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Features-page start
Features at-a-glance
Stochastic Polynomial Chaos Expansions

Stochastic polynomial chaos expansions (SPCE) are a recently introduced surrogate model that can reproduce the non-deterministic behavior of stochastic simulators. With the SPCE module, you can easily train an accurate emulator that can handle multi-modal stochastic models, without the need of expensive model replications.

  • Adaptive sparse construction of the SPCE model

  • Mean estimation

  • Multiple integration strategies

  • Postprocessing of the results



Generalized lambda models

Generalized lambda models (GLaM) allow the user to efficiently emulate the behavior of stochastic simulators, thanks to the flexible family of Lambda distributions. With this UQLab module, GLaMs can be easily deployed as all other surrogate models. 

  • Support for both replication-based and replication-free training strategies

  • Multiple lambda parameter estimation methods

  • Direct evaluation of the (semi-)analytical conditional response PDF, CDF and inverse CDF

  • Postprocessing of the results



Random fields
Random fields

Many uncertainty quantification problems feature random variables that vary in space and time. Such variables are known as random fields. UQLab offers an intuitive way to define random fields, discretize them and sample trajectories

  • Gaussian and non-Gaussian translation random fields

  • Conditional random fields

  • Expansion optimal linear estimation (EOLE)

  • Karhunen-Loève expansion (PCA-based discrete approach and Nyström method)

Stochastic spectral embedding

Stochastic spectral embedding is a novel metamodelling technique that gradually refines a global spectral expansion into a sequence of local expansions on increasingly smaller domains.

The UQLab SSE module provides the original (and so far unique) implementation of the method, both in its static and adaptive flavours.

  • Construct PCE-based SSE with only a few lines of code

  • Complete control over the embedded sparse PCE expansions

  • Support for all experimental sampling strategies included in UQLab

  • Adaptive experimental design refinement where the model complexity demands it

  • Efficient sparse tree representation and options for expansion flattening

  • Advanced visualization options in 1D and 2D

High-performance computing (HPC) dispatcher

The high-performance computing (HPC) dispatcher module allows one to connect UQLab to common distributed computing resources (e.g., HPC clusters), providing a convenient interface to set up, submit, and retrieve remote computations directly from within UQLab running on their PC.

  • Conveniently dispatch computations from within UQLab to distributed computing resources

  • Monitor and retrieve the results of dispatched computations directly from the command line

  • Support for dispatched evaluations of all UQLab models (including third-party software
    through UQLink)

  • Support for dispatched evaluations of generic MATLAB functions (e.g., for parametric studies)

  • Support for parallel computations without MATLAB or UQLab on the remote machine

  • Out-of-the-box support for various job schedulers (including SLURM, LSF, etc.);
    support for custom schedulers via user-defined settings

HPC Dispatcher
Reliability-based design optimization

The reliability-based design optimization (RBDO) module offers a set of state-of-the-art algorithms to solve various types of optimization problems under probabilistic constraints. They include:

  • Reliability index approach (RIA)

  • Performance measure approach (PMA)

  • Single loop approach (SLA)

  • Sequential optimization and reliability assessment (SORA)

On top of these well-known algorithms, the modular design of the RBDO module allows the user to set up customized solution schemes by combining all of the reliability, surrogate modeling, and optimization techniques available in UQLab.

Reliability-based design optimization
Bayesian inference for model calibration and inverse problems

Bayesian inference is a powerful tool for probabilistic model calibration and inverse problems. UQLab offers a flexible and intuitive way to set-up and solve Bayesian inverse problems.

  • Intuitive definition of prior knowledge, forward model and data

  • State-of-the-art Markov Chain Monte Carlo (MCMC) algorithms

  • Customizable discrepancy between model and measurements

  • Support for user-specified custom likelihood

  • Support for multiple forward models and multiple discrepancy models (joint inversion)

  • Fully integrated with UQLab (e.g. surrogate models, complex priors, etc.)

Regression parameters marginals

UQLib is a collection of general-purpose open-source MATLAB libraries that are useful in the context of uncertainty quantification. These functions are currently used across the scientific modules of UQLab, but they are designed for generic use.

  • Optimization (e.g., cross-entropy optimization, covariance matrix adaptation-evolution strategy and its constrained variant)

  • Differentiation (e.g., gradient computation)

  • Kernel (stationary and non-stationary kernel functions)

  • Input/output processing (e.g., subsampling)

Rosenbrock's function minimization
Sensitivity Analysis
Sensitivity analysis

The sensitivity analysis module contains sample-based, linearization, and global sensitivity analysis methods, that quantitatively measure the importance of each input parameter.

  • Sample-based methods (input/output correlation and standard regression coefficients,
    with their rank-based versions)

  • Linearization (perturbation) method

  • Screening methods (Morris' elementary effects, Cotter indices)

  • Moment-independent global method (Borgonovo indices)

  • Sobol' indices computed by Monte Carlo simulation or analytically
    (from polynomial chaos expansions and low-rank tensor approximations)

  • Generalization of Sobol' indices for dependent input parameters
    (Kucherenko and ANCOVA indices)

Sampling- vs.  PCE-based Sobol' indices

UQLink allows the seamless connection of third-party software to UQLab using universal "wrapping" of external codes through templates and a mark-up system.

  • Based on automated text input file generation

  • Available for all platforms (Windows, Linux, macOS)

  • Examples with simple C/C++ code and commercial finite element software

Beam structure
Support vector machines

Support vector machines (SVM) come from machine learning and allow one to build predictive models from data. In the context of uncertainty quantification, SVM for regression (SVR) can be used as surrogate models of complex simulators using designs of computer experiments. SVM for classification (SVC) can be used in the context of reliability analysis.

  • L1-SVR and L2-SVR formulation for regression

  • Soft-margin classification

  • Anisotropic and user-defined kernels

  • Leave-one-out error and span approximations

  • Multiple optimization algorithms (grid search, BFGS, cross-entropy, CMA-ES, etc.)

Classification using support vector machines
Probabilistic Modelling
Advanced probabilistic modeling

In many uncertainty quantification problems, the sources of uncertainty are represented by complex probabilistic models (random vectors). UQLab offers a powerful and extendible set of tools to represent, infer, and sample from complex multivariate distributions.

  • Extensive library of marginal distributions

  • Modeling dependence with Gaussian and Vine copulas

  • Statistical Inference of marginals and copulas from data

  • Advanced sampling strategies (space-filling), including Monte-Carlo sampling, optimized latin hypercube sampling (LHS), low-discrepancy series (Sobol' and Halton sequences)

  • Sampling enrichment (nested LHS)

  • Support for custom-defined and bounded marginals

  • Isoprobabilistic transform facilities

Random vector with Gaussian copula
Modeling Tools
Easy plug-in of MATLAB-based models

Uncertainty quantification aims at predicting the impact of input parameters uncertainty onto the predictions of a computational model. UQLab offers a simple infrastructure to handle analytical models and MATLAB-based computational models (solvers).

  • Built-in support for functions defined as strings, function handles and m-files

  • Intuitive integration of more complex codes

  • Support for stochastic simulators

Complex model response
Polynomial Chaos Expansions
Polynomial chaos expansions

Polynomial Chaos Expansions (PCE) are a metamodeling tool that enables the fast construction of surrogate models, which can be efficiently used for moment- and sensitivity analysis.

  • Full and sparse polynomial chaos expansions

  • Advanced truncation strategies (hyperbolic norms, max interaction, custom basis specification)

  • Quadrature/sparse Gaussian quadrature (based on Smolyak grids)

  • Regression algorithms: Ordinary least squares (OLS), Least Angle Regression (LARS), Orthogonal Matching Pursuit (OMP), Subspace Pursuit (SP), and Bayesian Compressive Sensing (BCS)

  • Degree-adaptive and q-norm-adaptive polynomial chaos expansions

  • Polynomials orthogonal to arbitrary distributions (via Stieltjes construction)

Full vs. Sparse PCE
Full vs. PCE response
Kriging (Gaussian process modeling)

Gaussian process modeling is a flexible and robust technique to build fast surrogate models based on small experimental designs

  • Simple, ordinary, and universal Kriging 

  • Highly customizable trend and correlation functions

  • Maximum-likelihood- and cross-validation-based hyperparameter estimation

  • Gradient-based, global, and hybrid optimization methods

  • Interpolation (noise-free response) and regression (noisy response) modes

  • Fast resampling though UQLab Random Fields module

Kriging predictor: the mean and variance
Polynomial Chaos-Kriging (PC-Kriging)

Polynomial Chaos-Kriging associates the global approximation behavior of polynomial chaos expansions and the local accuracy of Kriging to provide a highly accurate surrogate model at low computational costs.

  • Support for sequential and optimal construction of PC-Kriging

  • Full control on both levels of approximation: polynomial chaos expansions and Kriging directly use the corresponding dedicated UQLab modules

  • Support for sparse, adaptive and arbitrary polynomial chaos expansions

  • Gradient-based, global and hybrid optimization methods for Kriging

1D PC-Kriging example
Low-Rank Approximations
Canonical low-rank tensor polynomial approximations

Canonical low-rank approximations (LRA) are a powerful alternative to polynomial chaos expansions that are particularly effective in high dimension.

  • Low-rank basis construction based on orthonormal polynomials

  • Adaptive identification of maximum rank and polynomial degree via cross-validation

  • Alternate least-square calculation of basis elements and coefficients

  • Polynomials orthogonal to arbitrary distributions (via Stieltjes construction)

True vs. LRA responses
Reliability Analysis
Reliability analysis (rare event estimation)

When the performance of a system is affected by uncertainties on its characteristics and/or its environment, reliability can be assessed by computing probabilities of failure.
UQLab offers state-of-the-art reliability algorithms and a powerful modular framework
for active learning reliability.

  • FORM/SORM approximation methods

  • Sampling methods (Monte Carlo, importance sampling, subset simulation)

  • Modular framework to build custom active learning solution schemes

  • Kriging-based adaptive methods (AK-MCS, APCK-MCS)

  • Adaptive and Bayesian line sampling

Subset simulation for reliability analysis
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