9,777 research outputs found
Sum-factorization techniques in Isogeometric Analysis
The fast assembling of stiffness and mass matrices is a key issue in
isogeometric analysis, particularly if the spline degree is increased. We
present two algorithms based on the idea of sum factorization, one for matrix
assembling and one for matrix-free methods, and study the behavior of their
computational complexity in terms of the spline order . Opposed to the
standard approach, these algorithms do not apply the idea element-wise, but
globally or on macro-elements. If this approach is applied to Gauss quadrature,
the computational complexity grows as instead of as
previously achieved.Comment: 34 pages, 8 figure
Surface Networks
We study data-driven representations for three-dimensional triangle meshes,
which are one of the prevalent objects used to represent 3D geometry. Recent
works have developed models that exploit the intrinsic geometry of manifolds
and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants,
which learn from the local metric tensor via the Laplacian operator. Despite
offering excellent sample complexity and built-in invariances, intrinsic
geometry alone is invariant to isometric deformations, making it unsuitable for
many applications. To overcome this limitation, we propose several upgrades to
GNNs to leverage extrinsic differential geometry properties of
three-dimensional surfaces, increasing its modeling power.
In particular, we propose to exploit the Dirac operator, whose spectrum
detects principal curvature directions --- this is in stark contrast with the
classical Laplace operator, which directly measures mean curvature. We coin the
resulting models \emph{Surface Networks (SN)}. We prove that these models
define shape representations that are stable to deformation and to
discretization, and we demonstrate the efficiency and versatility of SNs on two
challenging tasks: temporal prediction of mesh deformations under non-linear
dynamics and generative models using a variational autoencoder framework with
encoders/decoders given by SNs
Hierarchical interpolative factorization for elliptic operators: integral equations
This paper introduces the hierarchical interpolative factorization for
integral equations (HIF-IE) associated with elliptic problems in two and three
dimensions. This factorization takes the form of an approximate generalized LU
decomposition that permits the efficient application of the discretized
operator and its inverse. HIF-IE is based on the recursive skeletonization
algorithm but incorporates a novel combination of two key features: (1) a
matrix factorization framework for sparsifying structured dense matrices and
(2) a recursive dimensional reduction strategy to decrease the cost. Thus,
higher-dimensional problems are effectively mapped to one dimension, and we
conjecture that constructing, applying, and inverting the factorization all
have linear or quasilinear complexity. Numerical experiments support this claim
and further demonstrate the performance of our algorithm as a generalized fast
multipole method, direct solver, and preconditioner. HIF-IE is compatible with
geometric adaptivity and can handle both boundary and volume problems. MATLAB
codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat
The diameter of KPKVB random graphs
We consider a model for complex networks that was recently proposed as a
model for complex networks by Krioukov et al. In this model, nodes are chosen
randomly inside a disk in the hyperbolic plane and two nodes are connected if
they are at most a certain hyperbolic distance from each other. It has been
previously shown that this model has various properties associated with complex
networks, including a power-law degree distribution and a strictly positive
clustering coefficient. The model is specified using three parameters : the
number of nodes , which we think of as going to infinity, and which we think of as constant. Roughly speaking controls the power
law exponent of the degree sequence and the average degree.
Earlier work of Kiwi and Mitsche has shown that when (which
corresponds to the exponent of the power law degree sequence being ) then
the diameter of the largest component is a.a.s.~polylogarithmic in .
Friedrich and Krohmer have shown it is a.a.s.~ and they
improved the exponent of the polynomial in in the upper bound. Here we
show the maximum diameter over all components is a.a.s.~ thus giving
a bound that is tight up to a multiplicative constant.Comment: very minor corrections since the last versio
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