Lassi Juhani Päivärinta

Inverse problems

Inverse problems constitute an interdisciplinary field of science concentrating on mathematical theory and practical interpretation of indirect measurements. Their applications include medical imaging, atmospheric remote sensing, industrial process monitoring, and astronomical imaging. By using the methods of inverse problems, it is possible to bring modern mathematics to a vast number of applied fields.

Scientific innovations found in mathematical research can be brought to real life applications through modelling. The solutions often combine recent theoretical and computational advances. The exciting but high-risk problems in the research plan of the PI include mathematics of invisibility cloaking, invisible patterns, practical algorithms for imaging, and random quantum systems. Progress in these problems could have a considerable impact in applications such as construction of metamaterials for invisible optic fibre cables, scopes for magnetic resonance imaging (MRI) devices, and early screening for breast cancer.

Results

Are Superman’s supervision and Harry Potter’s invisibility cloak really possible? Our eyes react only to the wavelength of visible light, an electromagnetic wave of an extremely small wavelength. A longer wavelength that could pass through barriers would make a supervision device possible. For an invisibility cloak, one must create a material that makes the waves bend around it, giving you the illusion that there is nothing between the object and the observer.

The limit of visibility and invisibility in specific settings was found. In electrodynamics, waves cannot enter a volume covered with a layer of superconductor, and no information about its interior can be recovered. What happens when the superconductive layer is infinitely thin? The results reveal that there is a certain speed of growth; if the speed is higher, no waves penetrate the body, and if it is slower, the waves permeate the layer, revealing the inside. This phenomenon of tunnelling was never observed in classical physics before.

Impact

There are several fields in science and technology where abstract mathematical results might lead to new discoveries or advances, e.g. in geophysical prospecting using seismic waves, medical imaging (CT, MRI, photo-acoustic tomography, etc.), and investigating the ozone layer using the light of distant stars.

Whenever one wants to interpret indirect measurements, one needs to solve an inverse problem. Since the results are very theoretical in nature, they may find application only decades later with the advancement of technology. For example, John Radon solved the following problem in 1917: if the integrals of the function f over all lines are known, determine the function f. In 1979, A. Cormack and G. Hounsfield received a Nobel Prize in medicine. They showed how x-ray measurements can be used to reconstruct a three-dimensional image of the inside of a human body. Their research marked the beginning of computerised tomography. The method they used was the same as Radon’s reconstruction 60 years earlier.