Open positions

Open positions for graduate/PhD students

I have a few positions open for graduate students who are interested in working on the mathematical/numerical analysis aspects of biomedical imaging applications, in particular in optical tomographic algorithms for breast cancer detection. If you are interested, you can get the background of this work and what exactly we are presently doing by looking at the papers on my publications page, and in particular by reading the articles listed at the bottom of this page.

With this close and interdisciplinary collaboration between mathematicians and biomedical scientists, we are among the international avantgarde of optical tomograph, and probably ahead of other groups by a year or two. However, in order to stay there, we need to develop our algorithms further.

What we are interested in

We are particularly interested in the following topics:

• Multigrid algorithms for inverse problems: In order to solve the imaging problem, we have to solve three-dimensional diffusion-type equations many times, possibly many thousands of times. We have to do this since the imaging problem is an optimization problem with the PDE as a constraint, and we then form the Schur complement of the optimality condition. In each CG iteration we then have to solve two diffusion PDEs. A multigrid algorithm would surely help to speed this up.
• Regularization strategies: Inverse problems are often ill-posed, i.e. a little bit of measurement noise leads to a significant change in the reproduced image. Tikhonov-type regularization can help here, but we need better strategies.
• Making our algorithms ready for the real world: We've been pretty good at making our programs run even for experimental data and setups, not only mathematical toy problems. However, some aspects are still not robust enough.

What's in it for you

The nice part about this project is that it spans a wide range of topics. You will work mathematically developing numerical algorithms, implement them in a program that already solves real-world problems, use real measurement data, and frequently travel to and work with my collaborators at Baylor Medical College in Houston who are handling the experimental side of this project. You will not have to excel at all these tasks, but if you like the idea of working in an interdisciplinary team, this is your chance.

The project is funded through one grant for about 3 million US$, so there are other people who find this work exciting. (Several other grants are presently pending, and it is our hope that the work on this and related projects will allow us to grow into a whole group of people doing similar things.) We crank out a lot of publications and would be happy to have contributing co-authors. So this is definitely not a dead-end, and you can go on into mathematical, software, or biomedical directions during and after your PhD time.

What you need to know

Ideally, you would be firm in finite element software (the theory of finite elements would be nice, too, but that's not the main focus of this work) and linear algebra, know plenty of C++ (and be using deal.II), have basics in inverse problems, and be fluent in optical physics to talk to our collaborators.

Realistically, you should have a subset of these skills, and be interested and open to learn about the other things. A PhD is a learning phase, so it is understood that you are not expected to know all this up front already.

Interested?

If you are interested, let me know: send me an email and explain to me what prior knowledge you have and why you are interested in this position.

Literature

If you want to get an overview of what exactly we are doing in biomedical imaging, you should take a look at two first of the following papers. Finally, the mathematical aspects of this work are best explained in this preprint below:

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