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December 02, 2005
New Math Models Pick Up Where Computers Fail
For all the advances in computer power of recent years, many real-world processes are still so complex that they defy the capability of even the most advanced supercomputers to describe them - and to address such problems, mathematicians are being called for help.
As part of that effort, Oregon State University recently received a $647,000 grant from the U.S. Department of Energy. It's one project in a national, $20-million initiative to have advanced mathematics pick up where sheer computing power is inadequate.
In this project, OSU mathematicians will be trying to model the flow of fluid through a porous medium, such as water through soil. It may sound simple, but in practice this can be so extraordinarily complex that there are still more questions than answers.
"The use of models that are suitable for laboratory experiments to describe processes on the scale of a watershed will bring any computer to its knees," said Ralph Showalter, professor and head of the OSU Department of Mathematics. "We're trying to connect information at the microscale to the big picture, and for that we need new mathematical systems that at least give the computers a chance."
This federal initiative will cover many topics, ranging from the production of energy to pollution cleanup, manufacturing smaller computer chips and making new "nanomaterials." OSU is one of 17 universities and eight Department of Energy participating laboratories, which include many of the most prestigious research and technology institutions in the country.
The program tackles problems of "multi-scale mathematics" - questions that span time scales from fractions of a second to years, and the atomic level to whole watersheds. The problems are so vast they cannot easily be broken down into simpler questions that could be solved using traditional mathematical techniques and models.
OSU's role will be to better describe fluid flow, which might relate to many topics, such as groundwater movement, blood flow through tissue or injection molding processes used in industry.
Even in the study of something as basic as water moving through soil, what you see depends on what window you look through, Showalter said.
"You look through a microscope at a liquid moving for a few moments between soil particles and you observe a certain behavior," he said. "Study the same process at the scale of a bucket or barrel, and longer time scales, and the picture is incredibly different. And for our purposes, we might need to effectively model this process on the scale of a reservoir or a polluted field of groundwater over a period of decades."
Showalter said that conceptually, it's similar to trying to describe the path of a butterfly on a long migration, rather than the up-and-down motion of its body with each cycle of its wings. Existing mathematics is able to do this averaging or "upscaling" in many cases, he said, but not yet in the more complex problems the DOE initiative plans to address.
Primary investigators on the OSU research are Showalter and Malgorzata Peszynska, an assistant professor of mathematics. They will try to create new mathematical models that are able to tackle these topics, and then do analysis and simulation to study their accuracy.
With success, they said, someday the problems may be simplified enough that a supercomputer can handle them.
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