# PhD Projects

Projects for incoming students to consider taking on

## Infinite-Dimensional Compressed Sensing and Frames in Medical Imaging

### Supervisors:

Anders Hansen

### Description:

Description: In Magnetic Resonance Imaging (MRI) the mathematical reconstruction problem is to reconstruct a function g as accurately as possible given only a limited number of point samples of its continuous Fourier spectrum. This definition holds, to a large degree, for many inverse problems. This project aims to combine the usage of function frames with the theory of Compressed Sensing in order to substantially improve the reconstruction of the continuous function g - rather than the traditional discrete version - and our results so far have shown tremendous potential. Given that an infinite-dimensional (i.e. continuous) model is used, both the analysis and computations are more difficult than the finite dimensional (i.e. discrete) methods. The project can be steered towards pure or applied to one's liking, and there are many open questions in both areas.

## Can everything be computed? - On the Solvability Complexity Index and Towers of Algorithms

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### Supervisors:

Anders Hansen, Bogdan Roman

### Description:

Description: One of the most famous computing problem is finding the zeros of a polynomial using only finitely many arithmetic operations and radicals of the coefficients. The negative answer to this problem for the quintic has had an enormous impact on the theory of computations. One had to resort to approximations and then limits. This was the birth of computational mathematics. This project addresses some of the fundamental barriers in the theory of computations via the concepts of towers of algorithms and the Solvability Complexity Index (SCI), the latter being the smallest number of limits needed to compute a desired quantity. Fascinatingly, there are several fundamental problems in computational mathematics that are still open, e.g. computing spectra of SchrÃ¶dinger operators. This rather theoretical project, relying on functional analysis and operator theory, will be about such fundamental problems in computational mathematics.