scholarly journals Approximation of Gaussian Curvature by the Angular Defect: An Error Analysis

2021 ◽  
Vol 26 (1) ◽  
pp. 15
Author(s):  
Marie-Sophie Hartig
Keyword(s):  
Error Analysis ◽  
Gaussian Curvature ◽  
Linear Convergence ◽  
Convergence Speed ◽  
Science And Engineering ◽  
Voronoi Region ◽  
General Validity ◽  
Spherical Surfaces ◽  
Weaker Conditions ◽  
Special Case

It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian curvature at a vertex on the surface is approximated by the quotient of the angular defect and the area of the Voronoi region. The Voronoi region is the subset of the mesh that contains all points that are closer to the vertex than to any other vertex. Numerical error analyses suggest that the Gauss–Bonnet scheme always converges with quadratic convergence speed. However, the general validity of this conclusion remains uncertain. We perform an analytical error analysis on the Gauss–Bonnet scheme. Under certain conditions on the mesh, we derive the convergence speed of the Gauss–Bonnet scheme as a function of the maximal distance between the vertices. We show that the conditions are sufficient and necessary for a linear convergence speed. For the special case of locally spherical surfaces, we find a better convergence speed under weaker conditions. Furthermore, our analysis shows that the Gauss–Bonnet scheme, while generally efficient and effective, can give erroneous results in some specific cases.

scholarly journals On the linear convergence of the general first order primal-dual algorithm

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kai Wang ◽  
Deren Han
Keyword(s):  
Linear Convergence ◽  
Analytical Framework ◽  
Dual Algorithm ◽  
Linear Rate ◽  
First Order ◽  
Convex Problems ◽  
Primal Dual ◽  
The One ◽  
Weaker Conditions ◽  
Special Case

<p style='text-indent:20px;'>In this paper, we consider the general first order primal-dual algorithm, which covers several recent popular algorithms such as the one proposed in [Chambolle, A. and Pock T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011) 120-145] as a special case. Under suitable conditions, we prove its global convergence and analyze its linear rate of convergence. As compared to the results in the literature, we derive the corresponding results for the general case and under weaker conditions. Furthermore, the global linear rate of the linearized primal-dual algorithm is established in the same analytical framework.</p>

scholarly journals Power convexity of a class of Hessian equations in the ball

2015 ◽  
Vol 3 (3) ◽  
pp. 134
Author(s):  
Yunhua Ye
Keyword(s):  
Lower Bound ◽  
Gaussian Curvature ◽  
Principal Curvature ◽  
Power Functions ◽  
Strictly Convex ◽  
Hessian Equations ◽  
Curvature Estimates ◽  
Admissible Solutions ◽  
Special Case

<p>Power convexities of a class of Hessian equations are considered in this paper. It is proved that some power functions of the smooth admissible solutions to the Hessian equations are strictly convex in the ball. For a special case of the equation, a lower bound principal curvature and Gaussian curvature estimates are given.</p>

Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

2016 ◽  
Vol 8 (4) ◽  
pp. 648-669 ◽  
Author(s):  
Xiulian Shi ◽  
Yanping Chen
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Approximate Solutions ◽  
Weakly Singular Kernels ◽  
Numerical Examples ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Solutions Of Equations ◽  
Rigorous Error ◽  
Special Case

AbstractA spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially inL∞norm and weightedL2norm. Finally, two numerical examples are presented to demonstrate our error analysis.

scholarly journals Dynamics of Planar Systems That Model Stage-Structured Populations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
N. Lazaryan ◽  
H. Sedaghat
Keyword(s):  
Global Convergence ◽  
Saddle Point ◽  
Structured Populations ◽  
Vital Rates ◽  
Planar System ◽  
Density Dependent ◽  
Planar Systems ◽  
High Level ◽  
Weaker Conditions ◽  
Special Case

We study a general discrete planar system for modeling stage-structured populations. Our results include conditions for the global convergence of orbits to zero (extinction) when the parameters (vital rates) are time and density dependent. When the parameters are periodic we obtain weaker conditions for extinction. We also study a rational special case of the system for Beverton-Holt type interactions and show that the persistence equilibrium (in the positive quadrant) may be globally attracting even in the presence of interstage competition. However, we determine that with a sufficiently high level of competition, the persistence equilibrium becomes unstable (a saddle point) and the system exhibits period two oscillations.

scholarly journals On ε-Optimality of the Pursuit Learning Algorithm

2012 ◽  
Vol 49 (3) ◽  
pp. 795-805 ◽  
Author(s):  
Ryan Martin ◽  
Omkar Tilak
Keyword(s):  
Learning Algorithm ◽  
Learning Automata ◽  
Tuning Parameter ◽  
Convergence Properties ◽  
Science And Engineering ◽  
Practical Applications ◽  
Tuning Parameters ◽  
Time Optimization ◽  
Real Time Optimization ◽  
Special Case

Estimator algorithms in learning automata are useful tools for adaptive, real-time optimization in computer science and engineering applications. In this paper we investigate theoretical convergence properties for a special case of estimator algorithms - the pursuit learning algorithm. We identify and fill a gap in existing proofs of probabilistic convergence for pursuit learning. It is tradition to take the pursuit learning tuning parameter to be fixed in practical applications, but our proof sheds light on the importance of a vanishing sequence of tuning parameters in a theoretical convergence analysis.

scholarly journals On ε-Optimality of the Pursuit Learning Algorithm

2012 ◽  
Vol 49 (03) ◽  
pp. 795-805 ◽  
Author(s):  
Ryan Martin ◽  
Omkar Tilak
Keyword(s):  
Learning Algorithm ◽  
Learning Automata ◽  
Tuning Parameter ◽  
Convergence Properties ◽  
Science And Engineering ◽  
Practical Applications ◽  
Tuning Parameters ◽  
Time Optimization ◽  
Real Time Optimization ◽  
Special Case

Estimator algorithms in learning automata are useful tools for adaptive, real-time optimization in computer science and engineering applications. In this paper we investigate theoretical convergence properties for a special case of estimator algorithms - the pursuit learning algorithm. We identify and fill a gap in existing proofs of probabilistic convergence for pursuit learning. It is tradition to take the pursuit learning tuning parameter to be fixed in practical applications, but our proof sheds light on the importance of a vanishing sequence of tuning parameters in a theoretical convergence analysis.

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