Course syllabus - Inverse Problems, 6 credits

Information about the course

Course code: FOUK036
Third-cycle subject: Mathematics/Applied Mathematics
School: School for Education, Culture and Communication
Responsible department: Department for Mathematics and Physics
Valid from: Autumn term 2025
Established by: The Dean of the School
Decision date: 2025-03-04
Level of education: Third cycle level

Course objective

The course aims to equip doctoral candidates with an understanding of the theory and numerical aspects of inverse problems, with a specific emphasis on regularization techniques. In particular, Tikhonov type regularization and iterative regularization methods for ill-posed inverse problems will be introduced and various inverse problems will be studied.

Course content

Introduction of basic concepts, well-posedness and ill-posedness, some examples of ill-posed inverse problems.
Ill-posed linear operator equations.
Tikhonov regularization method.
Iterative regularization methods.
Numerical realization of regularization methods.
Inverse source identification problems for PDEs with final overdetermination.

Intented learning outcomes

After passing the course the doctoral student should be able to

Demonstrate understanding of the nature and significance of inverse problems, as well as being able to apply key concepts and techniques to solve them.
Apply regularization techniques to solve a given selection of ill-posed inverse problems.
Solve various inverse problems for PDEs, such as source identification problems.
Implement numerical realization of regularization methods for inverse problems using MATLAB.

The intended qualitative targets in relation to the Higher Education Ordinance, appendix 2.

Knowledge and understanding

For the Degree of Doctor, the doctoral student shall demonstrate:

A1: broad knowledge and systematic understanding of the research field as well as advanced and up-to-date specialised knowledge in a limited area of this field, and
A2: familiarity with research methodology in general and the methods of the specific field of research in particular.

Competence and skills

For the Degree of Doctor, the doctoral student shall demonstrate

B1: the capacity for scholarly analysis and synthesis as well as to review and assess new and complex phenomena, issues, and situations autonomously and critically,
B2: the ability to identify and formulate issues with scholarly precision critically, autonomously, and creatively, and to plan and use appropriate methods to undertake research and other qualified tasks within predetermined time frames and to review and evaluate such work, and
B4: the ability in both national and international contexts to present and discuss research and research findings authoritatively in speech and writing and in dialogue with the academic community and society in general.

Teaching formats

Lectures and seminars.

Examination

SEM1, seminar, 2 credits, oral performance of selected theoretical section, related to learning outcomes 1-3, grade Fail (U) or Pass (G).

LAB1, written report, 2 credits, presenting analysis and implementation for a given project, related to learning outcomes 1-4, grade1 Fail (U) or Pass (G).

INL1, written assignment, 2 credits, assignment related to learning outcomes 1-3, grade Fail (U) or Pass (G).

Grade

Examinations included in the course are assessed according to a two-grade scale, fail or pass.

A person who has not passed the regular examination shall be given the opportunity to retake the test.

Requirements

To participate in the course and the examinations included in the course, the applicant must be admitted to doctoral studies.

Selection criteria

Selection of applicants will be made in accordance with the ranking below.

Doctoral students in mathematics/applied mathematics.
Doctoral students at Mälardalen University.
Doctoral students at other universities in Sweden.
Doctoral students at other higher education institutions outside Sweden.