scholarly journals TOPOLOGY-GUIDED TESSELLATION OF QUADRATIC ElEMENTS

2009 ◽  
Vol 19 (02) ◽  
pp. 195-211 ◽  
Author(s):  
SCOTT E. DILLARD ◽  
VIJAY NATARAJAN ◽  
GUNTHER H. WEBER ◽  
VALERIO PASCUCCI ◽  
BERND HAMANN
Keyword(s):  
Level Sets ◽  
Piecewise Linear ◽  
Quadratic Function ◽  
Linear Functions ◽  
Quadratic Functions ◽  
Piecewise Linear Functions ◽  
Reeb Graph ◽  
Triangle Meshes ◽  
Triangular Patch ◽  
Piecewise Quadratic Functions

Topology-based methods have been successfully used for the analysis and visualization of piecewise-linear functions defined on triangle meshes. This paper describes a mechanism for extending these methods to piecewise-quadratic functions defined on triangulations of surfaces. Each triangular patch is tessellated into monotone regions, so that existing algorithms for computing topological representations of piecewise-linear functions may be applied directly to the piecewise-quadratic function. In particular, the tessellation is used for computing the Reeb graph, a topological data structure that provides a succinct representation of level sets of the function.

scholarly journals The Reeb Graph Edit Distance is Universal

2020 ◽  
Author(s):  
Ulrich Bauer ◽  
Claudia Landi ◽  
Facundo Mémoli
Keyword(s):  
Edit Distance ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Supremum Norm ◽  
Piecewise Linear Functions ◽  
Reeb Graph ◽  
Graph Edit Distance ◽  
Reeb Graphs ◽  
Interleaving Distance ◽  
Specific Construction

AbstractWe consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.

Modelling the single-electron transistor with piecewise linear functions

2009 ◽  
Author(s):  
Arturo Sarmiento-Reyes ◽  
Luis Hernandez-Martinez ◽  
Miguel Angel Gutierrez de Anda ◽  
Francisco Javier Castro Gonzalez
Keyword(s):  
Piecewise Linear ◽  
Linear Functions ◽  
Single Electron ◽  
Single Electron Transistor ◽  
Piecewise Linear Functions

Mesh duality and Legendre duality

1990 ◽  
Vol 428 (1875) ◽  
pp. 351-377 ◽  
Keyword(s):  
Piecewise Linear ◽  
Voronoi Tessellation ◽  
Tangent Plane ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Dual Transformation ◽  
Delaunay Mesh ◽  
Definition Of

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

scholarly journals Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 166 ◽  
Author(s):  
Andreas Griewank ◽  
Andrea Walther
Keyword(s):  
Piecewise Linear ◽  
Linear Representation ◽  
Local Minimizer ◽  
Descent Method ◽  
Linear Functions ◽  
Steepest Descent Method ◽  
Piecewise Linear Functions ◽  
Piecewise Linearization ◽  
Algorithmic Differentiation ◽  
Concave Part

For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f ˇ and a concave part f ^ , including a pair of generalized gradients g ˇ ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how f ˇ and f ^ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g ˇ and − g ^ are then used to drive DCA algorithms, where the (convex) inner problem can be solved in finitely many steps, e.g., by a Simplex variant or the true steepest descent method. Using a reflection technique to update the gradients of the concave part, one can ensure finite convergence to a local minimizer of f, provided the Linear Independence Kink Qualification holds. For piecewise smooth objectives the approach can be used as an inner method for successive piecewise linearization.

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