Thomas F. Coleman Professor
Director, Advanced Computing Research Institute (ACRI)
Ph.D. University of Waterloo, 1979

Our research is concerned with the design and understanding of practical and efficient numerical algorithms for continuous optimization problems. Our primary focus has been on the development of algorithms for large-scale optimization: exploitation of scarcity and use of parallelism have been two dominant themes.

Many large-scale optimization problems display scarcity in both second derivative (Hessian) matrices and constraint matrices. We are developing techniques and tools for the efficient calculation of derivative matrices - often the most time-consuming, error-prone task - using the judicious application of methods from the blossoming subfield of automatic differentiation. Moreover, we are investigating how to extend our approach from explicit sparse problems to dense but structured problems. This work is joint with CS Ph.D. student Arun Verma. Structured iterative solution techniques are being developed by visiting Ph.D. student Shahadat Hossain.

A rich source of large-scale structured optimization problems are the related computational areas of discrete optimal control, dynamical systems, and boundary value problems involving systems of ordinary differ-ential equations (and differential-algebraic equations). A common thread in these important areas is that determination of the relevant derivative matrices is enormously expensive. However, it appears that structured application of automatic differentiation techniques can dramatically lower this cost. We are investigating this possibility with Applied Math Ph.D. student Gudbjorn Jonsson.

Various inverse problems, and image-en-hancement problems in particular, can be posed as continuous optimization problems. A popular approach has been to do exactly this using very smooth (least-squares) objective functions. However, we have discovered that significantly better "enhanced'' solutions can often be achieved using piecewise differentiable objective functions and simple constraints. We are developing new algorithms, including parallel approaches, for these problems. Applications range from clearing up noisy blurred photographs to medical ultrasonic imaging. Involved in this work are ACRI researchers Yuying Li and Chunguang Sun and Applied Math Ph.D. student Adrian Mariano.

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