Chapter 6 - Variational Schemes for Bounded Domains

Abstract

This chapter considers variational extensions of the differential method for the problem of generating subdivision rules along the boundaries of finite domains. The key to this approach is to develop a variational version of the original differential problem. Given this variational formulation, we construct an inner product matrix, Ek that plays a role analogous to dk [x] in the differential case. Remarkably, such inner products can be computed exactly for limit functions defined via subdivision, even if these functions are not piecewise polynomial. A matrix relation of the form Ek Sk-1 == Uk-1 Ek-1 is derived, where Uk-1 is an upsampling matrix. This new relation is a matrix version of the finite difference relation dk[x]Sk-1 [x] == 2dk-1 [x2] that characterized subdivision in the differential case. Solutions Sk-1 to this matrix equation yields subdivision schemes that converge to minimizers of the original variational problem. To illustrate these ideas, the problem of constructing subdivision schemes for two types of splines with simple variational definitions: natural cubic splines and bounded harmonic splines are considered. For bounded harmonic splines, no choice of finite element basis leads to locally supported subdivision rules. Instead, the heuristic of choosing the finite element basis with appropriate smoothness and support as small as possible is followed. In practice, this rule appears to lead to a variational scheme whose subdivision rules are highly localized.