3,023 research outputs found
A numerical algorithm for semi-discrete optimal transport in 3D
This paper introduces a numerical algorithm to compute the optimal
transport map between two measures and , where derives from a
density defined as a piecewise linear function (supported by a
tetrahedral mesh), and where is a sum of Dirac masses.
I first give an elementary presentation of some known results on optimal
transport and then observe a relation with another problem (optimal sampling).
This relation gives simple arguments to study the objective functions that
characterize both problems.
I then propose a practical algorithm to compute the optimal transport map
between a piecewise linear density and a sum of Dirac masses in 3D. In this
semi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on Computational
Geometry conf. proc.}, ACM (1992)] showed that the optimal transport map is
determined by the weights of a power diagram. The optimal weights are computed
by minimizing a convex objective function with a quasi-Newton method. To
evaluate the value and gradient of this objective function, I propose an
efficient and robust algorithm, that computes at each iteration the
intersection between a power diagram and the tetrahedral mesh that defines the
measure .
The numerical algorithm is experimented and evaluated on several datasets,
with up to hundred thousands tetrahedra and one million Dirac masses.Comment: 23 pages, 14 figure
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
Three-dimensional alpha shapes
Frequently, data in scientific computing is in its abstract form a finite
point set in space, and it is sometimes useful or required to compute what one
might call the ``shape'' of the set. For that purpose, this paper introduces
the formal notion of the family of -shapes of a finite point set in
\Real^3. Each shape is a well-defined polytope, derived from the Delaunay
triangulation of the point set, with a parameter \alpha \in \Real controlling
the desired level of detail. An algorithm is presented that constructs the
entire family of shapes for a given set of size in time , worst
case. A robust implementation of the algorithm is discussed and several
applications in the area of scientific computing are mentioned.Comment: 32 page
Optimizing the geometrical accuracy of curvilinear meshes
This paper presents a method to generate valid high order meshes with
optimized geometrical accuracy. The high order meshing procedure starts with a
linear mesh, that is subsequently curved without taking care of the validity of
the high order elements. An optimization procedure is then used to both
untangle invalid elements and optimize the geometrical accuracy of the mesh.
Standard measures of the distance between curves are considered to evaluate the
geometrical accuracy in planar two-dimensional meshes, but they prove
computationally too costly for optimization purposes. A fast estimate of the
geometrical accuracy, based on Taylor expansions of the curves, is introduced.
An unconstrained optimization procedure based on this estimate is shown to
yield significant improvements in the geometrical accuracy of high order
meshes, as measured by the standard Haudorff distance between the geometrical
model and the mesh. Several examples illustrate the beneficial impact of this
method on CFD solutions, with a particular role of the enhanced mesh boundary
smoothness.Comment: Submitted to JC
Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
For a set of points in , and parameters and \eps, we present
a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time.
Surprisingly, the space used by the data-structure is \Otilde (n /k); that
is, the space used is sublinear in the input size if is sufficiently large.
Our approach provides a novel way to summarize geometric data, such that
meaningful proximity queries on the data can be carried out using this sketch.
Using this, we provide a sublinear space data-structure that can estimate the
density of a point set under various measures, including:
\begin{inparaenum}[(i)]
\item sum of distances of closest points to the query point, and
\item sum of squared distances of closest points to the query point.
\end{inparaenum}
Our approach generalizes to other distance based estimation of densities of
similar flavor. We also study the problem of approximating some of these
quantities when using sampling. In particular, we show that a sample of size
\Otilde (n /k) is sufficient, in some restricted cases, to estimate the above
quantities. Remarkably, the sample size has only linear dependency on the
dimension
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