Walking in a Triangulation

Devillers, Olivier; Pion, Sylvain; Teillaud, Monique

doi:10.1142/s0129054102001047

Cited by 78 publications

(95 citation statements)

References 13 publications

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“…If no edge separates the point from the triangle, the triangle containing the point is found. Devillers et al (2002) prove that this stochastic walk terminates with probability one. Having found the triangle that contains the point p, the interpolation value at p is given as a linear combination of the function values (v 1 , v 2 , v 3 ) at the cor- …”

Section: Simplicial Interpolationmentioning

confidence: 90%

“…A standard approach to find the triangle where the interpolation point is located is the stochastic walk algorithm. As described in Devillers et al (2002), this algorithm 'walks' through the triangulation: Given a specific triangle, it randomly chooses one of its edges and checks whether the line supporting this edge separates the point from the triangle. If so, the point is obviously not contained in the triangle and the algorithms proceeds by considering the neighboring triangle that shares the chosen edge.…”

Section: Simplicial Interpolationmentioning

confidence: 99%

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Computing equilibria in dynamic models with occasionally binding constraints

Brumm

Grill

2014

Journal of Economic Dynamics and Control

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We propose a method to compute equilibria in dynamic models with several continuous state variables and occasionally binding constraints. These constraints induce non-differentiabilities in policy functions. We develop an interpolation technique that addresses this problem directly: It locates the non-differentiabilities and adds interpolation nodes there. To handle this flexible grid, it uses Delaunay interpolation, a simplicial interpolation technique. Hence, we call this method Adaptive Simplicial Interpolation (ASI). We embed ASI into a time iteration algorithm to compute recursive equilibria in an infinite horizon endowment economy where heterogeneous agents trade in a bond and a stock subject to various trading constraints. We show that this method computes equilibria accurately and outperforms other grid schemes by far. AbstractWe propose a method to compute equilibria in dynamic models with several continuous state variables and occasionally binding constraints. These constraints induce non-differentiabilities in policy functions. We develop an interpolation technique that addresses this problem directly: It locates the non-differentiabilities and adds interpolation nodes there. To handle this flexible grid, it uses Delaunay interpolation, a simplicial interpolation technique. Hence, we call this method Adaptive Simplicial Interpolation (ASI). We embed ASI into a time iteration algorithm to compute recursive equilibria in an infinite horizon endowment economy where heterogeneous agents trade in a bond and a stock subject to various trading constraints. We show that this method computes equilibria accurately and outperforms other grid schemes by far.

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Section: Simplicial Interpolationmentioning

confidence: 90%

Section: Simplicial Interpolationmentioning

confidence: 99%

Computing equilibria in dynamic models with occasionally binding constraints

Brumm

Grill

2014

Journal of Economic Dynamics and Control

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“…Once a domain has been te Delaunay elements (collectiv put PLC), some of these e ity and need to be improv rithms [6,11,21,22,29] inse bad elements with better one nay (or "almost Delaunay") tions. Fortunately, inserting a from, a constrained Delauna easy as in an ordinary Delau…”

Section: Vertex Insermentioning

confidence: 99%

“…In the cavity retriangulation algorithm (Section 3.1), the incremental insertion algorithm uses the simplest walking algorithm for point location [6], because the cavities are usually small.…”

Section: Implementation Notesmentioning

confidence: 99%

Incrementally Constructing and Updating Constrained Delaunay Tetrahedralizations with Finite Precision Coordinates

Shewchuk

2013

Proceedings of the 21st International Meshing Roundtable

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Summary. Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision coordinates yield vertices on a line that are not perfectly collinear and polygonal facets that are not perfectly flat. We experimentally compare two recent algorithms for inserting a polygonal facet into a CDT: a bistellar flip algorithm of Shewchuk (Proc. 19th Annual Symposium on Computational Geometry, June 2003) and a cavity retriangulation algorithm of Si and Gärtner (Proc. Fourteenth International Meshing Roundtable, September 2005). We modify these algorithms to succeed in practice for polygons whose vertices deviate from exact coplanarity.

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“…Hilbert Direct uses the same geometric principles as the "triangulation walking" studies [6] by the computational geometry community, where the main focus is the theoretical analysis of point query performance. A complete theoretical analysis is a separate area of study, where the additional assumption of uniformly distributed mesh nodes is necessary for tractability [7] (such a theoretical analysis is beyond the scope of this paper).…”

Section: Proximity Searchmentioning

confidence: 99%

Efficient query processing on unstructured tetrahedral meshes

Papadomanolakis

Ailamaki

López

et al. 2006

Proceedings of the 2006 ACM SIGMOD International Conference on Management of Data

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Modern scientific applications consume massive volumes of data produced by computer simulations. Such applications require new data management capabilities in order to scale to terabyte-scale data volumes [25,10]. The most common way to discretize the application domain is to decompose it into pyramids, forming an unstructured tetrahedral mesh. Modern simulations generate meshes of high resolution and precision, to be queried by a visualization or analysis tool. Tetrahedral meshes are extremely flexible and therefore vital to accurately model complex geometries, but also are difficult to index. To reduce query execution time, applications either use only subsets of the data or rely on different (less flexible) structures, thereby trading accuracy for speed. This paper presents efficient indexing techniques for generic spatial queries on tetrahedral meshes. Because the prevailing multidimensional indexing techniques attempt to approximate the tetrahedra using simpler shapes (rectangles) query performance deteriorates significantly as a function of the mesh's geometric complexity. We develop Directed Local Search (DLS), an efficient indexing algorithm based on mesh topology information that is practically insensitive to the geometric properties of meshes. We show how DLS can be easily and efficiently implemented within modern database systems without requiring new exotic index structures and complex preprocessing. Finally, we present a new data layout approach for tetrahedral mesh datasets that provides better performance compared to the traditional space filling curves. In our PostgreSQL implementation DLS reduces the number of disk page accesses and the query execution time each by 25% up to a factor of 4.

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Walking in a Triangulation

Abstract: Given a triangulation in the plane or a tetrahedralization in 3-space, we investigate the efficiency of locating a point by walking in the structure with different strategies.

Cited by 78 publications

References 13 publications

Computing equilibria in dynamic models with occasionally binding constraints

Computing equilibria in dynamic models with occasionally binding constraints

Incrementally Constructing and Updating Constrained Delaunay Tetrahedralizations with Finite Precision Coordinates

Efficient query processing on unstructured tetrahedral meshes

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