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Approximation of the gradient of a function on the basis of a special
class of triangulations
V. A. Klyachin Volgograd State University
Abstract:
We introduce the class of $\Phi$-triangulations of a finite set $P$ of points
in $\mathbb{R}^n$ analogous to the classical Delaunay triangulation.
Such triangulations can be constructed using the condition of empty
intersection of $P$ with the interior of every convex set in a given family
of bounded convex sets the boundary of which contains the vertices of a simplex
of the triangulation. In this case the classical Delaunay triangulation
corresponds to the family of all balls in $\mathbb{R}^n$. We show how
$\Phi$-triangulations can be used to obtain error bounds for an approximation
of the derivatives of $C^2$-smooth functions by piecewise linear functions.
Keywords:
Delaunay triangulation, empty sphere condition, families of convex sets,
piecewise linear approximation.
DOI:
https://doi.org/10.4213/im8691
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English version:
Izvestiya: Mathematics, 2018, 82:6, 1136–1147
Bibliographic databases:
UDC:
514.174.3+519.65
MSC: 65D25, 65D07 Received: 14.05.2017 Revised: 30.08.2017
Citation:
V. A. Klyachin, “Approximation of the gradient of a function on the basis of a special
class of triangulations”, Izv. RAN. Ser. Mat., 82:6 (2018), 65–77; Izv. Math., 82:6 (2018), 1136–1147
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http://mi.mathnet.ru/eng/izv8691https://doi.org/10.4213/im8691 http://mi.mathnet.ru/eng/izv/v82/i6/p65
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