Numerical methods and machine learning algorithms for solution of 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.
Course schedule: Lectures at Mondays, 13:15-15:00. The last lecture will be at 23 January 2023. Lectures at 31.10, 7.11, 14.11 will be given in Zoom but also in room Pascal at the Department of Mathematical Sciences.
Starting from 15 November 2022 lectures will be given only in Zoom.
Zoom Link for lectures in 2022
Meeting ID: 666 3890 0564
Passcode: 742617
The first lecture in 2022 (live/Zoom): 31 October, Time: 13:15-15:00.
Course schedule: Lectures at Mondays, 13:15-15:00. The last lecture will be at 23 January 2023. Lectures at 31.10, 7.11, 14.11 will be given also at room Pascal at the Department of Mathematical Sciences.
Lecture 1: 31.10.2022
Lecture 2: 07.11.2022
Lecture 3: 14.11.2022
Lecture 4: 21.11.2022
Lecture 5: 28.11.2022
Lecture 6: 05.12.2022
Lecture 7: 12.12.2022
Lecture 8: 09.01.2023
Lecture 9: 16.01.2023
Lecture 10: 30.01.2023
Course literature:
Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems
Authors: L. Beilina, M. V. Klibanov
Registration
The course gives 7.5 Hp.
Registration for PhD students at all universities: the course code is NFMV020 and registration is done via the link registration
Registration for Master program students at Chalmers: the course code is MVE065
Registration for Master program students at GU: the course code is MMF900 ( contact person for registration: Sonja Göc)
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Videos from the 30 Jyväskylä Summer school, 9.08.2021 – 13.08.2021
Organization of the course. Introduction to inverse and ill-posed problems. Physical formulations leading to inverse and ill-posed problems. Coefficient inverse problems.
Microwave medical imaging in monitoring of hyperthermia.
Physical formulations leading to inverse and ill-posed problems.
Classical and conditional correctness, Tikhonov's theorem, examples of ill-posed problems.
Methods of regularization of inverse problems. Tikhonovs' regularization,
Morozov's discrepancy principle and Balancing principle.
Linear and non-linear least squares problem. Normal equations. Data fitting.
Classification algorithms. Least squares and ML algorithms (perceptron learning, WINNOW ) for classification.
QR and SVD. Solution of rank-deficient least squares problems.
Principle component analysis (PCA). PCA for image recognition.
Kernel methods and support vector machines (SVM) for classicifaction.
Regularized and non-regularized neural networks.
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.
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Lectures for the course "Numerical methods and machine learning algorithms for solution of Inverse problems"
Lecture 1
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.
Lecture 2.
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.
Lecture 3.
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
Lecture notes on Helmholtz equation
Lecture 4.
Lagrangian approach for solution of time-harmonic CIP. Presentation and discussion of the course project “Solution of time-harmonic acoustic coefficient inverse problem”.
Lecture 5.
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.
Lecture 6.
Approximate global convergence and Adaptive finite element method for solution of hyperbolic CIP.
Lecture 7.
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".
Lecture 8.
Classification algorithms: linear and polynomial classifiers, linear and quadratic perceptron learning algorithm, WINNOW. Neural networks for classification.
Lecture 9.
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".
Lecture 10.
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.
Computer Projects
1. Project "Regularized least squares and machine learning algorithms for classification"
Paper "Numerical analysis of least squares and perceptron learning for classification problems"
Matlab program for classification of data from dataset iris.csv
Test dataset mnist_test_10.csv with handwritten digits
Train dataset mnist_train_10.csv with handwritten digits
Matlab program for reading datasets mnist*.csv
Matlab function used by the above program
2. Project"Principal component analysis for image recognition"
Matlab program with an example of using PCA
3. Project “Solution of time-harmonic acoustic coefficient inverse problem”
4. Project "Regularized adaptive algorithms for detection of tumours in microwave medical imaging"
Matlab code together with data used in algorithm for microwave medical imaging
6.Project "Regularized SVM algorithms for classification problems"
7.Project "Machine learning techniques for image recognition in medical diagnostics of skin cancer "
8. Project "Parameter identification in a mathematical model
describing tumour-macrophages interactions"
Link to MATLAB code for the project 8.
Reference literature:
- Learning Machine learning techniques
for image recognition in medical diagnostics of skin cancer 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
Course summary:
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