Detalles del proyecto
Descripción
ABSTRACT:
A mathematical model for a natural (e.g., the heart beat of a
human) or man-made process (e.g., the radio wave of a wireless
signal) is a mathematical expression or an algorithm that on
evaluation of system parameters (such as a point in time) yields
a model value (e.g., the amplitude of a wave). Models are
created by understanding the process, which suggests the form
of the expression, and by observing and measuring an actual
process. From those data points the best model is fitted by
a computation. Erich Kaltofen studies both how to fit to data
certain models, such as fractions of sparse polynomials, and
then how to certify that the computation has produced the best
possible model. The algorithms for creation of best fits and
subsequent certification of optimality can be compute-intensive
and require multi-processor computing environments.
Erich Kaltofen and his students and collaborators will design
algorithms for symbolic models such as sparse multivariate
rational functions and formulas with very large and even
parametric exponents. Our algorithms can work with both exact
and approximate data, the latter by hybrid symbolic/numeric
techniques. Computation with floating point scalars requires
a new kind of probabilistic analysis when randomization is
applied, and we will make use of recent results on estimating
the spectra and condition numbers of random matrices. One
application of such randomization is the efficient solution
of highly under- and overdetermined dense linear systems.
A new alternative to error analysis is the exact validation via
symbolic computation of the global optimality of our approximate
solutions. Semidefinite programming and Newton refinement
are used to compute a numerical sum-of-squares representation,
which is converted to an exact rational identity for a nearby
rational lower bound. Since the exact certificates leave no
doubt, the numeric heuristics need not be fully analyzed.
We will search for rationalizations that can validate very
large sums-of-squares and hence apply to large inputs. We will
develop parallel and distribute computing tools for the arising
symbolic and hybrid symbolic-numeric computation tasks.
Estado | Finalizado |
---|---|
Fecha de inicio/Fecha fin | 1/9/08 → 31/8/12 |
Enlaces | https://www.nsf.gov/awardsearch/showAward?AWD_ID=0830347 |
Financiación
- National Science Foundation: USD300,000.00
!!!ASJC Scopus Subject Areas
- Matemáticas (todo)
- Redes de ordenadores y comunicaciones
- Ingeniería eléctrica y electrónica
- Comunicación