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Design of Hermite Subdivision Schemes Aided by Spectral
Radius Optimization

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Bin Han, Michael L. Overton, and Thomas P.-Y. Yu

## Abstract

We present a method for constructing
multivariate refinable Hermite interpolants and their associated
subdivision algorithms based on a combination of analytical and
numerical approaches. Being the limit of a linear iterative
procedure, the critical $L^2$ Sobolev smoothness of a refinable
Hermite interpolant is given by the spectral radius of a matrix
dependent upon the {\it refinement mask}. The design question:
given certain constraints (support size, symmetry type, refinement
pattern etc.), how can one choose the refinement mask so that the
resulting refinable function has optimal smoothness? This question
naturally gives rise to a spectral radius optimization problem.
In general, the objective function is not convex, and may not be
differentiable, or even Lipschitz, at a local minimizer.
Nonetheless, a recently developed robust solver for nonsmooth
optimization problems may be applied to find local minimizers of
the spectral radius objective function. In fact, we find that in
specific cases that are of particular interest in the present
context, the objective function is smooth at local minimizers and
may be accurately minimized by standard techniques. We present two
necessary mathematical tricks that make the method practical: (i)
compression of matrix size based on symmetry; (ii) efficient
computation of gradients of the objective function. We conclude by
reporting some computational results.

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