MAGIC079: Inverse Problems

Course details

A core MAGIC course

Semester

Spring 2021
Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th

Hours

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

Timetable

Tuesdays
11:05 - 11:55 (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

Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and 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.

  • Inverse Heat Conduction (Beck et al., )
  • The Boundary Element Method for Solving Improperly Posed Problems (Ingham and Yuan, )
  • The Mollification Method and the Numerical Solution of Ill-Posed Problems (Murio, )
  • Inverse Problems for Partial Differential Equations (Isakov, )

Assessment

The assessment for this course will be released on Monday 10th May 2021 at 00:00 and is due in before Monday 24th May 2021 at 11:00.

There will be 4 questions and students will have 2 ways to prepare there answers and upload they pdf file solutions. The passing mark is 50%.

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

Files

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Lectures

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