Gathering echoes: Finding the hidden with math

Gathering echoes
Finding the hidden with math

BY PATRICK KURP
Special to the Rice News

Because Liliana Borcea specializes in inverse problems, a one-size-fits-most branch of applied mathematics, her work can show up in the most unlikely places.

LILIANA BORCEA

For instance, soldiers looking for land mines might rely on the algorithms she has devised. So might doctors examining unborn children for birth defects, engineers looking for structural flaws in concrete roads and bridges, oil companies locating underground deposits, and aerial reconnaissance teams on the lookout for troops and equipment in a dense forest. What do these things have in common?

”All of them are concealed,” said Borcea, professor of computational and applied mathematics (CAAM) at Rice. ”You cannot reach them directly, so you gather echoes. You send signals and you measure echoes.”

With radar, sonar and ultrasound, scientists use the inverse method to gather information about inaccessible or obscured regions by probing from the outside. It’s like working backward — deducing a cause from an effect.

”What is new and challenging is that we study these problems in cluttered media that occur naturally in applications,” Borcea said. ”The Earth, for example, has a very complicated, heterogeneous structure. Aging concrete is highly heterogeneous. Routinely, this clutter has been ignored, which gives unreliable results. Our work studies the effect of the small-scale heterogeneities on the imaging.”

Borcea graduated with a bachelor’s degree in applied physics from the University of Bucharest, Romania, in 1987, and she and her husband immigrated to the United States in 1990. They settled in California, and Borcea earned her master’s degree in scientific computing and computational mathematics at Stanford University in 1994 and her Ph.D. in the same field two years later. Her dissertation was titled ”Direct and Inverse Problems for Transport in High Contrast Media.”

Between 1996 and 1997, Borcea served as a National Science Foundation (NSF) postdoctoral fellow in the Applied Mathematics Department of the California Institute of Technology. In 1996, she joined the CAAM department at Rice as an assistant professor and was promoted to associate professor in 2001.
 
In collaboration with George Papanicolaou, the Robert Grimmett Professor in Mathematics at Stanford, and Chrysoula Tsogka, assistant professor of mathematics at the University of Chicago, she has devised a method for imaging in clutter called adaptive coherent interferometry. This work adapts methods from stochastic analysis, statistical signal processing, image processing, numerical wave propagation and optimization to obtain robust images in clutter.

Borcea’s research has been consistently funded by the NSF. In August 2005, her work on imaging in random media was selected by the NSF Applied Mathematics Program as a research highlight and used by the NSF in its budget request to Congress. She also received funding from the Office of Naval Research.

Borcea has collaborated with William Symes, the Noah Harding Professor of Computational and Applied Mathematics and associate director of the Rice Center for Computational Geophysics. They conduct research into seismic imaging for use in exploration geophysics. She has also worked on land mine imaging with Waymond Scott, professor of electrical and computer engineering at Georgia Tech.

”We are dealing with clutter and what we want is resolution. We want to develop reliable statistical methods for looking into the Earth, which is a very complicated medium,” said Borcea, who serves on the international advisory panel of the journal Inverse Problems. ”I know this sounds very abstract and difficult, but the work I am doing is important because it has so many applications in the real world where real people live.”