques, including the span leaveoneout and the crossvalidation error estimation methods, as well as various optimization algorithms, are available to estimate the hyperparameters.
Learn how to select different kernel families through an introductory SVR example.
Learn how to use the span estimate of the leaveoneout error or Kfold crossvalidation as well as various optimization strategies.
Apply SVR to a computational model with multiple outputs.
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!
Learn how to specify a random vector and draw samples using various sampling strategies.
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 mcode 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.
Learn different ways to define a computational model.
Learn how to pass parameters to a computational model.
Learn how to handle models that return vector outputs instead of scalars.
Polynomial chaos expansions
Modern computational models are often expensivetoevaluate. 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 50500 runs.
With this limited number of runs, a polynomial surrogate may be built, which mimics the behavior of the true model at a closetozero computational cost. Polynomial chaos expansions are a particularly powerful technique which makes use of socalled orthogonal polynomials with respect to the input probability distributions.
Learn how to deploy various strategies to compute PCE coefficients.
Try out various methods to generate an experimental design (DOE) and compute PCE coefficients with leastsquare minimization.
Apply sparse PCE for the estimation of the midspan deflection of a simply supported beam subject to a uniform random load.
Apply sparse PCE to surrogate a model with multiple outputs.
Build a sparse PCE surrogate model from existing data.
Build a sparse PCE using polynomials orthogonal to arbitrary distributions
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 onedimensional families (Gaussian, exponential, Matérn, or userdefined).
The hyperparameters can be estimated either using the MaximumLikelihood or the CrossValidation method using various optimization techniques (local and global). UQLab supports both interpolation and regression (Kriging with noisy model response) mode.
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Learn how to select different correlation families through an introductory Kriging example.
Learn how to use maximum likelihood or leaveoneout crossvalidation as well as various optimization strategies.
Learn how to specify various trend types.
Create a Kriging surrogate for a function with multiple input dimensions (the BraninHoo function).
Create a Kriging surrogate from existing data (the truss structure data set).
Create a Kriging surrogate from existing data (the Boston housing data set).
Apply Kriging to a computational model with multiple outputs.
Learn how to create a Gaussian process regression (GPR) model for model with noisy response.
Apply GPR to create a regression model for the Boston housing data set.
Polynomial ChaosKriging
Polynomial ChaosKriging (PCKriging) combines features from polynomial chaos expansions and Kriging in a single efficient surrogate.
A PCKriging 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, PCKriging can effectively reproduce the global approximation behavior typical of PCE, while at the same time retaining the interpolatory characteristics of Kriging.
The PCKriging implementation in UQLab leverages on the PCE and Kriging modules, to provide a fullyconfigurable tool that is seamlessly integrated with the whole metamodeling tools offered by the platform.
Learn how to create a PCKriging surrogate of a simple onedimensional function.
Learn how to create a PCKriging metamodel of a multidimensional function (the Ishigami function).
Apply PCKriging to a computational model with multiple outputs.
Create a PCKriging metamodel from existing data (the truss data set).
Lowrank approximations
Canonical lowrank 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 highdimensional model inputs. The key idea is to approximate a response quantity of interest with a sum of a small number of appropriate rankone 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 lowrank representations, hence offering similar flexibility in terms of supported polynomial types.
Create a canonical lowrank tensor approximation of a simple engineering model.
Create a canonical lowrank tensor approximation of a highly nonlinear function.
Create a canonical lowrank tensor approximation of a complex function.
Learn how to create a canonical lowrank tensor approximation of a model with multiple outputs.
Learn how to create a canonical lowrank 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 samplebased methods (e.g., input/output correlation), linearization methods (e.g., perturbation analysis), screening methods (e.g., Morris' elementary effects), momentindependent 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 samplingbased algorithms as well as some metamodelspecific techniques to extract the most accurate information based on relatively small sample size (e.g., polynomial chaos expansion and lowrankapproximationbased Sobol' indices).
Apply various sensitivity analysis techniques to a benchmark problem (borehole function).
Learn how to obtain the Sobol' indices using either the samplingbased or the PCE/LRAbased methods.
Discover the amazing efficiency of
PCEbased Sobol' analysis on
a 100dimensional example.
Apply sensitivity analysis techniques to model with multiple outputs.
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 MonteCarlo Sampling (MCS), Importance Sampling, and Subset Simulation) as well as the more recent surrogate modelingbased methods (e.g., Adaptive Kriging MonteCarlo simulation).
All the available algorithms take advantage of the high interoperability with the other modules of UQLab, hence making their deployment timeefficient.
Learn how to apply all of the available reliability analysis algorithms on a classic linear example: the RS case.
See how different simulationbased methods perform on a highly nonlinear limit state function.
Apply several different algorithms in a highdimensional, strongly nonlinear test case.
Perform reliability analysis on a parallel system with the FORM method.
Compute the outcrossing rate in a nonstationary timevariant reliability problem using the PHI2 method.