Optimized refinable enclosures of multivariate polynomial pieces

An enclosure is a two-sided approximation of a unior multivariate function by a pair of typically simpler functions such that over the domain of interest. Enclosures are optimized by minimizing the width and refined by enlarging the space . This paper develops a framework for efficiently computing enclosures for multivariate polynomials and, in particular, derives piecewise bilinear enclosures for bivariate polynomials in tensor-product Bézier form. Runtime computation of enclosures consists of looking up dim pre-optimized enclosures and linearly combining them with the second differences of . The width of these enclosures scales by a factor 1/4 under midpoint subdivision.