Course syllabus
Course literature
- JMSS: Jennifer Muller and Samuli Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, 2013. Use this link to access with your KTH ID.
- CC: Christian Clason, Regularization of Inverse Problems, ArXiv e-print:2001.00617, 2021.
- CCTV: Christian Clason and Tuomo Valkonen, Introduction to Nonsmooth Analysis and Optimization, ArXiv e-print:2001.00617, 2024.
- FNFW: Frank Natterer and Frank Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, 2001. Use this link to access with your KTH ID.
- CV: Curtis R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002. Use this link to access with your KTH ID.
| Week 35 | |
| 2025-08-25 Room: 3721 |
Lecture 1: Introduction
- Examples of inverse problems
- Formalisation as operator inversion
- Forward operator: Linear and non-linear inverse problems
- Inverse mapping
- Naive reconstructions and inverse crimes
Recommended reading:
- JMSS: 1, 2.1, 2.3 Recommended exercises: - JMSS: 2.3.3, 2.3.4, 2.3.5 |
| 2025-08-26 Room: 3721 |
Lecture 2: Linear inverse problems: Basic functional analysis - Hilbert spaces - Compact operators - Moore-Penrose Inverse - Spectral theorem & singular value decomposition (SVD) Recommended reading: - JMSS: 3.5, 3.6, 4.1 Additional reading: - CC: 1-2, 3.1, 3.2 Recommended exercises: - JMSS: 3.6.1 - Exercise sheet: Lecture 2 |
| 2025-08-29 Room: 3721 |
Exercise session 1 |
| Week 36 | |
| 2025-09-01 Room: 3721 |
Lecture 3: Ill-posedness & general theory of regularisation - Hadamard's conditions and ill-posedness - Regularisation and parameter selection rules - Convergence rates Recommended reading: - JMSS: 3.1, 3.3, 3.4 - 13.1-13.2, 13.3.7 in Inverse and Ill-Posed Problems [2025] Additional reading: - CC: 4.1, 4.2 Recommended exercises: - JMSS: 3.3.1, 3.3.2, 3.3.4, 3.3.5 - Exercise sheet: Lecture 3 |
| 2025-09-02 Room: 3721 |
Exercise session 2 |
| 2025-09-04 Room: 3721 |
Lecture 4: Spectral regularisation - A priori parameter choice rules - A posteriori parameter choice rules - Convergence rates Recommended reading: - JMSS: 4.2, 4.4, 5.4 - 13.3.1-13.3.2 in Inverse and Ill-Posed Problems [2025] Additional reading: - CC: 5.1, 5.2 Recommended exercises: - JMSS: 4.4.2, 4.4.3, 4.4.4, 5.4.4, 5.4.5, 5.4.6, 5.4.7 - Exercise sheet: Lecture 4 |
| 2025-09-05 Room: 3721 |
Exercise session 3 - Exercises from the sheet: (5), 8 |
| Week 37 | |
| 2025-09-08 Room: 3721 |
Lecture 5: Iterative methods (Lecture cancelled, self study) - Semi-convergence & early stopping - Landweber (SIRT) - Kacmartz (ART) - Conjugate gradient method Recommended reading: - JMSS: 5.5 - FNFW: 5.3.1, 5.7 - 13.3.3-13.3.4 in Inverse and Ill-Posed Problems [2025] Additional reading: - CC: 11.1 Recommended exercises: - JMSS: 5.5.1, 5.5.2 |
| 2025-09-09 Room: 3721 |
Exercise session 4 - Some ODL - Exercises from the sheet: (8), 9(a)(ii) (SIRT) |
| 2025-09-10 Room: 5O2Spo (Sporthallen) |
Computer lab 1: Introduction to ODL, function spaces and functions |
| 2025-09-11 Room: 3721 |
Lecture 6: Variational models - 1: Optimisation - Minimisation of functionals - Minimisation algorithms Recommended reading: - 4.1 and 4.3 in Optimization and Inverse Problems [2025] Recommended exercises: - Exercise sheet: Lecture 6 |
| 2025-09-12 Room: 3721 |
Exercise session 5 - ODL - Exercise from the sheet: 10 - Properties of an operator related to Tikhonov regularisation |
| Week 38 | |
| 2025-09-15 Room: 3721 |
Lecture 7: Variational models - 2: Quadratic Tikhonov regularisation - Existence and convergence - Relation to TSVD - Large-scale implementation Recommended reading: - JMSS: 5.1-5.3 Additional reading: - CC: 10 Recommended exercises: - JMSS: 5.1.1, 5.1.2, 5.3.1, 5.3.2, 5.3.4 |
| 2025-09-16 Room: 3721 |
Exercise session 6 - Exercise from the sheet: 12 - Landweber iteration |
| 2025-09-17 Room: 5O2Spo (Sporthallen) |
Computer lab 2: Introduction to the ODL tomography interface |
| 2025-09-18 Room: 3411 |
Lecture 8: Variational models - 3: Total variation - Functions of bounded variation - Existence and convergence Recommended reading: - JMSS: 6.1, 6.2 Additional reading: - A Guide to the TV Zoo [2013] - Total Variation in Imaging [2015] - An Introduction to Total Variation for Image Analysis [2010] Recommended exercises: - JMSS: 6.1.1, 6.2.1, 6.2.2 |
| 2025-09-19 Room: 3418 |
Exercise session 7 - Exercise from the sheet: 14 - Several smaller problems related to convexity and variational regularisation 10 |
| Week 39 | |
| 2025-09-22 Room: 3721 |
Lecture 9: Variational models - 4: Total variation - Non-smooth optimisation - Large-scale implementation Recommended reading: - JMSS: 6.3, 6.4 Additional reading: - Numerical Methods and Applications in Total Variation Image Restoration [2011] Recommended exercises: - JMSS: 6.4.1, 6.4.2 |
| 2025-09-23 Room: 3721 |
Exercise session 8 - Continuation of last session - Gateaux derivatives for some specific functionals |
| 2025-09-24 Room: 5O2Spo (Sporthallen) |
Computer lab 3: Landweber iteration and discrepancy principle |
| 2025-09-25 Room: 3721 |
Lecture 10: Variational models - 5: Higher order regularisers - TGV, Besov space norms - Statistical interpretation as MAP estimator - Discretisation invariance Recommended reading: - JMSS: 7, 8 Additional reading: - Higher-order total variation approaches and generalisations [2020] Recommended exercises: - JMSS: 7.3.1, 8.0.1, 8.1.1 |
| 2025-09-26 Room: 3721 |
Exercise session 9 |
| Week 40 | |
| 2025-09-29 Room: 3721 |
Lecture 11: Variational models - 6: Sparsity promoting regularisation - Sparse signals - Dictionary learning - Compressed sensing Recommended reading: - Dictionaries for Sparse Representation Modeling [2010] - Dictionary Learning [2011] - L1-L2 Optimization in Signal and Image Processing [2010] - Compressed Sensing Makes Every Pixel Count [2009] - Demystifying Compressive Sensing [2017] Additional reading: - Dictionary Learning [2015] - Sparsity and l^1-Regularization [2018] - Compressed sensing [2006] - Robust Uncertainty Principles. Exact Signal Reconstruction From Highly Incomplete Frequency Information [2006] - Total Variation Minimization in Compressed Sensing [2017] |
| 2025-09-30 Room: 3721 |
Exercise session 10 |
| 2025-10-01 Room: 4V2Röd (Röd) |
Computer lab 4: Tikhonov regularization, the conjugate gradient method and the L-curve |
| 2025-10-02 Room: 3721 |
Lecture 12: Statistical regularisation - 1 - Formalisation as statistical (Bayesian) inference - Notions of ill- and well-posedness - Estimators, connection to variational models Recommended reading: - CC: 12, 13 - Inverse problems: From regularization to Bayesian inference [2017] - Energy-based models for inverse imaging problems [2025] Additional reading: - The Bayesian Approach to Inverse Problems [2017] - On Bayesian Inference for Some Statistical Inverse Problems with Partial Differential Equations [2017] |
| 2025-10-03 | Exercise session 11 (Moved to 2025-10-08) |
| Week 41 | |
| 2025-10-06 Room: 3721 |
Lecture 13: Statistical regularisation - 2 - Survey of theoretical results - Posterior sampling techniques Additional reading: - The Bayesian Approach to Inverse Problems [2017] |
| 2025-10-07 Room: 3721 |
Exercise session 12 |
| 2025-10-08 Time: 10:00-12:00 Room: 3721 |
Exercise session: Exam preparation |
| 2025-10-09 Room: 3721 |
Lecture 14: Deep learning based approaches
- Neural network architectures
- Post-processing - Unrolling - Choice of loss function Recommended reading: |
| 2025-10-10 Room: 3721 |
Examination of project |
| Week 43 | |
| 2025-10-24 Room: E35, E36 |
Final exam (tentamen) |
| Week 51 | |
| 2025-12-15 Room: E52 |
Re-exam (omtentamen) |