By Edwin L. Aguirre
Two faculty members from the Department of Mathematical Sciences
, Asst. Profs. Min Hyung Cho
and Nilabja Guha
, have won grants totaling nearly $340,000 from the National Science Foundation (NSF) for research that could advance a range of other disciplines, from high-resolution imaging to economics and finance.
Examples of these models include electromagnetic and gravitational interactions in physics; interactions between atoms, molecules or cells in biology; and more general fractional differential equations and network models in materials science, quantum theory, ecology and social science.
According to Cho, today’s interaction models are becoming more complex, due to the massive amounts of data involved, and require faster, more accurate and more efficient computational methods to handle simulations of large-scale interactions on high-performance computers.
“Our project will introduce novel representations of interaction models that are suitable for accelerated computation, and will design efficient algorithms for evaluating the interactions,” he says.
Cho and his team – which includes Djeneba Kassambara and Jared Weed, who are both Ph.D. students in computational mathematics, and undergraduate math major Mark Vaccaro – are also developing advanced software packages for simulating interaction models as part of the study. The software, which will be released to the scientific community under open-source license agreements, can be used to create remote-sensing and medical imaging devices, as well as nanophotonic devices, such as one based on the reflective, prism-like hair of Saharan silver ants.
“We chose Saharan silver ants to demonstrate numerical simulations of light-wave scattering in layered media embedded with triangular objects,” says Cho. “Our goal is to understand how the dense, triangular hair of Saharan silver ants can enhance the reflection of sunlight and allow them to survive the harsh desert environment.”
Results from this study will help engineers design a passive photonic radiative cooling device for electronics that can reduce their internal temperature when exposed to direct sunlight. “This technology can potentially save on energy consumption, and thus benefit society and the environment,” Cho says.
New Statistical Tools
In his research, Guha will use the Bayesian method – a mathematical technique that applies probabilities or distributions to statistical problems and allows people to revise and update their assumptions after obtaining new experimental data – for analyzing changes in the underlying data-generating process.
According to Guha, almost all dynamic and random processes in nature go through sudden and significant structural changes. Often, the change is expressed in an observable quantity, such as the change in fuel prices, stock indices, crime activities or population density, which responds significantly to a change in an unobservable factor such as an economic downturn, a change in public policy or an outbreak of disease.
“Such change points are routinely observed across all scientific disciplines and applications, including economics, epidemiology, social sciences, cybersecurity, finance and others,” he says.
Guha notes that while there is substantial literature proposing elaborate methods for detecting change points in different settings, there has been limited consideration of change points in what he considers as “hierarchical models with complex dependence or sparsity structures.”
“My research fills this gap with new statistical tools motivated by specific, real-life applications, by developing a theoretical framework while retaining efficiency and usefulness in current applications,” he says.
Through the project, Guha hopes to outline a comprehensive framework for estimating change point in problems that may arise in different applications.
“They can include change points in data with massive dimensions, such as those one would encounter in, say, high-resolution imaging or complex connected graphs,” he says.