Uncertainty Quantification

Uncertainty Quantification

Uncertainty Quantification, Parametric Problems, and Model Reduction

General Information

Lecturer Prof. Hermann G. Matthies


AssistantDr. Truong-Vinh Hoang


Schedule

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

Exercises: Thu 13:15-14:15 o'clock in room 826, Mühlenpfordtstr. 23 (Computerpool WiRe)

Thursday, 11.04.2018

Syllabus

Lectures

  • Lecture 1: Introduction (uncertainty quantification, history of probability theory, inverse problems, introduction to probability theory, linear algebra essentials)
  • Lecture 2: Linear Algebra Essentials (vector space, norms, matrix functions, special types of matrices, matrix decompositions, eigenvalues-eigenvectors, condition number)
  • Lecture 3: Classical methods for parameter state estimation (some introductory toy examples, least squares and solving it by different decompositions)
  • Lecture 4-5: Statistical methods for parameter state estimation (statistical estimators, definitions of unbiased/consistent/linear/best/BLUE/OLS/LAD estimators, Gauss-Markov theorem, the maximum likelihood estimator
  • Lecture 6: The Bayes' theorem I - Bayes' rule, choosing prior (inproper prior and uninformed priors, the maximum entropy theorem)
  • Lecture 7: The Bayes' theorem II - Conjugate priors
  • Lecture 8-9: Markov-chains and the MCMC method, introduction to the Kalman-filter
  • Lecture 10: Kalman-filter
  • Lecture 11: Update methods based on the conditional expectation, summary

Literatures

1. Dongbin Xiu, Numerical Methods for Stochastic Computations, A spectral approach (2010)

2. Peter Whittel, Probability via Expectation (2000)

3. Gilbert Strang, Linear Algebra and Its Applications, Fourth Edition (2005)

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.

Software

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.

Tutorials

Tutorial 1-2: basics of probability theory

  • Clone the open source libary SGLIB

  • First tutorial

Tutorial 2: Linear algebra

Tutorial 3: Interpolation and regression

  • The least squre method using different decompositions (see MATLAB file sglib-testing/demo/least/squares/test_ls_1.m)

Tuturial 4: Statistical estimators

  • Estimating the mean and variance of a normally distributed random variable with different estimators

Tutorial 5-6:

  • Computing OLS and LAD for the Ascombe's quartet
  1. OLS
  2. LAD

Tutorial 7-8: Maximum Likelihood and the Bayesian update

  • Maximum Likelihood Estimate (MLE) and Maximum A-Posteriori (MAP) estimate- Example1: coin flipping, Example 2: oscillator

Tutorial 9: Markov Chain Monte Carlo method

  • Markov chains (see MATLAB file sglib-testing/demo/mcmc/test_markov_chain_weather.m)
  • The Metropolis-Hastings sampling (see MATLAB file sglib-testing/demo/mcmc/test_metropolis_hastings.m)
  • Bayesian inversion by the MCMC method:
  1. The basics, (see MATLAB file sglib-testing/demo/mcmc/test bayes_mcmc.m)
  2. Bayesian inversion on the oscillator toy example

Tutoria 10: Kalman-filter

Homework Assignments

Tests

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