Henrik Garde

Project title

Complex Calderón Problem: Interior Admittivity from Partial Surface Measurements

What is your project about?

Calderón’s problem is a famous and challenging mathematical problem. Based on boundary electrical measurements of an object (or a person), the aim is to reconstruct a three-dimensional image of the interior spatial structure. If measurements only pertain to a small subset of the object’s surface, and if a complex function must be determined relating to both the conductivity and the permittivity, then no mathematically proven reconstruction method yet exists.

The project takes a completely new approach, by transforming the measurement data using so-called functional calculus. This will be used to improve the modulus of continuity and reduce non-linearity, which are the barriers to proving the applicability of optimisation-based methods. Additionally, it is investigated to what extent the geometric shape and location of complex perturbations can be determined from the surface measurements.

How did you become interested in your particular field of research?

From childhood I have been curious about solving simple mathematical problems and teaching myself to program. My main interest was to find my own methods of solution, which requires a certain steadfastness. Mathematics has always seemed like the perfect combination of a creative subject which at the same time is very structured.

During my university education I developed a strong interest in harmonic analysis, inspired by my Bachelor supervisor Ole Christensen. My interest in inverse problems and partial differential equations was only sparked when writing my Master’s thesis supervised by Kim Knudsen, and it still remains my main research topic.

Subsequently, I have also developed a general interest in approximation theory, in particular related to models in mathematical physics, where harmonic analysis has once again been useful.

What are the scientific challenges and perspectives in your project?

The project will develop a new area of inverse problems research, combining several different branches of mathematics: Functional calculus and spectral theory which is mainly used in mathematical physics, properties of boundary operators from partial differential equations, and theory for optimisation-based methods on infinite-dimensional Banach spaces. All of this with the purpose of giving mathematical guarantees within e.g. medical imaging.

Much intuition can be gained from numerical computations. However, if the aim is to develop a general theory and give mathematical guarantees that the methods should work every time, this is only achievable using pen, paper, and persistence.

What is your estimate of the impact, which your project may have to society in the long term?

The project spans several methodologies that usually are part of pure mathematics, however here it will result in direct applications for medical imaging and materials science. In my opinion, this is exactly what an applied mathematics project should comprise: Investigating a concrete and practically relevant problem, where no easy proof exists, but requires the development of completely new mathematics.

Besides the direct applications of the research, my hope is that the project will inspire talented students to see the importance of applied mathematical analysis and strengthen this research area in general throughout Denmark.

Which impact do you expect the Sapere Aude programme will have on your career as a researcher?

It is an excellent opportunity to dedicate several years to an ambitious project solely based on my own research ideas. A project where it is necessary to have the aid of two young talented postdocs, and a close collaboration with my international colleagues.

It is a significant milestone in my research career, and a clear recognition of the quality and time that I have invested in my research. I am certain that this will give a lasting boost to applied mathematical analysis research at my department.

Background and personal life

I spend a lot of time on my research, and I actually prefer doing research in the evening when my thoughts should be elsewhere, or on longer research stays abroad. It can seem like an obsession, that only halts when I have tested the ideas that have materialised during the day, sometimes leading to rather late evenings.

When I am not thinking about my research, I enjoy visiting family on Zealand as frequently as possible, which is a fresh break from my daily life. I love to dine out with family and friends, and often experiment with new restaurants.

I consider myself a very creative thinker. This is also expressed in my sometimes peculiar taste for films, fiction, and humour.