Algorithms for Uncertainty Quantification - Summer 17

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Term
Summer 17
Lecturer
Dr. Tobias Neckel
Time and Place
Lecture: Tuesday, 14:15-15:45 MI 02.07.023
Tutorial: Wednesday, 12:15-13:45 MI 02.07.023
Audience
tba
Tutorials
Ionut Farcas
Exam
tba
Semesterwochenstunden / ECTS Credits
4 SWS (2V+2Ü) / 5 Credits
TUMonline
Algorithms for UQ



Contents

Computer simulations of different phenomena heavily rely on input data which – in many cases – are not known as exact values but face random effects. Uncertainty Quantification (UQ) is a cutting-edge research field that supports decision making under such uncertainties. Typical questions tackled in this course are “How to incorporate measurement errors into simulations and get a meaningful output?”, “What can I do to be 98.5% sure that my robot trajectory will be safe?”, “Which algorithms are available?”, “What is a good measure of complexity of UQ algorithms?”, “What is the potential for parallelization and High-Performance Computing of the different algorithms?”, or “Is there software available for UQ or do I need to program everything from scratch?”

In particular, this course will cover:

  • Brief repetition of basic probability theory and statistics
  • 1st class of algorithms: sampling methods for UQ (Monte Carlo): the brute-force approach
  • More advanced sampling methods: Quasi Monte Carlo & Co.
  • Relevant properties of interpolation & quadrature
  • 2nd class of algorithms: stochastic collocation via the pseudo-spectral approach: Is it possible to obtain accurate results with (much) less costs?
  • 3rd class of algorithms: stochastic Galerkin: Are we willing to (heavily) modify our software to gain accuracy?
  • Dimensionality reduction in UQ: apply hierarchical methodologies such as tree-based sparse grid quadrature. How does the connection to Machine Learning and classification problems look like?
  • Which parameters actually do matter? => sensitivity analysis (Sobol’ indices etc.)
  • What if there is an infinite amount of parameters? => approximation methods for random fields (KL expansion)
  • Software for UQ: What packages are available? What are the advantages and downsides of major players (such as chaospy, UQTk, and DAKOTA)
  • Outlook: inverse UQ problems, data aspects, real-world measurements

Announcements

  • The lecture scheduled on June 6 2017 is cancelled
  • The tutorial scheduled on June 7 2017 is cancelled

Lecture Slides

Lecture slides are published here successively.


Worksheets and Solutions

Number Topic Worksheet Tutorial Solution
1 Python overview Worksheet1 April 26 Assignment 1, 2, 3
2 Probability and statistics overview Worksheet2 May 03 Assignment 1 Assignment 6 Solution worksheet2
3 Standard Monte Carlo sampling Worksheet3 May 10 Assignment 1 Assignment 2 Assignment 3 Assignment 4
4 More advanced sampling techniques Worksheet4 May 17 Assignment 1 Assignment 2.1 Assignment 2.2
5 Aspects of interpolation and quadrature Worksheet5 May 24 Assignment 1 Assignment 2 Assignment 3
6 Polynomial Chaos 1: the pseudo-spectral approach Worksheet6 May 31 Assignment 1 Assignment 2
7 Polynomial Chaos 2: the stochastic Galerkin approach Worksheet7 June 14 Solution worksheet7
8 The sparse pseudo-spectral approach Worksheet8 June 21 Assignment 1 Assignment 2
9 Sobol' indices for global sensitivity analysis Worksheet9 June 28 Assignment 2.1 Assignment 2.2
10 Random fields in Uncertainty Quantification Worksheet10 July 05 Assignment 1 Assignment 2

Literature

  • R. C. Smith, Uncertainty Quantification – Theory, Implementation, and Applications, SIAM, 2014
  • D. Xiu, Numerical Methods for Stochastic Computations – A Spectral Method Approach, Princeton Univ. Press, 2010
  • T. J. Sullivan, Introduction to Uncertainty Quantification, Texts in Applied Mathematics 63, Springer, 2015
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