Inverse Problems (MAGIC079) |
GeneralThis course is part of the MAGIC core. Description
When it is possible to input the governing equation(s), shape(s) and size(s) of the domain(s), boundary and initial conditions, material properties of the media contained in the field, and forces or sources, then the analysis determining the unknown field is considered mathematically well-posed, i.e. the solution exists, is unique and it depends continuously on the data. If any of these elements are unknown or unavailable, then the field problem becomes improperly defined (ill-posed) and is of an indirect (or inverse) type. The course will give an introduction to Inverse Problems. Various mathematical, numerical and statistical techniques for solving inverse problems will be described.
SemesterSpring 2019 (Monday, January 21 to Friday, March 29) Hours
Timetable
PrerequisitesThere is a background level of linear algebra, partial differential equations, numerical and functional analysis for which there are general courses. Also just enough physics to understand the phenomena of heat conduction, fluid flow, acoustics, optics and electromagnetism used to formulate the forward problems.
Syllabus* Basic linear inverse problems - enough linear algebra and functional analysis to understand ill-conditioning and regularization of inverse problems.
* Basic techniques for linear inverse problems - truncated singular value decomposition, Tikhonov's regularization, parameter choice methods, etc.
* PDE theory for inverse problems - enough to read the main existence, uniqueness and stability papers, e.g. Isakov's book. Some mathematical techniques and concepts, e.g. Schauder fixed point theorem, contraction principle, Fredholm alternative, etc.
* Numerical methods for inverse problems including FEM and BEM for forward problem solution and iterative regularization methods. Level set method. Constrained minimization gradient based methods. Possibly Bayesian approach and introduction to MCMC.
Lecturers
BibliographyNote: Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.) AssessmentThe assessment for this course will be via a single take-home paper in April with a few weeks to complete and submit online. There will be 2-3 questions and you will need a mark of 50
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