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Inverse problems recover causes given the effects, reveal forces behind actions and allow to understand the underlying mechanisms behind physical and biological processes using mathematics. Mathematical techniques developed for solving inverse problems connect both pure and applied areas of mathematics. The applications range from geophysical oil prospecting to medical imaging, from communication theory to invisibility cloaking, radars, signal processing, remote sensing, non-destructive testing and others. The course will cover two themes: discrete inverse problems and continuous inverse problems (both linear and non-linear). Discrete problems related techniques will include weighted least squared, generalized inverses, maximum likelihood methods, Monte-Carlo, convolution methods, gradient methods etc., illustrated by examples from discrete acoustic tomography, X-ray imaging, image blurring-deblurring tools. Continuous inverse problems techniques will include Radon transform, integral equations, the Fréchet derivatives, Fourier analysis, PDEs and control theory, illustrated by examples from tomography and non-destructive testing. The ill-posed nature of inverse problems requires to consider controllability, observability, stability and regularisation methods, all of which will be discussed throughout the course.
This module is intended for students working in various areas of applied mathematics, data science or computer science, where inverse problems are merging also with machine learning techniques.