MAGIC079: Inverse Problems

Course details

A core MAGIC course

Semester

Spring 2022
Monday, January 31st to Friday, March 25th; Monday, April 25th to Friday, May 6th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Tuesdays
12:05 - 12:55 (UK)

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 and numerical techniques for solving inverse problems will be described. 

Prerequisites

There 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. 

Lecturers

  • DL

    Professor Daniel Lesnic

    University
    University of Leeds
    Role
    Main contact
  • SH

    Dr Sean Holman

    University
    University of Manchester

Bibliography

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Monday 9th May 2022 at 00:00 and is due in before Monday 23rd May 2022 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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