Uncertainty Quantification and Stochastic Modeling

Seminário sobre Quantificação de Incertezas e Modelagem Estocástica

Departamento de Engenharia Mecânica

PUC-Rio - Pontifícia Universidade Católica do Rio de Janeiro

Rio de Janeiro - RJ, Brasil

November 5-6, 2009

Professor Nicholas Zabaras

Lecture notes

These notes are work in progress and continuous updating, corrections and additions are to take place regularly. The lecture notes below have been compiled from our own research and key textbooks, journal papers and notes of related courses at several universities. These references are linked directly within the slides. Some of the links to publications will only work if your university has access to the publishers of the corresponding journals.

Stochastic Modeling:

Introduction to spectral stochastic finite element methods
Sparse grid (Smolyak) collocation methods
Stochastic Decomposition Methods (meGPCE, mePCM, HDMR, ANOVA, etc.)
Stochastic support space methods
Stochastic variational multiscale methods
Spectral stochastic projection (fractional step) methods for modeling fluid flow in porous media
Linear and non-linear model reduction strategies for generating data-driven stochastic input models
Maximum entropy methods for data-driven stochastic input models
Stochastic optimization using a support space approach
Reduced order modeling for SPDES
Summary of futuristic topics of mathematical and technological interest

Bayesian Computing and Inverse Problems:

Introduction to Probability and Statistics: Introduction to probability, random variables, mean/variance/moments, conditional probability density, marginals, correlation, independence, multivariate random variables, central Limit Theorem, Law of Large Numbers, introduction to Monte Carlo simulation, parametric and non-parametric estimation, inverse problems Maximum Likelihood Estimators (MLE), Examples.
Introduction to Bayesian computation: Bayes’ rule, prior, likelihood and posterior distributions, introduction to prior modeling, informative & non-informative priors, MaxEnt priors, smoothness priors, posterior calculation examples, Maximum a posteriori estimator (MAP), Hierarchical Bayesian models, Bayesian regression example.
Introduction to Stochastic Simulation: Markov Chains, Introduction to Metropolis Hastings algorithm, application to the Ising PDF.
Markov Chain Monte Carlo: Markov Chain, MCMC sampling algorithms.
Prior Models: Prior models for inverse problems, Markov Random Fields, regularization of inverse problems.
Computing Gaussian conditional densities and applications.
Posterior exploration: Gibbs sampling, etc.
Other topics: Hypermodels, prior-conditioning, Bayesian learning.
Sequential MC
Post-processing : of statistical exploration results.
Surrogate likelihood models: Using sparse grid collocation to create a surrogate stochastic likelihood model.

Top of UQ page

References: The course lectures are based on a number of key reference books and research papers that are linked directly in each lecture. These references should be consulted for more details on the topics discussed here. Top of UQ page

Course description

This short course on uncertainty quantification and stochastic modeling was offered by Prof. N. Zabaras at PUC-Rio and was organized by Professors Rubens Sampaio (PUC-Rio) and Fernando A. Rochinha (COPPE/UFRJ). It became possible with financial and organizational support from CAPES, CNPq, and FAPERJ. This follows a series of successful offerings of this course at other academic, federal and industrial institutions most recently at the University of Illinois at Urbana-Champaign (Computational Sciences and Engineering program) on June 11-12, 2007 (see this program for more details).

The course addresses uncertainty quantification and predictive modeling of engineering systems. Achieving predictive simulations of physical systems generally requires a concerted effort in verification and validation. In particular, assessment of model/code validity requires targeted comparisons against experimental measurements, with well characterized uncertainty/error bars in both experimental and computational results. This short course and workshop reviews recent developments in uncertainty quantification (UQ) in computational science, focusing on the utilization of generalized polynomial chaos expansions (GPCE) and collocation techniques for representation of random variables and processes, and the various means of forward propagation of uncertainty in systems governed by partial and ordinary differential equations (e.g. applications in chemistry, thermofluids, materials, etc.). We will review Galerkin modeling in stochastic spaces, computational solution aspects, error estimation, and post-processing techniques, and cover both non-intrusive (sampling-based) and intrusive (direct) UQ methods. We will also discuss the utilization of Bayesian methods for estimation of uncertain parameters from data. Parameter estimation is a crucial element of any overall predictive simulation strategy, as the determination of well characterized uncertainties in model input parameters is a key step towards reliable UQ for model predictions. A number of current research topics on uncertainty quantification will finally be discussed, including interfacing multiscale and stochastic modeling, GPCE and Bayesian based stochastic optimization problems for systems governed by stochastic partial differential equations (SPDEs), UQ in oscillatory dynamical systems and flow fields, and others. Top of UQ page