# Introduction to Uncertainty Quantification¶

## General information¶

• Disclaimer: These materials (including lecture notes, assignments, developed code, and recorded lecture videos) on uncertainty quantification have been developed for the course Math/Stat 579 at the University of New Mexico, Fall 2016. As a work in progress, these pages will be modified and supplemented over time. 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 and are released under the Creative Commons CC BY license. Interested students and researchers 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:

https://bitbucket.org/motamed/uq2017

• Description: The behavior of many physical systems can be mathematically modeled by deterministic ordinary or partial differential equations (ODEs or PDEs). These models however differ from reality for two main reasons: 1) an intrinsic variability of the physical system (aleatoric uncertainty); and/or 2) our inability to accurately characterize all parameters of the mathematical model (epistemic uncertainty). Uncertainty is therefore a fundamental feature of physical systems and 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, etc. In order to understand the uncertainties inherent in the model, and often represented in a probabilistic setting, we need a process called “uncertainty quantification” (UQ).

In this course we consider different types of ODEs/PDEs with stochastic input parameters (coefficients, forcing terms, initial/boundary conditions, etc.) and study various numerical techniques for solving both forward and inverse problems.

• 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 or R, is preferred. Past exposure to numerical analysis and computation is a plus.