# MA4M2 Mathematics of Inverse Problems

**Lecturer:** Florian Theil

**Term(s):** Term 2

**Status for Mathematics students:**

**Commitment:** 30 lectures

**Assessment:** 100% written examination

**Formal registration prerequisites: **None

**Assumed knowledge: **

- MA250 Introduction to PDEs: Fundamental module on PDEs where the most important solution techniques are introduced
- MA3G1 Theory of PDEs: Properties of the solutions of PDEs, initial conditions and boundary conditions
- MA4A2 Advanced PDEs: Functional analytic methods. Only the theory of Sobolev spaces is needed.

**Useful background:**

**Synergies: **

**Content:** Inverse problems play an increasingly important role for modern data oriented applications. Classical examples are medical imaging and tomography where one attempts to reconstruct the internal structure from transmission data.

Using the theory of partial differential equations it is possible to map the unknown internal structure to the observed data. The task of inverting this map is called 'Inverse Problem'.

We will study the mathematical theory that underpins the construction of the forward operator and devise regularisation techniques that will result in well posed inverse problems.

- Review of Functional Analysis and PDE theory
- Modelling of simple physical systems, Radon transform
- Loss functions and the direct method
- Regularisation: Tikhonov and Total Variation
- Convergence of solutions for vanishing noise

**Aims:** Students will be able to identify inverse problems in applications like accoustics. They will be become aware of the connections between the theory of partial differential equations and parameter estimation problems, as well as being able to devise regularisations for simple inverse problems so that the regularised problem admits solutions.

By the end of the module, students should be able to:

- Understand the difference between forward problems and inverse problem
- Derive the loss functional for specific applications
- Apply the direct method to establish the existence of solutions of regularised inverse problems
- Become competent in using methods from PDE theory
- Apply knowledge to model simple physical systems
- Be in a position to differentiate between noise and bias