An Approximation Algorithm for Optimal Piecewise Linear Interpolations of Bounded Variable Products

We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to c...

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Vydáno v:Journal of optimization theory and applications Ročník 199; číslo 2; s. 569 - 599
Hlavní autoři: Bärmann, Andreas, Burlacu, Robert, Hager, Lukas, Kutzer, Katja
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.11.2023
Springer Nature B.V
Témata:
Algorithms
Applications of Mathematics
Approximation
Bivariate analysis
Calculus of Variations and Optimal Control; Optimization
Engineering
Interpolation
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Quadratic programming
Theory of Computation
Triangulation
Weapons
Piecewise linear approximations
90C20
15A63
Nonconvex quadratic programming
90C11
90C59
Triangulations
Approximation algorithm
41A05
ISSN:0022-3239, 1573-2878
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Popis
Shrnutí:We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using these as building blocks, we construct a triangulation scheme called crossing swords that requires at most - times the number of simplices in any optimal triangulation. In other words, we derive an approximation algorithm for the optimal triangulation problem. We also show that crossing swords yields optimal triangulations in the case that each simplex has at least one axis-parallel edge. Furthermore, we present approximation guarantees for other well-known triangulation schemes, namely for the red refinement and longest-edge bisection strategies as well as for a generalized version of K1-triangulations. Thereby, we are able to show that our novel approach dominates previous triangulation schemes from the literature, which is underlined by illustrative numerical examples.
Bibliografie:ObjectType-Article-1
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-023-02292-3