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Shape Quality for Generalized Barycentric Interpolation
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
Andrew Gillette, Alexander Rand
We omit a full proof of Lemma 2.8 and instead provide a sketch of the key ideas; the interested reader can find relevant technical details in the proof of [164, Proposition 10]. In the mathematical analysis literature, the constant in the Sobolev embedding theorem is typically derived from the “cone condition” [3]. This condition requires the existence of a cone such that for any point x∈P $ \mathbf{x}\in P $ , the cone can be rotated and translated so that it lies entirely inside P with its apex at x $ \mathbf{x} $ . We can prove Lemma 2.8 by finding a single cone, depending only on the aspect ratio bound, such that the cone condition holds for any domain in pcvx∩par $ \mathbf{p}_{ \mathrm cvx }\cap \mathbf{p}_{ \mathrm ar } $ . In the polygonal case, a minimum interior angle bound is required so that the specific cone can be placed near the vertices, and Lemma 2.6 (2.1) provides this guarantee.
Flight Planning
Published in Yasmina Bestaoui Sebbane, Multi-UAV Planning and Task Allocation, 2020
The work presented in [373] is concerned with developing an algorithm that keeps a certain safety distance while passing an arbitrary number of possibly moving but nonmaneuvering obstacles. Starting from a 3D collision cone condition, input–output linearization is used to design a nonlinear guidance law [79]. The remaining design parameters are determined considering convergence and performance properties of the closed-loop guidance loop. Then, the guidance framework is developed in terms of a constrained optimization problem that can avoid multiple obstacles simultaneously while incorporating other mission objectives.
Theoretical Foundations of the Finite Element Method
Published in Sandeep Kumar, Ashish Pathak, Debashis Khan, Mathematical Theory of Subdivision, 2019
Sandeep Kumar, Ashish Pathak, Debashis Khan
For every point x∈Ω as a vertex, if we can form a cone of some arbitrarily small solid angle and orient it so that all the points of the cone lie in the domain, then we say that the open bounded domain satisfies the cone condition. A three-dimensional cone will look like an ice-cream cone. Domains with cusps will not satisfy the cone condition.
Simplified Levenberg–Marquardt method in Banach spaces for nonlinear ill-posed operator equations
Published in Applicable Analysis, 2023
Pallavi Mahale, Farheen M. Shaikh
From Assumption 3.1(d), we have for all . This shows that is continuous on . The smoothness condition on the space X in Assumption 3.1 (a) guarantees that the duality mapping is single-valued. In the literature on nonlinear ill-posed operator equation, the condition in Assumption 3.1 (d) is named as tangential cone condition. As mentioned earlier, in [19], Scherzer has constructed an operator close to Fréchet derivative satisfying assumption similar to Assumption 3.1 (d) for the class of parameter identification problems for elliptic equations. Recently in [24], Clason and Nhu have introduced Boudigand–Landweber iterative method for nonlinear ill-posed operator equation involving non-smooth operators. For the convergence analysis, they have used an assumption analogs to Assumption 3.1(d) and also constructed a family of operators for a class of parameter identification problem satisfying the assumption. In case when F is differentiable, we can take in Assumption 3.1 (c) and (d).
REGINN-IT method with general convex penalty terms for nonlinear inverse problems
Published in Applicable Analysis, 2022
F is weakly closed on , i.e. for any sequence , if and then there hold .There is a such that and .For each there is a continuous bounded linear operator such that (Tangential cone condition, TCC) There exists a constant such that for all .
Optimal control problems with control complementarity constraints: existence results, optimality conditions, and a penalty method
Published in Optimization Methods and Software, 2020
Christian Clason, Yu Deng, Patrick Mehlitz, Uwe Prüfert
The domain is nonempty, bounded, and satisfies the cone condition. Its boundary will be denoted by . Let the observation space as well as the state space be Hilbert spaces. The target will be fixed. The operator is an isomorphism while and are arbitrarily chosen. Finally, holds.