1,009 research outputs found
Efficient computation of partition of unity interpolants through a block-based searching technique
In this paper we propose a new efficient interpolation tool, extremely
suitable for large scattered data sets. The partition of unity method is used
and performed by blending Radial Basis Functions (RBFs) as local approximants
and using locally supported weight functions. In particular we present a new
space-partitioning data structure based on a partition of the underlying
generic domain in blocks. This approach allows us to examine only a reduced
number of blocks in the search process of the nearest neighbour points, leading
to an optimized searching routine. Complexity analysis and numerical
experiments in two- and three-dimensional interpolation support our findings.
Some applications to geometric modelling are also considered. Moreover, the
associated software package written in \textsc{Matlab} is here discussed and
made available to the scientific community
A sequential linear programming (SLP) approach for uncertainty analysis-based data-driven computational mechanics
In this article, an efficient sequential linear programming algorithm (SLP)
for uncertainty analysis-based data-driven computational mechanics (UA-DDCM) is
presented. By assuming that the uncertain constitutive relationship embedded
behind the prescribed data set can be characterized through a convex
combination of the local data points, the upper and lower bounds of structural
responses pertaining to the given data set, which are more valuable for making
decisions in engineering design, can be found by solving a sequential of linear
programming problems very efficiently. Numerical examples demonstrate the
effectiveness of the proposed approach on sparse data set and its robustness
with respect to the existence of noise and outliers in the data set
Partition of Unity Interpolation on Multivariate Convex Domains
In this paper we present a new algorithm for multivariate interpolation of
scattered data sets lying in convex domains \Omega \subseteq \RR^N, for any
. To organize the points in a multidimensional space, we build a
-tree space-partitioning data structure, which is used to efficiently apply
a partition of unity interpolant. This global scheme is combined with local
radial basis function approximants and compactly supported weight functions. A
detailed description of the algorithm for convex domains and a complexity
analysis of the computational procedures are also considered. Several numerical
experiments show the performances of the interpolation algorithm on various
sets of Halton data points contained in , where can be any
convex domain like a 2D polygon or a 3D polyhedron
SVM via Saddle Point Optimization: New Bounds and Distributed Algorithms
We study two important SVM variants: hard-margin SVM (for linearly separable
cases) and -SVM (for linearly non-separable cases). We propose new
algorithms from the perspective of saddle point optimization. Our algorithms
achieve -approximations with running time for both variants, where is the number of points and is
the dimensionality. To the best of our knowledge, the current best algorithm
for -SVM is based on quadratic programming approach which requires
time in worst case~\cite{joachims1998making,platt199912}. In
the paper, we provide the first nearly linear time algorithm for -SVM. The
current best algorithm for hard margin SVM achieved by Gilbert
algorithm~\cite{gartner2009coresets} requires time. Our
algorithm improves the running time by a factor of .
Moreover, our algorithms can be implemented in the distributed settings
naturally. We prove that our algorithms require communication cost, where is the number of clients,
which almost matches the theoretical lower bound. Numerical experiments support
our theory and show that our algorithms converge faster on high dimensional,
large and dense data sets, as compared to previous methods
Analysis and convergence of SMO-like decomposition and geometrical algorithms for support vector machines
Tesis doctoral inédita. Universidad Autónoma de Madrid, Escuela Politécnica Superior, noviembre de 201
A Convex Approach to Consensus on SO(n)
This paper introduces several new algorithms for consensus over the special
orthogonal group. By relying on a convex relaxation of the space of rotation
matrices, consensus over rotation elements is reduced to solving a convex
problem with a unique global solution. The consensus protocol is then
implemented as a distributed optimization using (i) dual decomposition, and
(ii) both semi and fully distributed variants of the alternating direction
method of multipliers technique -- all with strong convergence guarantees. The
convex relaxation is shown to be exact at all iterations of the dual
decomposition based method, and exact once consensus is reached in the case of
the alternating direction method of multipliers. Further, analytic and/or
efficient solutions are provided for each iteration of these distributed
computation schemes, allowing consensus to be reached without any online
optimization. Examples in satellite attitude alignment with up to 100 agents,
an estimation problem from computer vision, and a rotation averaging problem on
validate the approach.Comment: Accepted to 52nd Annual Allerton Conference on Communication,
Control, and Computin
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