Mathematical Sciences: Numerical Methods for Parameter Estimation in Distributed Parameter Systems

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9303222 Vogel The investigator combines Total Variation based techniques and parallel computing to develop improved parameter estimation methods. This proposal deals with computational methods for parameter estimation in distributed parameter systems. Parameter estimation is the process of determining coefficient functions, typically for partial differential equations, from observations of solutions. The work of this project was made possible by recent advances in scientific computing and image processing. In the field of image processing, the development of so-called ``Total Variation'' based techniques enables the reconstruction of much sharper images than was previously possible. The development of parallel computers allows the solution of complex 2- and 3-dimensional problems, which were previously intractable. Direct applications of research in parameter estimation include the improved reconstruction of images from electromagnetic, thermal, and acoustic data. Radar and sonar are perhaps the most well-known examples. Others include noninvasive medical imaging techniques such as CAT scans, sonograms, and Magnetic Resonance Imaging (MRI). Seismic data processing, where recordings of disturbances which have passed through the earth are used to infer the internal structure of the earth, is yet another important example. Parameter estimation also plays a key role in the development of mathematical models, i.e., mathematical descriptions of physical phenomena. These models are used to describe processes ranging from the transport of toxic chemicals through the soil to the spread of the AIDS epidemic. ***