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A multiscale approach to optimal transport. Comput
, 2011
"... Abstract. In this paper, we propose an improvement of an algorithm of Au-renhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Eu-clidean plane. Our algorithm exploits the multiscale nature of this optimal t ..."
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Cited by 19 (2 self)
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Abstract. In this paper, we propose an improvement of an algorithm of Au-renhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Eu-clidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances be-tween measures related to optimal transport (known as Earth-mover or Wasser-stein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart. 1.
Optimizing voronoi diagrams for polygonal finite element computations
- In Proceedings of the 19th International Meshing Roundtable
, 2010
"... Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectiv ..."
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Cited by 8 (0 self)
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Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities—short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number. 1
Kinetic convex hulls and delaunay triangulations in the black-box model
- In Proc. 27th Annu. Sympos. Comput. Geom
, 2011
"... Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the ob-jects are known in advance. This assumption severely limits the applicability ..."
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Cited by 6 (1 self)
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Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the ob-jects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the tradi-tional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum dis-placement of any point in one time step. We study the maintenance of the convex hull and the De-launay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ∈ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ∆k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(k∆k log 2 n) amortized time. For the Delaunay triangu-lation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2∆2k) at each time step.
Connectivity Oblivious Merging of Triangulations
"... Fig. 1: Merging a buddha tetrahedral mesh with a background grid. Our technique is able to handle meshes with distinct levels of refinement- observe how the internal tetrahedra of the buddha have not been refined. Abstract—Simplicial meshes are extremely useful as discrete approximations of continuo ..."
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Fig. 1: Merging a buddha tetrahedral mesh with a background grid. Our technique is able to handle meshes with distinct levels of refinement- observe how the internal tetrahedra of the buddha have not been refined. Abstract—Simplicial meshes are extremely useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, which tends to make the numerical simulations unstable. In this paper we propose an alternative technique for updating simplicial meshes that undergo geometric and topological changes. We exploit the property that a Weighted Delaunay Triangulation (WDT) can be used to implicitly define the connectivity of a mesh. Instead of explicitly maintaining connectivity information, we simply keep a collection of weights associated with each vertex. This approach allows for a simple way to merge triangulations, which we illustrate with examples in 2D and 3D. Keywords-Power diagram; Weighted Delaunay, Triangulations. I.
Journal of Computational Geometry jocg.org KINETIC CONVEX HULLS, DELAUNAY TRIANGULATIONS AND CONNECTIVITY STRUCTURES IN THE BLACK-BOX MODEL ∗
"... Abstract. Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the appl ..."
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Abstract. Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ∈ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ∆k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(min(n, k∆k log n) log n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2∆2 k) at each time step. 1
Fast Updating of Delaunay Triangulation of Moving Points by Bi-cell Filtering
"... Updating a Delaunay triangulation when data points are slightly moved is the bottleneck of computation time in variational methods for mesh generation and remeshing. Utilizing the connectivity coherence between two consecutive Delaunay triangulations for computation speedup is the key to solving thi ..."
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Updating a Delaunay triangulation when data points are slightly moved is the bottleneck of computation time in variational methods for mesh generation and remeshing. Utilizing the connectivity coherence between two consecutive Delaunay triangulations for computation speedup is the key to solving this problem. Our contribution is an effective filtering technique that confirms most bi-cells whose Delaunay connectivities remain unchanged after the points are perturbed. Based on bi-cell flipping, we present an efficient algorithm for updating two-dimensional and three-dimensional Delaunay triangulations of dynamic point sets. Experimental results show that our algorithm outperforms previous methods.
