Inverse Problems

News

• The slides of the twelfth and final lecture are available here (recording).
• A bonus exercise sheet is available here. Please note that these exercises will not be graded and do not need to be returned.

Dates

Please contact vesa.kaarnioja@fu-berlin.de in advance to organize a personal exam appointment.

General information

Description

Mathematical measurement models describe the causal effects of physical systems based on their material properties, initial conditions or other model parameters. In many practical problems, we have measurement data of the outcomes of these so-called "forward models" and we wish to infer the model parameters which caused the observations. This is an inverse problem.

Inverse problems are intrinsically ill-posed: the reconstruction of the unknown quantity may be highly sensitive to noise in the measurements, or a unique solution may not exist. For these reasons, regularization is an essential tool in order to find solutions to inverse problems. In this course, we will consider both deterministic regularization methods and statistical Bayesian inference. We will discuss the main challenges related to inverse problems as well as the main solution techniques.

Target audience

The course is intended for mathematics students at the Master's level.

Prerequisites

Multivariable calculus, linear algebra, basic probability theory, and MATLAB (or some other programming language).

Completing the course

Passing the course exam and completing weekly exercises.

Registration

• Please register to the course via Campus Management (CM), then you will be automatically registered in MyCampus/Whiteboard as well. Please note the deadlines indicated there. For further information and in case of any problems, please consult the Campus Management's Help for Students.
• Non-FU students should register to the course in KVV (Whiteboard).

Lecture notes

• Week 1: Introduction, Fredholm equation and its solvability, truncated SVD (files: recording)
• Week 2: Morozov discrepancy principle, Tikhonov regularization, and backward heat equation (files: week2.m)
• Week 3: Introduction to X-ray tomography, Landweber-Fridman iteration (files: tomodemo.m)
• Week 4: Conjugate gradient method (files: cgdemo.m)
• Week 5: Brief introduction to probability theory and Bayes' formula for inverse problems (files: week5.m)
• Week 6: Deconvolution example, Bayesian estimators, and well-posedness of Bayesian inverse problems (files: week6.m)
• Week 7: Sampling from Gaussian distributions, inverse transform sampling, prior modeling, and the linear Gaussian setting (files: priormodeling.m, week7.m)
• Week 8: Small noise limit of the posterior distribution, Monte Carlo, and importance sampling
• Week 9: Markov Chain Monte Carlo (files: mh.m, gibbs.m, autocovariance.m)
• Week 10: Optimization perspective and Gaussian approximation
• Week 11: Brief overview of Bayesian and Kalman filters (files: kfdemo.m)
• Week 12: Total variation regularization for X-ray tomography (files: recording, tvdemo.m, sinogram.mat)

Exercise sheets

• Exercise 1
• Exercise 2
• Exercise 3 (files: week3.mat)
• Exercise 4 (files: week3.mat)
• Exercise 5
• Exercise 6
• Exercise 7
• Exercise 8
• Exercise 9
• Exercise 10
• Exercise 11 (files: kfdemo.m)
• Bonus exercises (files: pde.mat)

Please note that the bonus exercises will not be graded and do not need to be returned.

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