University of Georgia
Mathematics Department Colloquium
John Lavery
Mathematical and Computer Sciences Division
Army Research Office
Research Triangle Park, NC
April 27 1999
Shape-preserving, Multiresolution, Piecewise-polynomial
Geometric Modeling without Human Interaction
Abstract: The Applied Analysis Program at the Army Research Office
supports research in mathematical analysis for advanced solid
materials for structures, armor and sensors, soil and granular
materials, fluid flow for chem/bio defense and diesel combustion,
photonic bandgaps, nonlinear dynamics for chaotic communication
devices, inverse scattering for landmine detection, data fusion and
modeling of irregular surfaces.
User-input-free, shape-preserving interpolation and approximation are
long-standing goals in geometric modeling. We discuss here a new class of
polynomial interpolating and approximating splines that preserve the shape
both of smooth data and of data with abrupt changes in magnitude or spacing.
The theoretical framework for these splines is straightforward and complete.
They have no ad hoc components: One convex nonlinear minimization principle
works for all cases (with only a "regularization" parameter needing to be
chosen). These splines do not require constraints, penalties, a posteriori
filtering or interaction with the user. Univariate and multivariate cases
and cubic and higher-degree splines are treated in a unified framework.
Finally, these splines can be implemented using efficient interior-point
methods for convex nonlinear optimization.