Single-Course English 5 ECTS

Introduction to uncertainty quantification for inverse problems

Overall Course Objectives

Uncertainty quantification (UQ) is the science of characterization and management of randomness in computational models of real-world applications. UQ blends theories and methods across stochastic analysis, statistical modeling and scientific computing.

This course introduces state-of-the-art numerical methods for quantification and reduction of uncertainties in computational models. UQ is paramount to enhance analysis and prediction tasks in multiple applications such as tomography, material science, spatial statistics, reliability, etc. Therefore, the course can be of interest to students from any discipline in applied mathematics and engineering. The course provides the mathematical background for theory and methods of UQ, which are illustrated via Python exercises.

Learning Objectives

  • Formulate and model inverse problems.
  • Apply Monte Carlo methods and evaluate their convergence.
  • Discretize infinite dimensional Gaussian random variables (e.g., Karhunen-Loéve expansion).
  • Apply statistical approaches to solve inverse problems (e.g., maximum likelihood estimation).
  • Describe the modeling and computational elements of the Bayesian approach to inverse problems.
  • Formulate different types of noise models and priors (e.g., conjugate priors).
  • Describe the relevant strategies and implement numerical methods for Bayesian computations to practical problems (e.g., Markov chain Monte Carlo).
  • Interpret and understand UQ results.
  • Use the software package CUQIpy.

Course Content

This short course will present fundamentals of uncertainty quantification for inverse problems. The idea is to cover both computational methods (e.g., Monte Carlo,kdiscretization of random fields, Markov chain Monte Carlo) and theoretical aspects (e.g., basic proofs, convergence properties, well-posedness).

We start by presenting an overall introduction to UQ and reviewing probability theory. This is performed with a very simple example that illustrates what is UQ and why is it good for us. Next, we explore Monte Carlo methods for the simulation of random variables and estimation of expectations. After these introductory topics, we cover essential elements of UQ for forward problems, especially the modeling and discretization of random variables and fields, as well as, forward well-posedness; the main numerical examples are also introduced in this section. Finally, we discuss the main component of the course, where we explore UQ for inverse problems. The idea is to formulate elemental inverse problems and present statistical approaches to solve them; we emphasize on Bayesian inference methods. Hence, we talk about likelihoods/noise models, and prior distributions as mechanism of regularization. The solution of the Bayesian inverse problem, given in terms of posterior statistics, is computed via Markov chain Monte Carlo sampling.

We describe how to formulated and solve inverse problems in a statistical setting by means of the software package CUQIpy, and we describe how the user can adjust the computational methods in this package.

The course aims at giving a hands-on experience, i.e., the student will learn how to apply a method and how to interpret the associated results. Therefore, lectures explaining the theory will be followed by exercise sessions.

Recommended prerequisites

Experience with probability theory (e.g., 02405), inverse problems (e.g., 02624), numerical computations (e.g., 02601), functional analysis (e.g., 01715) would be an advantage.

Teaching Method

Lectures and exercises (theory and computations)

See course in the course database.





3 weeks




DTU Lyngby Campus

Course code 02975
Course type PhD
Semester start Week 23
Semester end Week 26
Days Mon-fri 8:00-17:00

10.600,00 DKK