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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.
 Understanding 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.
 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."
 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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