# Introduction to Uncertainty Quantification¶

## General information¶

This is a crash course on uncertainty quantification, given at Uppsala University, Sweden, in Fall 2018. The target participants are PhD students from various departments at Uppsala University, KTH Royal Institute of Technology, Linköping University, and a few other institutes in Scandinavia. The course flyer can be found here.

**Instructor:**Mohammad Motamed**Disclaimer:**These materials (including lecture notes, assignments, developed code, and recorded lecture videos) on uncertainty quantification have been developed over the past five years at the University of New Mexico and Uppsala University. It is not intended to be a complete textbook on the subject, by any means. The goal is to get the student started with a few key concepts and then encourage further reading elsewhere.**License:**These materials are being made freely available to registered students and are released under the Creative Commons CC BY license. Interested students are welcome to use the materils (lecture notes, assignments, codes, and recorded lectures) and quote from them as long as they give appropriate attribution. The codes developed for this course are accessible through the follwoing public Bitbucket repository:**Description:**Deterministic differential equations (ODEs/PDEs) are often used as mathematical models to describe complex physical systems. Such deterministic models however differ from reality due to the inevitable presence of uncertainty in the systems.We can distinguish two types of uncertainty: 1) aleatoric uncertainty due to inherent randomness or variability in a system; and/or 2) epistemic uncertainty due to limited and/or inaccurate information about a system, for instance from insufficient, noisy data. It may hence be impossible to accurately characterize all model parameters by deterministic quantities. In fact, uncertainty is a fundamental feature of physical systems and hence needs to be taken into account when studying complex systems. Examples appear in climate modeling, the description of flows in porous media, behavior of living tissues, combustion problems, deformation of composite materials, earthquake motions, and many more.Uncertainty quantification (UQ) is a process that aims at quantitatively describing the origin, characterization, and propagation of different sources of uncertainty in complex systems. Uncertainty propagation relates to the solution of so-called forward problems, where the uncertainty in the input parameters is propagated through the model to give information about uncertain outputs. Uncertainty characterization is associated with the solution of so-called inverse problems, where the noisy experimental measurements are combined with a model in order to characterize uncertainty in the model input parameters.

In this introductory course we focus mainly on the forward propagation of uncertainty through different types of ODEs/PDEs with stochastic input parameters (coefficients, forcing terms, initial/boundary conditions, etc.). We will study various numerical techniques for solving forward problems. If time allows, we will also review the inverse propagation of uncertainty.

**Background:**- Required: Students should be comfortable with undergraduate mathematics and statistics, particularly calculus, linear algebra, differential equations, and basic probability. Experience writing and debugging computer programs is also required.
- Recommended: Experience with mathematical/statistical computing, for example in Matlab, is preferred. Past exposure to numerical analysis and computation is a plus.

## Course topics¶

- Introductory Overview
- Basics of Probability Theory
- Karhunen-Loeve expansion
- Stochastic Differential Equations
- Monte Carlo Sampling
- Multi-level Monte Carlo Sampling
- Multi-order Monte Carlo Sampling
- Stochastic Polynomial Approximation
- Stochastic Collocation
- Stochastic Galerkin
- Bayesian Inversion
- Markov Chain Monte Carlo
- Gibbs Sampling
- Matropolis-Hastings Sampling
- Metropolis-within-Gibbs Sampling
- References

## Homework assignments¶

# News¶

`November 4th, 2018`

- The fourth and final homework is available here.

`October 18th, 2018`

- Next week will be the final week, and we will meet two times:
- Tuesday Oct. 23 at 13.15 in Rum 2344
- Thursday Oct. 25 at 13.15 in Rum 4307 (regular place)

- Lecture notes for the last two classes are available here and here. They are password protected, and you will get the password in class today. For next meeting on Tuesday, try to read at least the first two notes, and then for Thursday try to read all four notes.
- About final project: Those who would like to do a final project and get 2.5 extra credit hours would need to discuss with me (either in class or outside class) before next Thursday Oct. 25.

- Next week will be the final week, and we will meet two times:
`October 12th, 2018`

- Lecture notes for the next class is available here. It is password protected, and you already have the password.
- HW3 is now available here. Note that it is NOT due next Thursday. This homework covers two subjects: MLMC and MOMC. I suggest that you start with the MLMC part now, and then after the lecture on MOMC next Thursday, do the MOMC part. The due is Tuesday October 23rd. We will hold one extra lecture on that day. I will inform you about the time and location next week. This means that in the last week of October, we will have two lectures, one on Tuesday and one on Thursday.
- I will not hold office hours next week. If you have question, please email me to make an appointment.

`October 5th, 2018`

- Lecture notes for the next class is available here. It is password protected, and you already have the password.
- I will not hold office hours next week. If you have question, please email me to make an appointment.
- On Thursday Oct. 11th there is a seminar talk on Neural networks and partial differential equations (Plats: ITC 1211, Tid: 13.15). We will therefore meet at 14.15 instead of 13.15 so that you can attend the seminar if you would like to.

`September 27th, 2018`

- Lecture notes for the next class is available here. Please try to take a careful look before class. Note that it is password protected. I will email you the password. If you do not receive the password by the end of today, please email me.
- I will hold office hours on Monday (October 1st) between 10.00 and 12.00 at Rum 2348.

`September 24th, 2018`

- Today I will hold office hours between 10.00 and 12.00 at Rum 2348.
- Lecture notes for the next class is available here. Please try to take a look before class.

`September 19th, 2018`

- Lecture notes for today’s lecture are now posted. See here and here.
- Try to read the lecture notes and do HW1 by next Thursday. You can work on the homework in groups of two and hand in a single report in class. If you think you need more time, shoot me an email (Write UQ in the title fo your email).
- Lecture notes for the next class on Thursday (on Stochastic Differential Equations) will be uploaded soon.

`September 12th, 2018`

- We will not have class this Thursday September 13th. Instead I will hold an extra lecture. The time will be announced later.
- Next week, the class will hold on Wednesday (September 19th), instead of Thursday (September 20th), the same time, the same place.