Geometric discretization through primal-dual meshes

This thesis introduces new tools for geometric discretization in computer graphics and computational physics. Our work builds upon the duality between weighted triangulations and power diagrams to provide concise, yet expressive discretization of manifolds and differential operators. Our exposition begins with a review of the construction of power diagrams, followed by novel optimization procedures to fully control the local volume and spatial distribution of power cells. Based on this power diagram framework, we develop a new family of discrete differential operators, an effective stippling algorithm, as well as a new fluid solver for Lagrangian particles. We then turn our attention to applications in geometry processing. We show that orthogonal primal-dual meshes augment the notion of local metric in non-flat discrete surfaces. In particular, we introduce a reduced set of coordinates for the construction of orthogonal primal-dual structures of arbitrary topology, and provide alternative metric characterizations through convex optimizations. We finally leverage these novel theoretical contributions to generate well-centered primal-dual meshes, sphere packing on surfaces, and self-supporting triangulations.

A Direct Approach to Robustness Optimization

This dissertation reformulates and streamlines the core tools of robustness analysis for linear time invariant systems using now-standard methods in convex optimization. In particular, robust performance analysis can be formulated as a primal convex optimization in the form of a semidefinite program using a semidefinite representation of a set of Gramians. The same approach with semidefinite programming duality is applied to develop a linear matrix inequality test for well-connectedness analysis, and many existing results such as the Kalman-Yakubovich--Popov lemma and various scaled small gain tests are derived in an elegant fashion. More importantly, unlike the classical approach, a decision variable in this novel optimization framework contains all inner products of signals in a system, and an algorithm for constructing an input and state pair of a system corresponding to the optimal solution of robustness optimization is presented based on this information. This insight may open up new research directions, and as one such example, this dissertation proposes a semidefinite programming relaxation of a cardinality constrained variant of the H ∞ norm, which we term sparse H ∞ analysis, where an adversarial disturbance can use only a limited number of channels. Finally, sparse H ∞ analysis is applied to the linearized swing dynamics in order to detect potential vulnerable spots in power networks.