Numerical methods and machine learning algorithms for solution of Inverse problems

Third-cycle level | 7.5 credits | Course code: NFMV020
HT 2021
Study period: 2021-11-02 - 2021-12-03
LANGUAGE OF INSTRUCTION: The course is given in English
Application period: 2021-08-25 - 2021-10-26
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Course description

In a first step, we will begin to give a short survey on the domain of inverse and ill-posed problems.
In the second part of the course will be considered physical formulations leading to ill- and well-posed problems, methods of regularization of inverse problems and numerical methods of solution of inverse and ill-posed problems, such that Lagrangian approach and adaptive optimization, methods of analytical reconstruction and layer-stripping algorithms, least squares algorithms and classification algorithms. Numerical solution of ill-posed problems including methods of image reconstruction with applications in image deblurring and magnetic resonance imaging (MRI) will be presented. Machine learning classification algorithms for solution of
inverse and ill-posed problems will be also studied as additional tool for improving of solutions obtained in all above described numerical methods.
This course includes the course project consisting of several assignments where some inverse or ill-posed problem should be solved
in Matlab or in C++/PETSc by algorithms studied in the course.

Requirements and Selection

Entry requirements

Numerical analysis, partial differential equations,  programming in Matlab.

Selection

Not relevant

Other information

Inverse and ill-posed problems arise in many real-world applications including medical microwave, optical and ultrasound imaging, MRT, MRI, oil prospecting and shape reconstruction, nondestructive testing of materials and detection of explosives, seeing through the walls and constructing of new materials.
Physical formulations leading to ill- and well-posed problems, methods of regularization of inverse problems and numerical methods of
solution of inverse and ill-posed problems, such that Lagrangian approach and adaptive optimization, methods of analytical reconstruction and layer-stripping algorithms will be addressed. Numerical solution of ill-posed problems including methods of image reconstruction with applications in image deblurring and magnetic resonance imaging (MRI) will be presented.
Machine learning classification algorithms such that regularized and non-regularized least squares and perceptron, SVM and Kernel methods
will be studied. Application of these algorithms for solution of inverse and ill-posed problems will be considered.

Link to website

https://canvas.gu.se/courses/49154

Course syllabus

NFMV020

Department

Department of Mathematical Sciences

Subject

Natural Science and Mathematics

Type of course

Subject area course

Keywords

inverse and ill posed problems, machine learning algorithms for classification, optimization, adaptive finite element method, image restoration, acoustic and electromagnetic scattering problems

CONTACTLarisa Beilina
0046 31 772 3567
larisa@chalmers.se