Section: Application Domains

Biology

OT methods have been introduced in biology via gradient flows in the Wasserstein metric. Writing certain chemotaxis systems in variational form allowed to prove sharp estimates on the long time asymptotics of the bacterial aggregation. This application had a surprising payback on the theory: it lead to a better understanding and novel proofs of important functional inequalities, like the logarithmic Hardy-Littlewood-Sobolev inequality. Further applications followed, like transport models for species that avoid over-crowding, or cross-diffusion equations for the description of biologic segregation. The inclusion of dissipative cross-diffusion systems into the framework of gradient flows in OT-like metrics appears to be one of the main challenges for the future development of the theory. This extension is not only relevant for biological applications, but is clearly of interest to participants with primary interest in physics or chemistry as well.

Further applications include the connection of OT with game theory, following the idea that many selection processes are based on competition. The ansatz is quite universal and has been used in other areas of the life sciences as well, like for the modeling of personal income in economics.

Another application of our methods is the use of inverse problems in measure spaces for microscopy imaging. The Single Molecule Microscopy Imaging techniques such as PALM  [62] or STORM  [164] have yielded a breakthrough in fluorescence microscopy, improving the typical resolution of conventional microscopes (250 nm) by an order of magnitude (20 nm). These techniques convert the problems of full image reconstruction into a family of sparse spike reconstructions. Our variational methods, which take advantage of the sparsity of the signals to recover, are much more powerful than the usual methods used by biologists for sparse recovery. They promise to release the full potential of PALM and STORM in terms of resolution and speed of acquisition.