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Journal articles on the topic 'Polyhedral approximation'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 May 2024

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1

Mokhnacheva, A. A., K. V. Gerasimova, and D. N. Ibragimov. "Methods of Numerical Simulation of 0-Controllable Sets of a Linear Discrete Dynamical System with Limited Control Based on Polyhedral Approximation Algorithms." Моделирование и анализ данных 13, no. 4 (December 28, 2023): 84–110. http://dx.doi.org/10.17759/mda.2023130405.

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<p>The article deals with the problem of constructing a polyhedral approximation of the 0-controllable sets of a linear discrete-time system with linear control constraints. To carry out the approximation, it is proposed to use two heuristic algorithms aimed at reducing the number of vertices of an arbitrary polyhedron while maintaining the accuracy of the description in the sense of the Hausdorff distance. The reduction of the problem of calculating the distance between nested polyhedra to the problem of convex programming is demonstrated. The issues of optimality of obtained approximations are investigated. Examples are given.</p>

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2

Horst, R., Ng V. Thoai, and H. Tuy. "Outer approximation by polyhedral convex sets." OR Spektrum 9, no. 3 (September 1987): 153–59. http://dx.doi.org/10.1007/bf01721096.

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3

Fonf, Vladimir P., Joram Lindenstrauss, and Libor Veselý. "Best approximation in polyhedral Banach spaces." Journal of Approximation Theory 163, no. 11 (November 2011): 1748–71. http://dx.doi.org/10.1016/j.jat.2011.06.011.

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4

Mitchell, Joseph S. B., and Subhash Suri. "Separation and approximation of polyhedral objects." Computational Geometry 5, no. 2 (September 1995): 95–114. http://dx.doi.org/10.1016/0925-7721(95)00006-u.

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5

Pugach, P. A., and V. A. Shlyk. "Piecewise Linear Approximation and Polyhedral Surfaces." Journal of Mathematical Sciences 200, no. 5 (July 1, 2014): 617–23. http://dx.doi.org/10.1007/s10958-014-1951-7.

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6

Schneider, Rolf. "Polyhedral approximation of smooth convex bodies." Journal of Mathematical Analysis and Applications 128, no. 2 (December 1987): 470–74. http://dx.doi.org/10.1016/0022-247x(87)90197-1.

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7

O’Dell, Brian D., and Eduardo A. Misawa. "Semi-Ellipsoidal Controlled Invariant Sets for Constrained Linear Systems." Journal of Dynamic Systems, Measurement, and Control 124, no. 1 (April 17, 2000): 98–103. http://dx.doi.org/10.1115/1.1434269.

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This paper investigates an alternative approximation to the maximal viability set for linear systems with constrained states and input. Current ellipsoidal and polyhedral approximations are either too conservative or too complex for many applications. As the primary contribution, it is shown that the intersection of a controlled invariant ellipsoid and a set of state constraints (referred to as a semi-ellipsoidal set) is itself controlled invariant under certain conditions. The proposed semi-ellipsoidal approach is less conservative than the ellipsoidal method but simpler than the polyhedral method. Two examples serve as proof-of-concept of the approach.

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8

Neyrinck, Mark C. "An Origami Approximation to the Cosmic Web." Proceedings of the International Astronomical Union 11, S308 (June 2014): 97–102. http://dx.doi.org/10.1017/s1743921316009686.

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AbstractThe powerful Lagrangian view of structure formation was essentially introduced to cosmology by Zel'dovich. In the current cosmological paradigm, a dark-matter-sheet 3D manifold, inhabiting 6D position-velocity phase space, was flat (with vanishing velocity) at the big bang. Afterward, gravity stretched and bunched the sheet together in different places, forming a cosmic web when projected to the position coordinates.Here, I explain some properties of an origami approximation, in which the sheet does not stretch or contract (an assumption that is false in general), but is allowed to fold. Even without stretching, the sheet can form an idealized cosmic web, with convex polyhedral voids separated by straight walls and filaments, joined by convex polyhedral nodes. The nodes form in ‘polygonal’ or ‘polyhedral’ collapse, somewhat like spherical/ellipsoidal collapse, except incorporating simultaneous filament and wall formation. The origami approximation allows phase-space geometries of nodes, filaments, and walls to be more easily understood, and may aid in understanding spin correlations between nearby galaxies. This contribution explores kinematic origami-approximation models giving velocity fields for the first time.

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9

Di Pietro, Daniele A., Jérôme Droniou, and Francesca Rapetti. "Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra." Mathematical Models and Methods in Applied Sciences 30, no. 09 (August 2020): 1809–55. http://dx.doi.org/10.1142/s0218202520500372.

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In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving the exactness, we show that the usual three-dimensional sequence of trimmed Finite Element (FE) spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete [Formula: see text]-products completes the exposition.

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10

Deville, Robert, Vladimir Fonf, and Petr Hájek. "Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces." Israel Journal of Mathematics 105, no. 1 (December 1998): 139–54. http://dx.doi.org/10.1007/bf02780326.

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11

Zahariuta, V. "On approximation by special analytic polyhedral pairs." Annales Polonici Mathematici 80 (2003): 243–56. http://dx.doi.org/10.4064/ap80-0-22.

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12

SUGITA, Mikio, and Ryoji OHBA. "Three dimensional surface determination by polyhedral approximation." Journal of the Japan society of photogrammetry and remote sensing 31, no. 2 (1992): 4–10. http://dx.doi.org/10.4287/jsprs.31.2_4.

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13

Frédéric Bonnans, J., and Marc Lebelle. "Explicit polyhedral approximation of the Euclidean ball." RAIRO - Operations Research 44, no. 1 (January 2010): 45–59. http://dx.doi.org/10.1051/ro/2010003.

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14

Valkonen, Tuomo. "Strong polyhedral approximation of simple jump sets." Nonlinear Analysis: Theory, Methods & Applications 75, no. 8 (May 2012): 3641–71. http://dx.doi.org/10.1016/j.na.2012.01.022.

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15

Lubin, Miles, Emre Yamangil, Russell Bent, and Juan Pablo Vielma. "Polyhedral approximation in mixed-integer convex optimization." Mathematical Programming 172, no. 1-2 (September 14, 2017): 139–68. http://dx.doi.org/10.1007/s10107-017-1191-y.

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16

Koide, Akio, Akio Doi, and Koichi Kajioka. "Polyhedral approximation approach to molecular orbital graphics." Journal of Molecular Graphics 4, no. 3 (September 1986): 149–55. http://dx.doi.org/10.1016/0263-7855(86)80016-9.

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17

Bible, Victor, and Richard J. Smith. "Smooth and polyhedral approximation in Banach spaces." Journal of Mathematical Analysis and Applications 435, no. 2 (March 2016): 1262–72. http://dx.doi.org/10.1016/j.jmaa.2015.11.018.

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18

Schütt, Carsten. "The convex floating body and polyhedral approximation." Israel Journal of Mathematics 73, no. 1 (February 1991): 65–77. http://dx.doi.org/10.1007/bf02773425.

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19

Brieden, Andreas, and Peter Gritzmann. "On Clustering Bodies: Geometry and Polyhedral Approximation." Discrete & Computational Geometry 44, no. 3 (October 3, 2009): 508–34. http://dx.doi.org/10.1007/s00454-009-9226-7.

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20

Varadhan, Gokul, and Dinesh Manocha. "Accurate Minkowski sum approximation of polyhedral models." Graphical Models 68, no. 4 (July 2006): 343–55. http://dx.doi.org/10.1016/j.gmod.2005.11.003.

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21

Lopez, Mario A., and Shlomo Reisner. "Algorithms for Polyhedral Approximation of Multidimensional Ellipsoids." Journal of Algorithms 33, no. 1 (October 1999): 140–65. http://dx.doi.org/10.1006/jagm.1999.1031.

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22

Khachay, Michael. "Committee polyhedral separability: complexity and polynomial approximation." Machine Learning 101, no. 1-3 (May 28, 2015): 231–51. http://dx.doi.org/10.1007/s10994-015-5505-0.

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23

Rollin, Yann. "Polyhedral approximation by Lagrangian and isotropic tori." Journal of Symplectic Geometry 20, no. 6 (2022): 1349–83. http://dx.doi.org/10.4310/jsg.2022.v20.n6.a4.

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24

Podivilova, E. O., and V. I. Shiryaev. "DYNAMIC SYSTEMS STATE, DISTURBANCES AND NOISES SET-VALUED ESTIMATION UNDER CONDITIONS OF INCOMPLETE INFORMATION." Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control & Radioelectronics 21, no. 1 (February 2021): 23–34. http://dx.doi.org/10.14529/ctcr210103.

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The paper considers the problem of set-valued dynamic systems state estimation under conditions of uncertainty, when the sets of disturbances and noises possible values are known and statistical information about them is absent or cannot be obtained. An algorithm for feasible set polyhedral approximation is described, when the sets of possible values of disturbances and noises are polyhedra. The algorithm is based on the implicit description of the information set with linear equations and inequalities systems and solving a number of linear programming problems. Methods for increasing the estimation accuracy by taking into account additional information about disturbances and noises models are considered. Set-valued estimation of the dynamical system state vector is described when the disturbances are given as a system of functions with unknown coefficients. In this case, due to the use of information that the coefficients are constant, the dynamic system state estimates are more accurate than in the case when the disturbances are known up to a set of possible values. A numerical example is presented to demonstrate the algorithm performance. Aim. The aim of the research is to develop dynamic system state, disturbance and noises set-valued estimation algorithms. Research methods. Methods of optimization theory, filtering, linear algebra, MATLAB software package were used in the work. Results. Dynamic system state estimation algorithm was described. The algorithm takes into account additional information about disturbances and noises models. A method of feasible set polyhedral approximation is described, which makes it possible to obtain a set-valued estimate of a state vector, a vector of disturbances and noises, and an evolution of reachable sets. It can be used in the adaptive estimation and control algorithms development. The algorithm for set-valued estimation of the system state vector and coefficients in the disturbance decomposition as a system of given functions is developed. Conclusion. An algorithm for feasible set polyhedral approximation was described.The numerical example was performed and the analysis of the estimateswas presented.

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25

Chave, Florent, Daniele A. Di Pietro, and Simon Lemaire. "A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics." Mathematical Models and Methods in Applied Sciences 32, no. 01 (January 2022): 175–207. http://dx.doi.org/10.1142/s0218202522500051.

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We prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. We then introduce two Hybrid High-Order methods for the approximation of the magnetostatics model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, we perform a comprehensive analysis of the two methods. We finally validate them on a set of test-cases.

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26

Bertsekas, Dimitri P., and Huizhen Yu. "A Unifying Polyhedral Approximation Framework for Convex Optimization." SIAM Journal on Optimization 21, no. 1 (January 2011): 333–60. http://dx.doi.org/10.1137/090772204.

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27

Kim, Young J., Gokul Varadhan, Ming C. Lin, and Dinesh Manocha. "Fast swept volume approximation of complex polyhedral models." Computer-Aided Design 36, no. 11 (September 2004): 1013–27. http://dx.doi.org/10.1016/j.cad.2004.01.004.

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28

De Pauw, Thierry. "Approximation by Polyhedral $G$ Chains in Banach Spaces." Zeitschrift für Analysis und ihre Anwendungen 33, no. 3 (2014): 311–34. http://dx.doi.org/10.4171/zaa/1514.

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29

Zabotin, I. Ya, K. E. Kazaeva, and O. N. Shulgina. "A cutting-plane method with internal iteration points for the general convex programming problem." Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki 165, no. 3 (January 11, 2024): 208–18. http://dx.doi.org/10.26907/2541-7746.2023.3.208-218.

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A cutting method for solving the problem of convex programming was proposed. The method calculates iteration points based on approximation by polyhedral sets of the constraint region and the epigraph of the objective function. Its distinguishing feature is that the main sequence of approximations is constructed within the admissible region. At each step, it is also possible to assess how close the current value of the function is to the optimal value. The convergence of the method was proved. A few of its implementations were outlined.

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30

Davis, W. E., and J. Craig Yacoe. "A New Polyhedral Approximation to an Ellipsoid of Revolution." International Journal of Space Structures 5, no. 3-4 (September 1990): 187–95. http://dx.doi.org/10.1177/026635119000500304.

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31

Kamenev, G. K., and A. I. Pospelov. "Polyhedral approximation of convex compact bodies by filling methods." Computational Mathematics and Mathematical Physics 52, no. 5 (May 2012): 680–90. http://dx.doi.org/10.1134/s0965542512050119.

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32

Watson, G. A. "Linear best approximation using a class of polyhedral norms." Numerical Algorithms 2, no. 3 (October 1992): 321–35. http://dx.doi.org/10.1007/bf02139472.

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33

Fukushima, Kenichi, and Yuichiro Waki. "A polyhedral approximation approach to concave numerical dynamic programming." Journal of Economic Dynamics and Control 37, no. 11 (November 2013): 2322–35. http://dx.doi.org/10.1016/j.jedc.2013.06.001.

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34

Horst, Reiner, Nguyen V. Thoai, and Harold P. Benson. "Concave minimization via conical partitions and polyhedral outer approximation." Mathematical Programming 50, no. 1-3 (March 1991): 259–74. http://dx.doi.org/10.1007/bf01594938.

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35

Aceti, John K., and Jeremy Brazas. "Elements of higher homotopy groups undetectable by polyhedral approximation." Pacific Journal of Mathematics 322, no. 2 (May 23, 2023): 221–42. http://dx.doi.org/10.2140/pjm.2023.322.221.

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36

Murota, Kazuo. "ON POLYHEDRAL APPROXIMATION OF L-CONVEX AND M-CONVEX FUNCTIONS." Journal of the Operations Research Society of Japan 58, no. 3 (2015): 291–305. http://dx.doi.org/10.15807/jorsj.58.291.

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37

Burger, Mathias, Giuseppe Notarstefano, and Frank Allgower. "A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization." IEEE Transactions on Automatic Control 59, no. 2 (February 2014): 384–95. http://dx.doi.org/10.1109/tac.2013.2281883.

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38

Hu, Renhong, Lizhen Shao, and Yuhao Cong. "Polyhedral approximation method for reachable sets of linear delay systems." IET Control Theory & Applications 14, no. 12 (August 13, 2020): 1548–56. http://dx.doi.org/10.1049/iet-cta.2019.0841.

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39

Lotov, A. V. "Decomposition methods for polyhedral approximation of the Edgeworth–Pareto hull." Doklady Mathematics 92, no. 3 (November 2015): 784–87. http://dx.doi.org/10.1134/s1064562415060113.

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40

Koutroumbas, K. "On the Partitioning Capabilities of Feedforward Neural Networks with Sigmoid Nodes." Neural Computation 15, no. 10 (October 1, 2003): 2457–81. http://dx.doi.org/10.1162/089976603322362437.

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In this letter, the capabilities of feedforward neural networks (FNNs) on the realization and approximation of functions of the form g: R1 → A, which partition the R1 space into polyhedral sets, each one being assigned to one out of the c classes of A, are investigated. More specifically, a constructive proof is given for the fact that FNNs consisting of nodes having sigmoid output functions are capable of approximating any function g with arbitrary accuracy. Also, the capabilities of FNNs consisting of nodes having the hard limiter as output function are reviewed. In both cases, the two-class as well as the multiclass cases are considered.

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41

Demir Sağlam, Sevilay, and Elimhan Mahmudov. "Optimality conditions for higher order polyhedral discrete and differential inclusions." Filomat 34, no. 13 (2020): 4533–53. http://dx.doi.org/10.2298/fil2013533d.

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The problems considered in this paper are described in polyhedral multi-valued mappings for higher order(s-th) discrete (PDSIs) and differential inclusions (PDFIs). The present paper focuses on the necessary and sufficient conditions of optimality for optimization of these problems. By converting the PDSIs problem into a geometric constraint problem, we formulate the necessary and sufficient conditions of optimality for a convex minimization problem with linear inequality constraints. Then, in terms of the Euler-Lagrange type PDSIs and the specially formulated transversality conditions, we are able to obtain conditions of optimality for the PDSIs. In order to obtain the necessary and sufficient conditions of optimality for the discrete-approximation problem PDSIs, we reduce this problem to the form of a problem with higher order discrete inclusions. Finally, by formally passing to the limit, we establish the sufficient conditions of optimality for the problem with higher order PDFIs. Numerical approach is developed to solve a polyhedral problem with second order polyhedral discrete inclusions.

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42

Iqbal, Muhammad Faisal, and Faizan Ahmed. "Approximation Hierarchies for the Copositive Tensor Cone and Their Application to the Polynomial Optimization over the Simplex." Mathematics 10, no. 10 (May 14, 2022): 1683. http://dx.doi.org/10.3390/math10101683.

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In this paper, we discuss the cone of copositive tensors and its approximation. We describe some basic properties of copositive tensors and positive semidefinite tensors. Specifically, we show that a non-positive tensor (or Z-tensor) is copositive if and only if it is positive semidefinite. We also describe cone hierarchies that approximate the copositive cone. These hierarchies are based on the sum of squares conditions and the non-negativity of polynomial coefficients. We provide a compact representation for the approximation based on the non-negativity of polynomial coefficients. As an immediate consequence of this representation, we show that the approximation based on the non-negativity of polynomial coefficients is polyhedral. Furthermore, these hierarchies are used to provide approximation results for optimizing a (homogeneous) polynomial over the simplex.

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43

Baturin, V. A., E. V. Goncharova, F. L. Pereira, and J. B. Sousa. "Measure-controlled dynamic systems: Polyhedral approximation of their reachable set boundary." Automation and Remote Control 67, no. 3 (March 2006): 350–60. http://dx.doi.org/10.1134/s0005117906030027.

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44

Hassan, A. S. O., H. L. Abdel-Malek, and A. A. Rabie. "Design centering and polyhedral region approximation via parallel-cuts ellipsoidal technique." Engineering Optimization 36, no. 1 (February 2004): 37–49. http://dx.doi.org/10.1080/03052150310001634880.

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45

Yang, Xunnian, and Jianmin Zheng. "Shape aware normal interpolation for curved surface shading from polyhedral approximation." Visual Computer 29, no. 3 (April 28, 2012): 189–201. http://dx.doi.org/10.1007/s00371-012-0715-y.

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46

Toriya, H., T. Takamura, T. Satoh, and H. Chiyokura. "Boolean operations for solids with free-form surfaces through polyhedral approximation." Visual Computer 7, no. 2-3 (March 1991): 87–96. http://dx.doi.org/10.1007/bf01901179.

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47

Dello Russo, Anahí, and Ana Alonso. "Finite element approximation of Maxwell eigenproblems on curved Lipschitz polyhedral domains." Applied Numerical Mathematics 59, no. 8 (August 2009): 1796–822. http://dx.doi.org/10.1016/j.apnum.2009.01.007.

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48

Jabr, R. A. "Tight polyhedral approximation for mixed-integer linear programming unit commitment formulations." IET Generation, Transmission & Distribution 6, no. 11 (November 1, 2012): 1104–11. http://dx.doi.org/10.1049/iet-gtd.2012.0218.

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49

Lubuma, Jean M. S., and Serge Nicaise. "Dirichlet problems in polyhedral domains II: Approximation by FEM and BEM." Journal of Computational and Applied Mathematics 61, no. 1 (July 1995): 13–27. http://dx.doi.org/10.1016/0377-0427(94)00050-b.

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50

Lazimy, Rafael. "Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems." Annals of Operations Research 210, no. 1 (August 1, 2012): 73–99. http://dx.doi.org/10.1007/s10479-012-1190-6.

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