32,227 research outputs found
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method
Computational tools for characterizing electromagnetic scattering from
objects with uncertain shapes are needed in various applications ranging from
remote sensing at microwave frequencies to Raman spectroscopy at optical
frequencies. Often, such computational tools use the Monte Carlo (MC) method to
sample a parametric space describing geometric uncertainties. For each sample,
which corresponds to a realization of the geometry, a deterministic
electromagnetic solver computes the scattered fields. However, for an accurate
statistical characterization the number of MC samples has to be large. In this
work, to address this challenge, the continuation multilevel Monte Carlo
(CMLMC) method is used together with a surface integral equation solver. The
CMLMC method optimally balances statistical errors due to sampling of the
parametric space, and numerical errors due to the discretization of the
geometry using a hierarchy of discretizations, from coarse to fine. The number
of realizations of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational cost.
Consequently, the total execution time is significantly reduced, in comparison
to the standard MC scheme.Comment: 25 pages, 10 Figure
Subdivision Shell Elements with Anisotropic Growth
A thin shell finite element approach based on Loop's subdivision surfaces is
proposed, capable of dealing with large deformations and anisotropic growth. To
this end, the Kirchhoff-Love theory of thin shells is derived and extended to
allow for arbitrary in-plane growth. The simplicity and computational
efficiency of the subdivision thin shell elements is outstanding, which is
demonstrated on a few standard loading benchmarks. With this powerful tool at
hand, we demonstrate the broad range of possible applications by numerical
solution of several growth scenarios, ranging from the uniform growth of a
sphere, to boundary instabilities induced by large anisotropic growth. Finally,
it is shown that the problem of a slowly and uniformly growing sheet confined
in a fixed hollow sphere is equivalent to the inverse process where a sheet of
fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless,
quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl
A Stable Multi-Scale Kernel for Topological Machine Learning
Topological data analysis offers a rich source of valuable information to
study vision problems. Yet, so far we lack a theoretically sound connection to
popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In
this work, we establish such a connection by designing a multi-scale kernel for
persistence diagrams, a stable summary representation of topological features
in data. We show that this kernel is positive definite and prove its stability
with respect to the 1-Wasserstein distance. Experiments on two benchmark
datasets for 3D shape classification/retrieval and texture recognition show
considerable performance gains of the proposed method compared to an
alternative approach that is based on the recently introduced persistence
landscapes
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