Uncertainty Quantification

Lecturer Dr. E. Zander


Assistant Dr. N. Friedman


Schedule

Lectures: Thu 11:30-13:00 o'clock in room 812, Mühlenpfordtstr. 23 (Seminarraum WiRe)

LAST LECTURES: 12.07.2016 Tuesday 18:00-19:30 (Seminarraum WiRe), 14.07.2016 Thursday 11:30-13:00

Exercises: Thu 14:00-15:30 o'clock in room 826, Mühlenpfordtstr. 23 (Computerpool WiRe)

LAST EXERCISE: 13.07.2016. Wednesday 13:15-14:45 (Computerpool WiRe)

14.04.2016

The content of the lecture is available here: http://www.biblio.tu-bs.de/semapp/ > Prof. Hermann G. Matthies: "Quantifizierung von Unsicherheiten...". (Password will be given during the lecture)

• Lecture 1: Introduction, basics of probability theory
• Lecture 2: Basics of probability theory
• Lecture 3:Lebesgue integration, moments, MC integration, weak low of large numbers
• Lecture 4: Different type of equivalencies of RVs, and convergences, Latin Hypercubic Sampling and the QMC and their convergence
• Lecture 5: QMC, direct integration : Gauß quadrature
• Lecture 6: Orthogonal polynomials, direct integration in multiple dimension - the full tensor grid and the sparse grid integration rules
• Lecture 7: Spectral methods - preliminaries (separability, function approximationdenseness), multivariate gPCEs, multiindices, statistics from gPCE
• Lecture 8: Sparse integration rules, response surface methods: orthogonal projection
• Lecture 9: Response surface methods: projection theory, interpolation, regression, stochastic Galerkin method
• Lecture 10-11: Low-rank representation of random fields (processes) - KLE, Fourier...
• Lecture 12: Parameter estimation - introduction, the Bayes theorem, Markov Chains, Markov property
• Lecture 13: Parameter estimation (MCMC)
• Lecture 14: Parameter estimation (MMSE)

There is no script yet, but an ongoing effort to create one alongside with the lecture. You can see it here, but be aware that it's far from finished. If you spot any errors or inaccuracies or you have any suggestions, please send them to the lecturer of this course.

You can download the software for this course by issuing the following command on the command line:

git clone git://github.com/ezander/sglib-testing

This will create a directory sglib-testing for you, which will contain all the necessary files. Please start matlab from this directory, so that sglib can do its initialisation.

Tutorial 0 - Introduction to SGLIB and Probability Theory

Tutorial 1 - PDFs, CDFs, sampling from different distributions, transformation of Random Variables

• Example1 (Approximation of CDF and PDF from sampling)
• Example2 (CDFs, PDFs, moments and sampling with the GENDIST functions)
• Example3 (Transformation of Random Variables)

Tutorial 2,3 - Monte Carlo and Quasi Monte Carlo Integrations

• Example1 (Sampling with SimParameter and SimParamset)
• Example2 (Convergence of the Monte Carlo, and the Quasi Monte Carlo)

Tutorial 4,5 - Direct integration method, abstract framework of UQ and its implementation

• Generating integration points I. (version in a more educative form)
• Generating integration points II. (version which is more practical)
• Compare sparse and full tensor grids
• Direct integration on a toy example
• Some small presentation of the idea of sparse grid

Tutorial 6,7 - Nonintrusive response surface methods

The followings are examples for general Polynomial Chaos Expansion of the oscillating mass on a damped spring

• Plot true response surface (reference surface)
• gPCE of response by Pseudo-Galerkin projection (or orthogonal projection)
• gPCE by interpolation (collocation)
• gPCE by regression (collocation)

Tutorial 8,9,10 - The stochastic Galerkin method, stochastic fields, basics before parameter estimation (Sigma Algebra, measrue space, measureable function, etc)

• Toy example for stochastic Galerkin method
• KLE

Tutorial 11: Parameter estimation (the inverse method)

Assignment 1: Convergence of the MC method (due date: 12.05.2016.)

Assignment 2: Extend assignment 1 with QMC and make a 5 minute presentation of your work (due date, and presentations: 26.05.2016.)

Assignment 3: Calculate the expected value of the position (x) and the velocity (v) of the mass in the oscillating mass example (solved by the function sglib-testing/demo/models/spring/spring_solve) supposing that k~U(0.5, 2.5), m~lnN(0.3, 0.35) are random variables and x0=1, v0=0, T=10, d=0 are deterministic input parameters. Define the mean values by using direct integration method using quadrature rule. You can get lot of help from the examples under Tutorial 4,5. Due date: 09.06.2016.

Whether there will be a test or oral exams will be decided in the course of this semester.