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Structure-Preserving Algorithms and Model Reduction in the Natural Sciences

A multi-disciplinary team of researchers from the California Institute of Technology and the University of California at Santa Barbara, Davis, and San Diego, is utilizing advanced theoretical and computational techniques for accurate simulation and modeling of various physical systems—ranging from ocean and atmospheric flows to chemical reactions, elastic structures, and the trajectories of spacecraft.

In addition to Caltech, the effort funded by the National Science Foundation has included engineers, scientists, and mathematicians from the Santa Barbara and Davis campuses of the University of California, the Scripps Institute of Oceanography, and the Jet Propulsion Laboratory.

Image of five snapshots from a high velocity sphere plate impact simulationUnderstanding and controlling many physical systems typically requires numerical simulations of multi-scale dynamics that occur over a wide range of time and space scales. Generally it is prohibitively expensive—requiring a great deal of computational time—to simulate all of the scales involved in many naturally occurring phenomena. Scales outside the computable range must be described by a reduced model.

The leader of the research team, Jerrold E. Marsden, professor of control and dynamical systems at Caltech, says the KDI grant project has had three main thrusts — accurate simulation of physical systems, model reduction, and optimization.

In the accurate simulation part, he says, what we developed are called variational integrators. What this means is that physical systems are described typically by minimization principles. This goes by the name of Hamilton's principle. So lots of mechanical systems in nature are governed by these variational principles. The basic idea of these variational integrators is to develop discretization methods, based on discretizing the variational principle rather than discretizing the equation, which is the usual approach people take in numerical analysis. This does result in a discretization of the equation, but it's what we call a structured discretization of it that preserves the basic properties of mechanics, such as conservation models of various sorts, and it builds them right into the algorithm.

Graphical representation of the performance of the variational integration.Marsden explains that variational integrators are "computer algorithms for computing the motion of physical systems. They are algorithms of a particular sort that are obtained not by using the usual discretization methods on the equations, but rather by using discretization methods on the underlying variational principle. Virtually all physical systems are governed by what physicists call variational principles, whereby something is optimized when a system moves from here to there—the action integral is optimized. So, instead of discretizing the equations, like f = ma, one discretizes the action principle."

Model reduction, according to Marsden, is another important computational tool because "many systems—fluid systems, chemical reactions, other complicated systems—have huge numbers of variables. You want to get out the key aspects of these systems without having to solve some of the equations, because often that's just not feasible. So model reduction is a technique for extracting key information, key low-dimensional models from otherwise complicated and high-dimensional systems."

Image depicting genesis orbit.In the optimization area, the researchers applied advanced computational tools to a U.S. spacecraft called the Genesis Discovery mission, which is designed to collect samples of solar matter for analysis in terrestrial laboratories. "We studied the optimal use of fuel for ensuring and guaranteeing that the spacecraft would get to where it's supposed to go, even in the presence of various errors, such as launch errors," Marsden says. "That was one of the new ideas and new directions that was spawned by this grant. It was a very nice example of collaborative, interdisciplinary effort."

Other practical applications of the research findings are in mechanical engineering, where model reduction applies to vibrations and oscillations of elastic structures, and in the chemical engineering field.

Senior researchers working with Marsden at Caltech have included Professor Linda Petzold, a computational science and engineering expert at UC Santa Barbara; Professor Steve Shkoller, a mathematician at UC Davis; and Dr. Martin Lo, a space scientist at JPL. The team has been augmented by postdoctoral fellows and graduate students.


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