# Numerical methods and machine learning algorithms for solution of Inverse problems

## 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.

PhD position in the field of Coefficient Inverse Problems

All lectures in November - December 2021 will be given in Zoom via the link

Meeting ID:  666 3890 0564

Passcode: 742617

The first lecture in 2021:  2 November, Time: 13:15-15:00.

Course schedule: Lectures at Tuesdays and Thursdays, starting from 2 November,  13:15-15:00.

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: sent mail to larisa@chalmers.se

Registration for Master program students at GU: the course code is MMF900

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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 1

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 2

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.

Lecture 3, part 1

Microwave Imaging in monitoring of hyperthermia

Lecture 3, part 2

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 4

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 5

Lecture 6.

Approximate global convergence and Adaptive finite element method for solution of  hyperbolic CIP.

Lecture 6

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 7

Lecture 8.

Classification algorithms: linear and polynomial classifiers, linear and quadratic perceptron  learning algorithm, WINNOW. Neural networks for classification.

Lecture 8

Lecture 9.

Linear models   for regression. Regularized and non-regularized neural networks. Kernel methods. Support Vector Machines. Kernel perceptron for classificaton.

Lecture 9 - SVM

Presentation of the course project "Regularized Least squares and machine learning algorithms for classification problem".

Lecture 9

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.

Lecture 10

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### Computer  Projects

Dataset  iris.csv

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

Matlab program with an example of using PCA

Link to the Matlab code for solution of Poisson's equation  with homogeneous boundary conditions  in 2D

Matlab code together with data used in algorithm for microwave medical imaging

5. Project  "Sensitivity study of parameters in a mathematical model of Intracellular SARS-CoV-2 replication"

#### Reference literature:

1. 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.
2. 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.
3. 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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