The aim of the module is to introduce the students to inverse problems via a rigorous treatment of ill-posed problems and the concepts of existence, uniqueness and stability in mathematical analysis. The module will provide the students with a solid foundation in inverse problems that, based on classical concepts in functional analysis, will also introduce the students to modern, cutting-edge, research-led mathematical techniques that are at the basis of imaging.

The following syllabus is indicative of the content to be addressed in this module.  1. Sobolev spaces:  - Measure theory (measurable sets, measurable functions)  - Weak derivatives and Sobolev spaces  - Approximation and density theorems  - Poincaré inequality, Sobolev embedding theorems.  2. Weak solvability of second order elliptic boundary value problems:  - Variational formulation of Dirichlet and Neumann problems   - The Dirichlet-to-Neumann map.   3. Introduction to inverse problems:  - Ill-posed problems in the sense of Hadamard  - Uniqueness and stability in inverse problems: a-priori assumptions and regularisation.  4. Inverse problems of physical interest:  - The inverse conductivity problem (Electrical Impedance Tomography)  - Optical tomography  - Inverse scattering problems in the frequency domain  - Non-uniqueness for anisotropic inverse problems.  5. Analytical and computational techniques in inverse problems:  - Stability estimates in inverse problems - Imaging: an introduction to reconstruction techniques in inverse problems. 

On successful completion of this module, students will be able to: 1. explain in professional reports the concepts of weak differentiability and Sobolev spaces that are fundamental in inverse and ill-posed problems; 2. differentiate between a forward and an inverse problem; 3. apply techniques of functional analysis to formulate an inverse problem; 4. develop logical arguments to prove uniqueness and stability in inverse problems. 5. initiate real-world projects on inverse problems, by developing projects scope statements and a work breakdown structure.

On successful completion of this module, students will be able to: 1. advocate the key role of inverse and ill-posed problems in real-world applications like imaging.

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The module will be taught by introducing an environment for the students that is experiential, collaborative and challenge-driven, beyond the conventional lecture format.  As the module will be embedded in a research-led postgraduate programme, students will develop a research-led foundation on inverse problems via continuous assessment. Reflecting the CHALLENGE-DRIVEN & EXPERIENTIAL nature of the programme, the continuous assessment will involve collaboration between peers. The module, embedded in a COLLABORATIVE & CROSS-DISCIPLINARY postgraduate programme, will encourage critical thinking and collaboration across the scientific disciplines of mathematics and physics. It will include authentic learning experiences drawn from real-world examples in inverse problems and imaging. The module will provide a learning experience for the students that will stimulate them to stay CURIOUS (think logically, analytically and critically and demonstrate strong problem-solving skills); to be COURAGEOUS (can undertake complex problems with diligent organisation and resilience, and communicate abstract concepts and highly technical problems effectively in a wide variety of contexts); to be ARTICULATE (can work and communicate effectively and efficiently on trans- and inter-disciplinary projects and teams); to be AGILE (can take a high-level view of a problem and formulate relevant models and modelling strategies that are flexible and generalisable to a wide variety of settings); to be RESPONSIBLE (able to challenge and question the appropriate use of a rigorous analytical technique, modelling, and the responsible implementation of these).