joint work with Behrend Heeren, Klaus Hildebrandt, Martin Rumpf
We consider Nonlinear Rotation-Invariant Coordinates (NRIC) representing triangle meshes with fixed combinatorics as a vector stacking all edge lengths and dihedral angles. Previously, conditions for the existence of vertex positions matching given NRIC have been established. We develop the machinery needed to use NRIC for solving geometric optimization problems and introduce a fast and robust algorithm that reconstructs vertex positions from close-to integrable NRIC. Comparisons to alternatives indicate that NRIC-based optimization is particularly effective for near-isometric problems.
joint work with Maurício Louzeiro, Leandro Prudente
The steepest descent method for multiobjective optimization on Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. The aim of the paper is twofold. Firstly, an asymptotic analysis of the method is presented. The second aim is to present iteration-complexity bounds for the method with these three stepsizes. In addition, some examples are presented to emphasize the importance of working in this new context. Numerical experiments are provided to illustrate the effectiveness of the method in this new setting and certify the obtained theoretical results.