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This work considers initial value problems (IVPs)
for ordinary differential equations (ODEs) where
some of the data is uncertain and given by intervals
as is the case in many areas of science and
engineering. Interval methods provide a way to
approach these problems but they raise fundamental
challenges in obtaining high accuracy and low
computation costs. This work introduces a constraint
satisfaction approach to these problems which
enhances traditional interval methods with a pruning
step based on a global relaxation of the ODE. The
relaxation uses Hermite interpolation polynomials
and enclosures of their error terms to approximate
the ODE. Our work also shows how to find an
evaluation time for the relaxation that minimizes
its local error. Theoretical and experimental
results show that the approach produces significant
improvements in accuracy over the best interval
methods for the same computation costs. The results
also indicate that the new algorithm should be
significantly faster when the ODE contains many
operations.
for ordinary differential equations (ODEs) where
some of the data is uncertain and given by intervals
as is the case in many areas of science and
engineering. Interval methods provide a way to
approach these problems but they raise fundamental
challenges in obtaining high accuracy and low
computation costs. This work introduces a constraint
satisfaction approach to these problems which
enhances traditional interval methods with a pruning
step based on a global relaxation of the ODE. The
relaxation uses Hermite interpolation polynomials
and enclosures of their error terms to approximate
the ODE. Our work also shows how to find an
evaluation time for the relaxation that minimizes
its local error. Theoretical and experimental
results show that the approach produces significant
improvements in accuracy over the best interval
methods for the same computation costs. The results
also indicate that the new algorithm should be
significantly faster when the ODE contains many
operations.