Dates

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

The conditions for completing this course are successfully completing and submitting at least 60% of the course's exercises and successfully passing the course exam.

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

Lecture notes will be published here after each week's lecture.

• Week 1: Introduction, preliminary functional analysis
• Week 2: Spectral theory of compact operators
• Week 3: Introduction to X-ray tomography (files: tomodemo.py / tomodemo.m)
• Week 4: Fredholm equation and its solvability, truncated SVD, and the Morozov discrepancy principle (files: tomo_tsvd.py, heateq_tsvd.py / tomo_tsvd.m, heateq_tsvd.m)
• Week 5: Tikhonov regularization (files: heateq_tikhonov.py / heateq_tikhonov.m)
• Week 6: Truncated iterative methods: Landweber–Fridman iteration and the conjugate gradient method (files: heateq_cg.py / heateq_cg.m)
• Week 7: Total variation regularization for X-ray tomography (files: sino.mat, tvdemo.py / tvdemo.m)
• Week 8: Brief overview of probability theory and Bayes' formula for inverse problems (files: source.py / source.m)
• Week 9: Deconvolution example, Bayesian estimators, and well-posedness of Bayesian inverse problems
• Week 10: Sampling from Gaussian distributions, inverse transform sampling, prior modeling, and the linear Gaussian setting (files: priormodeling.py, deconv.py / priormodeling.m, deconv.m)
• Week 11: Small noise limit of the posterior distribution, Monte Carlo method, and importance sampling
• Week 12: Markov Chain Monte Carlo (files: mh.py, gibbs.py / mh.m, gibbs.m, autocovariance.m)
• Week 13: Gaussian approximation

Exercise sheets

Weekly exercises will be published here after each lecture.

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

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

Contact

Literature

The course will mainly follow the following texts:

• J. Kaipio and E. Somersalo (2005). Statistical and Computational Inverse Problems. Springer, New York, NY.
• D. Sanz-Alonso,  A. M. Stuart, and A. Taeb (2018). Inverse Problems and Data Assimilation. https://arxiv.org/abs/1810.06191
• D. Calvetti and E. Somersalo (2007). Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing. Springer, New York, NY.

Further topics:

• O. P. Le Maître and O. M. Knio. Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Scientific Computation. Springer, New York, 2010.
• R. C. Smith. Uncertainty Quantification: Theory, Implementation, and Applications, volume 12 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014.
• T. J. Sullivan. Introduction to Uncertainty Quantification. Springer, New York, in press.
• D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton, NJ, 2010.