Uncertainty Quantification in Computational Sciences

Cornell University, 2007

Professor Nicholas Zabaras

Lecture notes

Introduction to spectral stochastic finite element methods
Sparse grid (Smolyak) collocation methods
Stochastic support space methods
Multielement generalized polynomial chaos methods megPC
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
Bayesian computation for modeling (stochastic) inverse problems
Reduced order modeling for SPDES
Summary of futuristic topics of mathematical and technological interest

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

These notes were used for a short course on UQ offered at UIUC (Computational Sciences and Engineering program) on June 11-12, 2007 together with Dr. Habib N. Najm, Sandia National Laboratories. The extended abstract for this course is given here: 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 workshop will describe recent developments in uncertainty quantification (UQ) in computational science, focusing on the utilization of generalized polynomial chaos expansions (GPCE) for representation of random variables and processes, and the various means of forward propagation of uncertainty in the GPCE context 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