Research Projects

Mathematical Biology

Richard Braun, Tobin Driscoll

​Every time you blink, a thin fluid film is left behind that covers the front of your eye. This tear film provides a smooth optical surface,  defense against inflammation and foreign particles, and lubricates the eye’s surface. When the tear film is not healthy, a variety of maladies may occur, including a collection of symptoms known as dry eye. Dry eye may arise from a shortage of tear fluid for each blink, from too much evaporation of the tear film, or a combination of both.If this shortage of tear fluid persists, pain and inflammation of the eye follow. Understanding the dynamics of healthy and unhealthy tear films may help lead to better understanding of the progression and treatment of dry eye and other conditions that afflict millions.

To learn more, visit: http://www.math.udel.edu/~braun/eyes.html​​

Discrete Mathematics

Sebastian Cioaba

​​Graph theory has come a long way since its beginnings with the work of Euler in the 1736 solving puzzles involving bridges, islands and rivers in the city of Köningsberg, Prussia (now Kaliningrad, Russia) to its modern days development as a robust discipline of mathematics and computer science having important interactions with other areas of mathematics and science and many real-life applications such as web-page ranking, clustering, error-correcting codes and network design.

Numerical Analysis and Scientific Computing, Scattering and Inverse Scattering

Francisco-Javier Sayas

​Our group deals with different numerical schemes for the scattering and propagation of transient and time-harmonic waves. We combine the creation of new simulation tools for waves with their numerical analysis. Among our favorite methods there are integral equation techniques (Galerkin boundary elements, Dirac delta methods, convolution quadrature), volume formulations (finite element methods and hybridizable discontinuous Galerkin methods), and different forms to couple both. Applications to direct and inverse problems for propagation of acoustic waves and their interaction with mechanical vibrations are part of our goals.​

Mathematical Biology

Robert Gilbert, Philippe Guyenne, Yvonne Ou, In collaboration with George Hsiao

We develop a mathematical background of ultrasound methodology for the diagnosis of bone brittleness. Such research is usually connected to an evaluation of the microstructure of cancellous/ trabecular bone.  We construct  artificial CT scans to enlarge our catalogue of bone samples to be used for the inverse problem associated with unknown bone parameters. This entails devising an efficient numerical scheme to produce realistic, orthotropic trabecular bone structures.   Our work on bone remodeling involves investigating the entry of 1,25D through the lipid bi-layer membrane of the stem cell and its entry and release into the cytoplasm.  This requires modeling the cell's membrane enclosure of 1,25D by adhering to binding sites to form a vesicle, the separation of the vesicle from the lipid bilayer and entry of a coated vesicle into the cytoplasm.  In order to understand the mechanical processes in bone remodeling we ned to investigate the flow of ions in vivo bone.  To this end, we use homogenization/mixture theory to derive the acoustic response for an in vivo model of wet bone.

Scattering and Inverse Scattering

David Colton, Peter Monk

​The field of inverse scattering theory has been a particularly active field in applied mathematics for the past thirty years. The aim of research in this field has been to not only detect but also to identify unknown objects through the use of acoustic, electromagnetic or elastic waves. Although the success of such techniques as ultrasound and x-ray tomography in medical imaging has been truly spectacular, progress has lagged in other areas of application which are forced to rely on different modalities using limited data in complex environments. Indeed it is often said that "Target identification is the great unsolved problem.We detect almost everything, we identify nothing". Until a few years ago, all existing algorithms for target identification were based on either a linearizing weak scattering approximation or on the use of nonlinear optimization techniques. However, as the demands of imaging increased, it became clear that incorrect model assumptions inherent in weak scattering approximations imposed severe limitations on when reliable reconstructions were possible. On the other hand, it was also realized that for many practical applications nonlinear optimization techniques required information that is in general not available. Hence, in recent years, alternative methods for imaging have been developed that avoid incorrect model assumptions but, as opposed to nonlinear optimization techniques, also avoid strong a priori assumptions about the scattering object. Such methods come under the general title of qualitative methods in inverse scattering theory and are based on the development and use of linear sampling methods and transmission eigenvalues, both of which were discovered and developed here at Delaware in collaboration with researchers in Germany and France. In particular, such methods are based on solving a linear integral equation for a range of "sampling points" and frequencies which are not "transmission eigenvalues" and lead to an approximation to the shape of the scattering object together with limited information about the material properties of the scatterer. Such an approach is remarkable since the inverse scattering problem itself is nonlinear but no linearizing assumptions have been made in the derivation of the above mentioned linear integral equation. This research project with AFOSR is devoted to the further development and application of this new approach in inverse scattering theory.