Barycentric coordinates for convex sets

Given a smooth, strictly convex planar domain, we investigate point-wise convergence of the sequence of Wachspress coordinates defined over finer and finer inscribed polygonal approximations of the domain. Based on a relation between the discrete Wachspress case and the limit smooth case given by the Wachspress kernel defined by Warren et al., we show that the corresponding sequences of Wachspress interpolants and mappings converge as O(h 2 ) for a sampling step size h of the boundary curve of the domain as h → 0. Several examples are shown to numerically validate the results and to visualise the behaviour of discrete interpolants and mappings as they converge to their smooth counterparts. Empirically, the same convergence order is observed also for mean value coordinates. Moreover, our numerical tests suggest that the convergence of interpolants and mappings is uniform both in the Wachspress and mean value cases.

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“…This is a special case of a general formula for barycentric kernels in any dimension introduced by Warren et al [15].…”

Section: The Wachspress Kernelmentioning

confidence: 93%

“…Examples of such kernels are the kernel defined by Warren et al [15], which we refer to as the Wachspress kernel, and the mean value kernel [1].…”

Section: Introductionmentioning

confidence: 99%

Convergence of Wachspress coordinates: from polygons to curved domains

Kosinka

Bartoň

2014

Adv Comput Math

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“…Often, when dealing with simplex shapes, the barycentric coordinate frame is preferred in which the location of a point within a simplex shape is defined as a weighted measure to each of the vertices, also referred to as areal coordinates when restricted to the two-dimensional simplex [20].…”

Section: Coordinate Transformationmentioning

confidence: 99%

Simplex Solutions for Optimal Control Flight Paths in Urban Environments

Zollars¹,

Rg²,

Dj³

2017

J Aeronaut Aerospace Eng

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This paper identifies feasible fight paths for Small Unmanned Aircraft Systems in a highly constrained environment. Optimal control software has long been used for vehicle path planning and has proven most successful when an adequate initial guess is presented flight to an optimal control solver. Leveragingfast geometric planning techniques, a large search space is discretized into a set of simplexes where a Dubins path solution is generated and contained in a polygonal search corridor free of path constraints. Direct optimal control methods are then used to determine the optimal flight path through the newly defined search corridor. Two scenarios are evaluated. The first is limited to heading rate control only, requiring the air vehicle to maintain constant speed. The second allows for velocity control which permits slower speeds, reducing the vehicles minimum turn radius and increasing the search domain. Results illustrate the benefits gained when including speed control to path planning algorithms by comparing trajectory and convergence times, resulting in a reliable, hybrid solution method to the SUAS constrained optimal control problem.

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“…Since the approach proceeds in an incremental greedyoptimal fashion, it is possible to stop the process when any desired level of complexity, or approximation accuracy, is reached. The control law is then derived from this lower bound using the barycentric technique proposed in Warren et al [2007]. The result is a nonlinear and smooth piece-wise polynomial control law.…”

Section: Approximated Explicit Predictive Policymentioning

confidence: 99%

Online thermal control methods for multiprocessor systems

Zanini

Atienza

Jones

et al. 2013

ACM Trans. Des. Autom. Electron. Syst.

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With technological advances, the number of cores integrated on a chip is increasing. This in turn is leading to thermal constraints and thermal design challenges. Temperature gradients and hotspots not only affect the performance of the system but also lead to unreliable circuit operation and affect the lifetime of the chip. Meeting temperature constraints and reducing hotspots are critical for achieving reliable and efficient operation of complex multi-core systems.In this article, we analyze the use of four of the most promising families of online control techniques for thermal management of multiprocessors system-on-chip (MPSoC). In particular, in our exploration, we aim at achieving an online smooth thermal control action that minimizes the performance loss as well as the computational and hardware overhead of embedding a thermal management system inside the MPSoC. The definition of the optimization problem to tackle in this work considers the thermal profile of the system, its evolution over time, and current time-varying workload requirements. Thus, this problem is formulated as a finite-horizon optimal control problem, and we analyze the control features of different online thermal control approaches. In addition, we implemented the policies on an MPSoC hardware simulation platform and performed experiments on a cycle-accurate model of the eight-core Niagara multi-core architecture using benchmarks ranging from Web-accessing to playing multimedia. Results show different trade-offs among the analyzed techniques regarding the thermal profile, the frequency setting, the power consumption, and the implementation complexity.

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Barycentric coordinates for convex sets

Cited by 177 publications

References 16 publications

Convergence of Wachspress coordinates: from polygons to curved domains

Convergence of Wachspress coordinates: from polygons to curved domains

Simplex Solutions for Optimal Control Flight Paths in Urban Environments

Online thermal control methods for multiprocessor systems

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