UD professors receive federal grants to support science and engineering advancements

Dominique Guillot, an associate professor in the University of Delaware's Department of Mathematical Sciences, was recently awarded the grant "Polynomial approximation in spaces of analytic functions" by the NSF Analysis program. The main goal of the research is to construct approximations of complicated functions using simpler ones, a problem of fundamental importance in many fields of science and engineering. Such approximations are used, for example, to simplify calculations and make computer algorithms faster and more efficient. This is particularly important for real-time systems (flight control systems, medical devices, smartphones, sensors, etc.) where numerical calculations need to be performed as quickly as possible.

Polynomials play a crucial role in constructing such approximations. They are used across many fields due to their versatility in simplifying complex functions and providing accurate estimations. In many instances, however, building explicit polynomial approximation schemes remains an open problem. In the project, Guillot studies new methods to construct polynomial approximations for functions belonging to well-known spaces of functions. The research advances our knowledge of how to efficiently build such approximations and also connects the problem to other fields in mathematics such as matrix analysis and operator theory. The project also supports education by training one Ph.D. student to become a new expert in the field. Guillot also plans to engage undergraduate and high school students in the research through various outreach activities.

Jingmei Qiu, a Unidel Professor of Mathematics at UD, is part of a six-institution consortium that received a grant from the Department of Defense’s (DoD) highly competitive Multidisciplinary University Research Initiative (MURI) program to study "Tensor Approaches for Simulating Kinetic Systems (TASKS)."

Modern challenges demand a deep understanding of high-dimensional complex systems characterized by non-stationary partial differential equations (PDEs). Examples include the kinetic theory in modeling probability density functions arising from describing particle distributions in nonequilibrium plasma. Tensor network approximation stands out as a promising method to combat the curse of dimensionality, which causes systems to become unmanageably large as the number of variables increases.

The team proposes to systematically develop an interlocking research framework to advance tensor network methods for simulating high-dimensional kinetic models. The underlying methods are relevant to a wide range of applications, such as clean energy systems and hypersonic flight.