17,402 research outputs found
A Discretize-Then-Optimize Approach to PDE-Constrained Shape Optimization
We consider discretized two-dimensional PDE-constrained shape optimization
problems, in which shapes are represented by triangular meshes. Given the
connectivity, the space of admissible vertex positions was recently identified
to be a smooth manifold, termed the manifold of planar triangular meshes. The
latter can be endowed with a complete Riemannian metric, which allows large
mesh deformations without jeopardizing mesh quality; see arXiv:2012.05624.
Nonetheless, the discrete shape optimization problem of finding optimal vertex
positions does not, in general, possess a globally optimal solution. To
overcome this ill-possedness, we propose to add a mesh quality penalization
term to the objective function. This allows us to simultaneously render the
shape optimization problem solvable, and keep track of the mesh quality. We
prove the existence of a globally optimal solution for the penalized problem
and establish first-order necessary optimality conditions independently of the
chosen Riemannian metric.
Because of the independence of the existence results of the choice of the
Riemannian metric, we can numerically study the impact of different Riemannian
metrics on the steepest descent method. We compare the Euclidean, elasticity,
and a novel complete metric, combined with Euclidean and geodesic retractions
to perform the mesh deformation
Why Use Sobolev Metrics on the Space of Curves
We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extremely desirable goal
Empirical geodesic graphs and CAT(k) metrics for data analysis
A methodology is developed for data analysis based on empirically constructed
geodesic metric spaces. For a probability distribution, the length along a path
between two points can be defined as the amount of probability mass accumulated
along the path. The geodesic, then, is the shortest such path and defines a
geodesic metric. Such metrics are transformed in a number of ways to produce
parametrised families of geodesic metric spaces, empirical versions of which
allow computation of intrinsic means and associated measures of dispersion.
These reveal properties of the data, based on geometry, such as those that are
difficult to see from the raw Euclidean distances. Examples of application
include clustering and classification. For certain parameter ranges, the spaces
become CAT(0) spaces and the intrinsic means are unique. In one case, a minimal
spanning tree of a graph based on the data becomes CAT(0). In another, a
so-called "metric cone" construction allows extension to CAT() spaces. It is
shown how to empirically tune the parameters of the metrics, making it possible
to apply them to a number of real cases.Comment: Statistics and Computing, 201
- …