Topics for projects - Markus Grasmair

I am mainly working in the theory of inverse problems. Below you will be able to find in due time some concrete topics for a master projects. Feel free to come with your own proposals, though. In any case, if you are potentially interested in a thesis in inverse problems, please take contact with me, so that we can arrange a meeting.

Hvis du er interessert i en bacheloroppgave, send meg en e-post, så vi kan avtale et møte og diskutere mulige tema.

Inverse problems are typically concerned with the solution of operator equations (usually integral or differential operators) where the solution is extremely sensitive with respect to noise (data and/or modeling errors). A classical example is the inversion of the Radon transform, which is the basis of computerised tomography (CT). Another example is deblurring, which is required for obtaining clear images both for imaging at the largest scales in astronomy and the smallest scales in microscopy. Another class of examples is concerned with parameter identification problems for PDEs where one wants to reconstruct some parameters (e.g. the heat source or a spatially varying conductivity) from the solution of a PDE.

Abstractly, an inverse problem can be formulated as the problem of solving an equation \(F(u) = v^\delta\) for \(u\), given some noisy measurement data \(v^\delta\) with noise level \(\delta\). Here \(F \colon U \to V\) is a possibly non-linear mapping between the Hilbert or Banach spaces \(U\) and \(V\). Because of the ill-posedness of the problem (that is, discontinuous dependence of the solution on the data \(v^\delta\)), a direct solution does not make sense. Instead, it is necessary to introduce some type of regularisation in the solution process that is based on prior knowledge of qualitative properties of the true solution \(u^\dagger\) of the problem.

Roughly spoken, a (convergent) regularisation method for the solution of \(F(u) = v^\delta\) is a parameter dependent mapping \[ G \colon V \times \mathbb{R}_{> 0} \to U \] with the following properties:

• \(G\) is continuous with respect to suitable topologies on \(V\) and \(U\).
• There exists a parameter choice rule \(\alpha \colon \mathbb{R}_{>0} \times V \to \mathbb{R}_{>0}\) such that the following holds: Whenever \(v \in V\), \(\delta \to 0^+\), and \(v^\delta \in V\) satisfy \(\lVert v^\delta-v\rVert < \delta\), we have that \(G(v^\delta,\alpha(\delta,v^\delta)) \to u^\dagger\), where \(u^\dagger \in U\) is a suitable solution of the equation \(F(u) = v\).

Amongst the most prominent regularisation methods are the following:

• Tikhonov regularisation: Here \(G(v,\alpha)\) is any selection of the minimisers of the optimisation problem \[\min_{u \in U} \frac{1}{2}\lVert F(u) - v^\delta\rVert^2 + \alpha\mathcal{R}(u).\] Under quite general assumptions on \(F\) and the regularisation function \(\mathcal{R} \colon U \to \mathbb{R}_{\ge 0} \cup \{+\infty\}\) this can be shown to be a regularisation with, for instance, the parameter choice rule \(\alpha(\delta,v) := \delta\).
• Landweber iteration: Here we define \(G(v,\alpha)\) as the \(n^{\text{th}}\) step of the iteration \[u_{k+1} = u_k - F'(u_k)^*(F(u_k) - v^\delta)\] with (e.g.) initialisation \(u_0 = 0\), where \(n = \lceil 1/\alpha \rceil\). Again, this can be shown to be a regularisation method with, for instance, the parameter choice rule \(\alpha = \delta\).

Given a convergent regularisation method, one can then ask the question of how fast \(G(v^\delta,\alpha)\) converges to the "true" solution \(u^\dagger\) as the noise level \(\delta\) tends to \(0\). Clearly, this will depend on the concrete parameter choice rule. However, one can in addition show that, apart from trivial situations, the speed will also depend on properties of the true solution (usually termed smoothness with respect to the regularisation method), and the convergence will be arbitrarily slow for low smoothness solutions. The study of these convergence rates and smoothness classes is the main focus of my research.

A possible master thesis in this topic would consist in the first half (project thesis for students in Industrial Mathematics) of a literature study of the necessary background in regularisation theory. The second half (actual master thesis for Industrial Mathematics) can then follow several possible directions. Possibilities include, but are not limited to, the study of a concrete inverse problem or a concrete regularisation method—of particular interest can be parameter identification problems for PDEs and different non-smooth regularisation methods enforcing different types of sparsity in the solutions—, the geometric interpretation of smoothness classes for total variation based regularisation methods, the study of convergence rates in the low smoothness regime where the true solution is not contained in the range of the regularisation method, or the numerical verification of convergence rates results for a specific regularisation method.

Prerequisite knowledge includes the courses TMA4230 - Functional Analysis and TMA4225 - Foundations of Analysis or similar. Previous knowledge of inverse problems is a distinct advantage, but can be gained during a specialisation course. Depending on the choice of direction for the actual master thesis, additional courses like TMA4305 Partial Differential Equations, TMA4220 Finite Elements, TMA4205 Numerical Linear Algebra, TMA4170 Fourier Analysis, or TMA4180 Optimisation 1 may be advantageous.

I am also open for suggestions of other possible topics. Please take contact with me to arrange a meeting if you are potentially interested in writing a master thesis or bachelor thesis under my supervision.