157 research outputs found
Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis
We propose a strategy for approximating Pareto optimal sets based on the
global analysis framework proposed by Smale (Dynamical systems, New York, 1973,
pp. 531-544). The method highlights and exploits the underlying manifold
structure of the Pareto sets, approximating Pareto optima by means of
simplicial complexes. The method distinguishes the hierarchy between singular
set, Pareto critical set and stable Pareto critical set, and can handle the
problem of superposition of local Pareto fronts, occurring in the general
nonconvex case. Furthermore, a quadratic convergence result in a suitable
set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure
Meshing Deforming Spacetime for Visualization and Analysis
We introduce a novel paradigm that simplifies the visualization and analysis
of data that have a spatially/temporally varying frame of reference. The
primary application driver is tokamak fusion plasma, where science variables
(e.g., density and temperature) are interpolated in a complex magnetic
field-line-following coordinate system. We also see a similar challenge in
rotational fluid mechanics, cosmology, and Lagrangian ocean analysis; such
physics implies a deforming spacetime and induces high complexity in volume
rendering, isosurfacing, and feature tracking, among various visualization
tasks. Without loss of generality, this paper proposes an algorithm to build a
simplicial complex -- a tetrahedral mesh, for the deforming 3D spacetime
derived from two 2D triangular meshes representing consecutive timesteps.
Without introducing new nodes, the resulting mesh fills the gap between 2D
meshes with tetrahedral cells while satisfying given constraints on how nodes
connect between the two input meshes. In the algorithm we first divide the
spacetime into smaller partitions to reduce complexity based on the input
geometries and constraints. We then independently search for a feasible
tessellation of each partition taking nonconvexity into consideration. We
demonstrate multiple use cases for a simplified visualization analysis scheme
with a synthetic case and fusion plasma applications
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
- …