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Algorithms for Uncertainty Quantification - Summer 18

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Term
Summer 18
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
Master students, e.g. of CSE, mathematics, informatics, data science, data engineering and analytics, physics,...
Tutorials
Friedrich Menhorn
Exam
Location: 1450, Willy-Messerschmitt-Zeichensaal (5504.01.450)
Time: 01.08.2018, 11:00-12:15
Semesterwochenstunden / ECTS Credits
4 SWS (2V+2Ü) / 5 Credits
TUMonline
Algorithms for UQ (IN2345)



Contents

Announcements

  • Review: 02.07.023, Aug 06, 2018, 12:00 - 12:45
  • Decision on allowed exam material: No additional material needed/allowed!
  • Old exam from summer 17 has been added to Exam section
  • As announced in the tutorial we will swap lecture and tutorial in week 24. That means: Tutorial: June 12, 14:00-16:00; Lecture: June 13, 12:00-14:00.
  • Evaluation of the lecture takes place during the lecture on June 13 2018. Please bring your laptop.
  • Typos in the slides of §6 have been fixed. A print version of the slides is now also available
  • The first lecture takes place on April 10 2018.

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

Lecture Slides

Worksheets and Solutions

Number Topic Worksheet Tutorial Code Solution
1 Python overview Worksheet1 April 11 Template Solution 1 Solution 2 Solution 3
2 Probability and statistics overview Worksheet2 May 02 Solution 1 Solution 6 Solution.pdf
3 Standard Monte Carlo sampling Worksheet3 May 9 Template Solution 2 Solution 3 Solution 4 Solution 5 Solution.pdf
4 More advanced sampling techniques Worksheet4 May 16 Template

Solution 2 Solution 3.1 Solution 3.2 Solution.pdf

5 Aspects of interpolation and quadrature Worksheet5 May 30 Template

Solution 1 Solution 1 Optional Solution 2 Solution 3

6 Polynomial Chaos 1: the pseudo-spectral approach Worksheet6 June 06 Template

Solution 1 Solution 3 Solution.pdf

7 Polynomial Chaos 2: the stochastic Galerkin approach Worksheet7 June 12

Solution.pdf

8 The sparse pseudo-spectral approach Worksheet8 June 20 Template

Solution 1 Solution 2

9 Sobol' indices for global sensitivity analysis Worksheet9 June 27 Ex2 Template

Solution Solution 2

10 Random fields in Uncertainty Quantification Worksheet10 July 04 Template

Solution 1 Solution 2 Optional 3

11 Software for Uncertainty Quantification & Old Exam Worksheet11 July 11

Solution

Exam

  • Review: 02.07.023, Aug 06, 2018, 12:00 - 12:45
  • first exam (preliminary, check TUMonline):
    • WED, Aug 01, 2018, 11:00-12:15 (75 min)
    • room: 1450, Willy-Messerschmitt-Zeichensaal
  • covered topics (preliminary): everything except:
    • inverse problems (lecture 12)
    • details of pure python programming
    • specific API of chaospy (or other packages)
  • style of exam exercises: similar to tutorials
  • allowed material: no material allowed
  • Written exam.

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