As computer hardware technology becomes more powerful, there is a corresponding growth in the demand for more efficient algorithms to solve large-scale scientific problems. My research is on the design and analysis of such algorithms. One area of interest is algorithms and complexity issues for numerical optimization. For example, some recent work focuses on linear programming. With Yinyu Ye of the University of Iowa, we have developed a new kind of "interior point" method that converges in a finite number of steps, independent of the numerical data in the objective function and right-hand side vector. This new method appears to be efficient in practice and improves on the known complexity of linear programming.
Also, PhD student Patty Hough and I have developed a new algorithm for solving the system of linear equations arising in interior point methods. The new algorithm has a guaranteed accuracy bound that was not possible for prior algorithms.
I am also interested in computational aspects of differential equations. With Scott Mitchell of Sandia I developed an algorithm for mesh generation for finite element analysis. Unlike previous algorithms, the new algorithm is based on computational geometry and hence comes with certain theoretical guarantees about robustness. Implementation of the algorithm is available via anonymous FTP. Research on other geometric and computational aspects of differential equations continues, including work on the boundary element method with PhD student Dave Bond.
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