# Introduction to Uncertainty Quantification (UQ)

This course is given at the graduate level and corresponds to 5 hp. The goal is to give an introduction to the rapidly expanding field of uncertainty quantification which builds on probability, statistics, computation and large scale simulations. There is uncertainty in the mathematical model, in the parameters, and in the initial and boundary data. How do these uncertainties propagate to the predictions of the computations? In the inverse problem, parameters are determined from measured data. What is the effect of the errors in the data on the parameter estimation? Some basic knowledge of random processes and differential equations is useful. Parameters in mathematical models based on differential equations will be estimated using frequentist and Bayesian techniques. We will follow the book in the literature list below. There will be lectures and homework problems from the exercises in the book. If UQ is applied to a problem related the student's own research project and a report is written about it, the credit will be 2.5 extra hp.

##### literature

R. C. Smith, Uncertainty Quantification, Theory, Implementation and Applications, SIAM, Philadelphia, 2014.

The author's homepage is here.

The book's homepage is here. There is a list of misprints there.

##### teacher

## Contents

Chapters of the book define the course.

- Introduction to UQ with examples (Ch 1, 2)
- Fundamentals of probability (Ch 4)
- Representation of random inputs (Ch 5)
- Parameter selection (Ch 6)
- Frequentist view of parameter estimation (Ch 7)
- Bayesian view of parameter estimation (Ch 8)
- Uncertainty propagation in models (Ch 9)
- Model discrepancies (Ch 12)
- Surrogate models (Ch 13)
- Local sensitivity analysis (Ch 14)

## Schedule

- Tuesday, March 3, 2015, 13.15-15, room 2214, Ch 1,2
- Tuesday, March 10, 2015, 13.15-15, room 2446, Ch 4
- Tuesday, March 24, 2015, 13.15-15, room 2344, Ch 4, 5, 6
- Tuesday, March 31, 2015, 13.15-15, room 2344, Ch 7
- Tuesday, April 7, 2015, 13.15-15, room 2344, Ch 8
- Tuesday, April 14, 2015, 13.15-15, room 2344, Ch 9, 12
- Friday, April 24, 2015, 13.15-15, room 2345, Ch 12, 13
- Wednesday, April 29, 2015, 15.15-17, room 2345, Ch 13
- Thursday, May 7, 2015, 15.15-17, room 2344, Ch 14

## Lecture notes

## Homework from the book

- Exercises 4.4, 4.5, 4.9, 6.3, 7.6, 8.2, 9.1, 13.1, 13.2, 13.3, 13.4, 14.2

## Place

Polacksbacken, ITC, Uppsala University

## Examination

There will be theoretical and computational assignments after every or every second lecture. For 5 hp credit for the course, all assignments should be solved satisfactorily. In a final project, you should apply UQ techniques to your own research problem and solve something of interest in your project.