Mathematics

My research is motivated by:

Image processing and inverse problems Imaging devices used in science, engineering and medicine, do not directly produce images. They produce measurements that have to be “inverted” to produce images. Often the data available is partial and corrupted. Therefore advanced mathematical models are necessary to remove the corruption and add missing information based on our visual or physical expectations of what the solution should be.

Data science Interpreting images by human eyes is hard work. Therefore we would like to automate that task, let the computer try to tell what objects are in an image. We would like to do the same with many other types of data. Data science produces mathematical models to extract this interpretation or other interesting information out of vast swathes of data.

Most work that I do is in the following theoretical and generally applicable areas of mathematics:

Optimisation methods The above image processing and data science tasks often involving finding the minimum or a maximum of a mathematical function, like finding the lowest point of a valley, or the peak of a mountain. Sometimes one wants to find the saddle point, which also has its geographical analog. However, the functions of interest in these problems do not live in our three-dimensional world, like mountains and valleys. The dimensions are in the millions and milliards. It is not easy to find that valley in such a high-dimensional space, especially when the functions involve are non-smooth: there are cliffs and other difficult terrain. I work on algorithms that can nevertheless do this, fast.

Variational analysis In order to develop those algorithms, and to study the stability of solutions to the models themselves under perturbations of data—important for reliability of the models—very careful mathematical analysis is required. When the functions involved are non-smooth, as our functions are, we have to do some involved set-valued analysis, also called variational analysis.

Geometric measure theory What about the images that we have reconstructed with our algorithms? Do they make any sense? Do they contain artefacts? Such artefacts would be very undesirable, as they can affect the interpretation of the image. These properties can be studied with the tools of geometric measure theory.

If you wish to learn more about these topics, please see my publications.