Algorithms with an edge

Photo by Tom Cogill

By Charles Feigenoff

An artist can take a pencil and sketch a flowing river with a series of lines. An engineer looking at that same river can create a computerized simulation that reflects the physical laws governing the fluid motion in real life. While artists’ tools are up to the task, the formulas or algorithms engineers use have their limitations.

Mathematicians have long used partial differential equations as a tool for creating accurate models. However, the complexity of the resulting mathematical algorithms tends to increase dramatically as more details, such as the actual shape of a riverbed, are taken into account. Artificial simplifications are often made, for instance by assuming that all objects have smooth boundaries, like ellipsoids.

A great deal of effort currently goes into finding new mathematical theories capable of handling more complex data. By combining harmonic analysis with partial differential equations, assistant professor Irina Mitrea is one of a group of mathematicians developing techniques to improve the applicability of the algorithms used to describe phenomena involving objects whose boundaries contain irregularities, such as corners or edges. This is especially important, Mitrea points out, because most realistic physical models involve some sort of roughness such as cracks, microscopic asperity, uneven physical characteristics, and other types of discontinuities.

The focus of Mitrea’s research is the study of spectral properties of the integral operators that naturally enter these types of algorithms. “The goal of my research is to develop ways, not only to guarantee the existence of a solution, but also to find effective means of computing it,” she says. Her work has applications to computer graphics, solid mechanics, thermal radiation, fluid flow, and elasticity.

Mitrea is also a recipient of one of the University’s highly competitive Fund for Excellence in Science and Technology (FEST) awards for young faculty. These are provided by the Office of the Vice President for Research and Graduate Studies and attest to the high standing of her research both inside and outside the University. Her research is currently supported from a grant from the National Science Foundation.