Quasi-Newton Exploration of Implicitly Constrained Manifolds

• Main Author: Tang, Chengcheng
• Other Authors: Pottmann, Helmut
• Language: en
• Published: 2012
• Online Access: Tang, C. (2011). Quasi-Newton Exploration of Implicitly Constrained Manifolds. KAUST Research Repository. https://doi.org/10.25781/KAUST-C94C8; http://hdl.handle.net/10754/209387

A family of methods for the efficient update of second order approximations of a constraint manifold is proposed in this thesis. The concept of such a constraint manifold corresponds to an abstract space prescribed by implicit nonlinear constraints, which can be a set of objects satisfying certain desired properties. This concept has a variety of applications, and it has been successfully introduced to fabrication-aware architectural design as a shape space consisting of all the implementable designs. The local approximation of such a manifold can be first order, in the tangent space, or second order, in the osculating surface, with higher precision. For a nonlinearly constrained manifold with rather high dimension and codimension, the computation of second order approximants (osculants) is time consuming. In this thesis, a type of so-called quasi-Newton manifold exploration methods which approximate the new osculants by updating the ones of a neighbor point by 1st-order information is introduced. The procedures are discussed in detail and the examples implemented to visually verify the methods are illustrated.