Machine learning algorithms for inverse problems.
This page contains the program of the course: lectures, exercise sessions and computer labs. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.
All lectures in January - February 2021 will be given in Zoom via the link
The first lecture in 2021: 12 January, Time: 13:15-14:00.
Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems
Authors: L. Beilina, M. V. Klibanov
The course gives 7.5 Hp.
Registration for PhD students and for students at Master program at Chalmers : sent mail to email@example.com
Registration for Master program students at GU: the course code is MMF900
Introduction. Physical formulations leading to ill- and well-posed problems. Compact set and compact operator. Definitions of well and ill-posed problems. Classical and conditional correctness. Concept of Tikhonov and Tikhonov's theorem. Quasi-solution. Examples of ill-posed problems. Model inverse problems: elliptic inverse Cauchy problem.
Physical formulations leading to ill- and well-posed problems.
Model inverse problems: elliptic inverse problems (Cauchy problem, Inverse source problem, Inverse spectral problem); Hyperbolic and Parabolic CIPs.
Methods for image reconstruction and image deblurring. Solution of a Fredholm integral equation of the first kind as an ill-posed problem. Bayesian approach. Adaptive finite element method.
Microwave Imaging in monitoring of hyperthermia
Lagrangian approach for solution of time-harmonic CIP. Presentation and discussion of the course project “Solution of time-harmonic acoustic coefficient inverse problem”.
Methods of regularization of inverse problems. The Tikhonov regularization functional. The accuracy of the regularized solution. The local strong convexity of the Tikhonov functional.
Methods of regularization of inverse problems: Morozov's discrepancy, balancing principle.
Linear and non-linear least squares problems. Gauss-Newton and Levenberg-Marquardt iterative methods for solution of nonlinear least squares problems.
QR and SVD. Solution of rank-deficient problems. Principal Component Analysis (PCA) for image compression and image recognition. Presentation of the course project "Principal Component Analysis for recognition of handwritten digits".
Classification algorithms: linear and polynomial classifiers, linear and quadratic perceptron learning algorithm, WINNOW. Neural networks for classification.
Linear models for regression. Regularized and non-regularized neural networks. Kernel methods. Support Vector Machines. Kernel perceptron for classificaton.
Presentation of the course project "Regularized Least squares and machine learning algorithms for classification problem".
Lagrangian approach and adaptive FEM for solution of parameter identification problem for system of ODE. Application of adaptive FEM for determination of drug efficacy in the model of HIV infection.
- Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
Physical Modeling in MATLAB 3/E, Allen B. Downey
The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.
Matlab and C++ programs for examples in the book Numerical Linear Algebra: Theory and Applications, Authors: Beilina, L., Karchevskii, E., Karchevskii, M. are available for download from the GitHub Page
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