Title: Scientific Discovery via Advanced Discretization and Optimization Methods Speaker: Denis Ridzal, Sandia National Laboratories Date/Time: February 12, 2008 at 2:00 - 3:00pm Brief Abstract: Recent advances in the development, analysis, and implementation of compatible discretization and embedded optimization methods are expanding and redefining the nature of questions that can be answered by scientific computing. From the point of view of engineering design, this shift has prompted the fundamental distinction between the optimization of a modest number of parameters within conventional mathematical models, typically the realm of black-box methods, and the discovery of radically new, often counterintuitive design patterns, enabled by recent research in function-space and topology optimization. This talk reviews several numerical methods and tools essential to promoting the science of computing to a means of scientific discovery. A unique aspect of our work is the focus on their theoretical and practical merging into integrated discovery environments. This process brings about new and unexpected mathematical and software challenges that drive our research and call for unorthodox approaches. Numerical methods for the solution of partial differential equations (PDEs) are seen as a foundation of the discovery loop. We will show how our recent theoretical insights in the area of compatible (or mimetic) PDE discretizations have helped guide the development of a next generation of software tools, which in turn prompted further mathematical questions and opened up new research directions. At the same time, the need to solve optimization problems with very large design spaces (for example, those arising in function-space optimization) has motivated the development of optimization algorithms that dynamically manage convergence indicators such as linear solver tolerances. Unlike conventional algorithms, the self-governing optimization schemes can take advantage of very coarse linear representations of the original problem, thereby significantly reducing computational costs. Finally, the theoretical merging of advanced discretization and optimization techniques, as well as their practical use, have helped us gain critical insight into their mathematical interaction. We present a key result indicating that the compatibility of a discretization with respect to a PDE need not imply stable and accurate solution of an optimization problem governed by that PDE, and offer alternatives. CSRI POC: James Stewart, (505) 844-8630 |