HPCwire: Solving Ill-Posed Problems in Scientific Computing

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May 05, 2006

Solving Ill-Posed Problems in Scientific Computing

, NCSA Access magazine editor

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It's only fair that a puzzle have only one solution, right? That's the great thing about a crossword or one of those Sudoku number games that the lifestyle pages can't stop talking about right now. There's only one way to fill in the boxes and get a complete answer.

Well, nobody said science was always fair. In what are known as ill-posed problems, there is no unique solution. A slight change in the data fed into the system of functions that rule a given ill-posed problem can produce a large, unpredictable change in the results.

"In the late 1800s, a scientist named Hadamard proposed that ill-posed problems didn't exist [or those that did weren't scientifically significant]. He was totally wrong. They're everywhere," says Rebecca Hartman-Baker, who recently completed her PhD in computer science at the University of Illinois at Urbana-Champaign.

They're found, for example, in medical imaging, financial modeling, environmental modeling, and astronomy. Though Hartman-Baker's PhD thesis focused on an ill-posed problem found in the field of geoprospecting, the approach applies to any of those fields.

"Any ill-posed problem for which you have an educated guess of where to start could use this," says Hartman-Baker, now a post-doc at Oak Ridge National Laboratory. "As a scientific computing type, the thing that brings me the most joy is to contribute to real-world scientific problems through my little thing. I want to get people who didn't know they needed to be involved in computer science to be in it."

Making hundreds of runs on NCSA's Platinum and Tungsten clusters, she developed a selection method for choosing the parameters that go into the problem and an optimization method for finding the ideal result among a sea of solutions.

Geoprospecting typically involves placing transmitters and receivers deep in the earth. The transmitter projects electromagnetic energy, or in some cases sound, that is picked up by the receiver miles away. The electromagnetic energy is altered in transit, based on the conductivity of the rock, water, oil, or other materials in the ground. From the data collected by the receiver, researchers can deduce what lies between it and the transmitter.

The challenge is -- and this is where the ill-posed problem comes in -- that different sizes, shapes, and orientations of underground deposits can produce the same data profile.

"Basically, you're trying to find the size and shape of an ellipse [that represents the deposit that a geoprospector might be targeting, such as oil]. Where the center is at. The rotation or orientation. How you do that is really indirect -- kinda backhanded," Hartman-Baker says.

The traditional method of solving this sort of ill-posed problem, known as Tikhonov regularization, gives a blurry picture of this ellipse. It stabilizes the problem around a single solution by adding things like smoothness constraints to the functions. But in situations like this, researchers tend to prefer distinct boundaries. To get these boundaries, Hartman-Baker proposed another class of stabilizing, known as selection methods. With selection methods, the solution is limited to some reasonable set of possible solutions, and the parameters fed into the problem are reduced to a manageable number (about 10 in the case of Hartman-Baker's work). These decisions limit the computational expense of solving the problem and provide a distinct ellipse.

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