Optimized refinable enclosures of multivariate polynomial pieces

An enclosure is a two-sided approximation of a uni- or multivariate functio n b is an element of B by a pair of typically simpler functions b(+), b(-) is an element of H not equal B such that b(-) less than or equal to b less than or equal to b(+) over the domain U of interest. Enclosures are optimiz ed by minimizing the width max(U) b(+) - b(-) and refined by enlarging the space R. This paper develops a framework for efficiently computing enclosur es for multivariate polynomials and, in particular, derives piecewise bilin ear enclosures for bivariate polynomials in tensor-product Bezier form. Run time computation of enclosures consists of looking up s < dim B pre-optimiz ed enclosures and linearly combining them with the second differences of b. The width of these enclosures scales by a factor 1/4 under midpoint subdiv ision. (C) 2001 Elsevier Science B.V. All rights reserved.