scholarly journals The Mask of Odd Pointsn-Ary Interpolating Subdivision Scheme

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Ghulam Mustafa ◽  
Jiansong Deng ◽  
Pakeeza Ashraf ◽  
Najma Abdul Rehman
Keyword(s):  
Explicit Formula ◽  
Error Bounds ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Total Absolute Curvature ◽  
Absolute Curvature ◽  
Special Cases ◽  
Preservation Property ◽  
Subdivision Curves ◽  
And Control

We present an explicit formula for the mask of odd pointsn-ary, for any oddn⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-pointa-ary schemes introduced by Lian, 2008, and (2m+1)-pointa-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd pointn-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented.

scholarly journals (2n-1)-Point Ternary Approximating and Interpolating Subdivision Schemes

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Muhammad Aslam ◽  
Ghulam Mustafa ◽  
Abdul Ghaffar
Keyword(s):  
Explicit Formula ◽  
Error Bounds ◽  
Computational Cost ◽  
Subdivision Schemes ◽  
Special Cases ◽  
Better Than

We present an explicit formula which unifies the mask of(2n-1)-point ternary interpolating as well as approximating subdivision schemes. We observe that the odd point ternary interpolating and approximating schemes introduced by Lian (2009), Siddiqi and Rehan (2010, 2009) and Hassan and Dodgson (2003) are special cases of our proposed masks/schemes. Moreover, schemes introduced by Zheng et al. (2009) can easily be generated by our proposed masks. It is also proved from comparison that(2n-1)-point schemes are better than2n-scheme in the sense of computational cost, support and error bounds.

Designing General p-Ary n-Point Smooth Subdivision Schemes

2014 ◽  
Vol 472 ◽  
pp. 510-515 ◽  
Author(s):  
Hong Chan Zheng ◽  
Qian Song
Keyword(s):  
Smooth Curve ◽  
Approximation Order ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Special Cases ◽  
Two Parameters ◽  
Interpolatory Subdivision

In this paper, in order to produce smooth curve, we design a class of n-point p-ary smooth interpolatory subdivision schemes that can reproduce polynomials of degree n-1 with approximation order n. Many classical interpolatory subdivision schemes are special cases of this kind of subdivision. We illustrate the approach with a new 5-point quaternary interpolatory subdivision scheme with two parameters, which reproduces polynomial of degree 4 with approximation order of 5 and can generate new interpolatory curves.

scholarly journals (n-2)-tightness and curvature of submanifolds with boundary

1978 ◽  
Vol 1 (4) ◽  
pp. 421-431 ◽  
Author(s):  
Wolfgang Kühnel
Keyword(s):  
Euclidean Space ◽  
Geometric Inequality ◽  
Total Absolute Curvature ◽  
Absolute Curvature

The purpose of this note is to establish a connection between the notion of(n−2)-tightness in the sense of N.H. Kuiper and T.F. Banchoff and the total absolute curvature of compact submanifolds-with-boundary of even dimension in Euclidean space. The argument used is a certain geometric inequality similar to that of S.S. Chern and R.K. Lashof where equality characterizes(n−2)-tightness.

scholarly journals A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 66 ◽  
Author(s):  
Aamir Shahzad ◽  
Faheem Khan ◽  
Abdul Ghaffar ◽  
Ghulam Mustafa ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Numerical Algorithm ◽  
Subdivision Schemes ◽  
Practical Applications ◽  
Curves And Surfaces ◽  
Geometrical Shapes ◽  
Subdivision Depth ◽  
Subdivision Curves

Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization.

Variance Regularization in Sequential Bayesian Optimization

2020 ◽  
Vol 45 (3) ◽  
pp. 966-992
Author(s):  
Michael Jong Kim
Keyword(s):  
Error Bounds ◽  
Broad Class ◽  
Planning Horizon ◽  
Bayesian Optimization ◽  
Model Parameters ◽  
Regularization Term ◽  
Problem Class ◽  
Special Cases ◽  
The Impact ◽  
Bayesian Dynamic Programming

Sequential Bayesian optimization constitutes an important and broad class of problems where model parameters are not known a priori but need to be learned over time using Bayesian updating. It is known that the solution to these problems can in principle be obtained by solving the Bayesian dynamic programming (BDP) equation. Although the BDP equation can be solved in certain special cases (for example, when posteriors have low-dimensional representations), solving this equation in general is computationally intractable and remains an open problem. A second unresolved issue with the BDP equation lies in its (rather generic) interpretation. Beyond the standard narrative of balancing immediate versus future costs—an interpretation common to all dynamic programs with or without learning—the BDP equation does not provide much insight into the underlying mechanism by which sequential Bayesian optimization trades off between learning (exploration) and optimization (exploitation), the distinguishing feature of this problem class. The goal of this paper is to develop good approximations (with error bounds) to the BDP equation that help address the issues of computation and interpretation. To this end, we show how the BDP equation can be represented as a tractable single-stage optimization problem that trades off between a myopic term and a “variance regularization” term that measures the total solution variability over the remaining planning horizon. Intuitively, the myopic term can be regarded as a pure exploitation objective that ignores the impact of future learning, whereas the variance regularization term captures a pure exploration objective that only puts value on solutions that resolve statistical uncertainty. We develop quantitative error bounds for this representation and prove that the error tends to zero like o(n-1) almost surely in the number of stages n, which as a corollary, establishes strong consistency of the approximate solution.

scholarly journals Fractal Behavior of a Ternary 4-Point Rational Interpolation Subdivision Scheme

2018 ◽  
Vol 23 (4) ◽  
pp. 65 ◽  
Author(s):  
Kaijun Peng ◽  
Jieqing Tan ◽  
Zhiming Li ◽  
Li Zhang
Keyword(s):  
Singular Point ◽  
Sufficient Conditions ◽  
Rational Interpolation ◽  
Subdivision Scheme ◽  
Fractal Curve ◽  
Rational Scheme ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Parameter Values ◽  
Selection Of

In this paper, a ternary 4-point rational interpolation subdivision scheme is presented, and the necessary and sufficient conditions of the continuity are analyzed. The generalization incorporates existing schemes as special cases: Hassan–Ivrissimtzis’s scheme, Siddiqi–Rehan’s scheme, and Siddiqi–Ahmad’s scheme. Furthermore, the fractal behavior of the scheme is investigated and analyzed, and the range of the parameter of the fractal curve is the neighborhood of the singular point of the rational scheme. When the fractal curve and surface are reconstructed, it is convenient for the selection of parameter values.

scholarly journals Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

2019 ◽  
Vol 181 (2) ◽  
pp. 473-507 ◽  
Author(s):  
E. Ruben van Beesten ◽  
Ward Romeijnders
Keyword(s):  
At Risk ◽  
Error Bounds ◽  
Value At Risk ◽  
Risk Measure ◽  
Mixed Integer ◽  
Conditional Value At Risk ◽  
Two Stage ◽  
Special Cases ◽  
Integer Recourse ◽  
Mean Risk

Abstract In traditional two-stage mixed-integer recourse models, the expected value of the total costs is minimized. In order to address risk-averse attitudes of decision makers, we consider a weighted mean-risk objective instead. Conditional value-at-risk is used as our risk measure. Integrality conditions on decision variables make the model non-convex and hence, hard to solve. To tackle this problem, we derive convex approximation models and corresponding error bounds, that depend on the total variations of the density functions of the random right-hand side variables in the model. We show that the error bounds converge to zero if these total variations go to zero. In addition, for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds.

Enumeration of Singular Configurations for Robotic Manipulators

1991 ◽  
Vol 113 (3) ◽  
pp. 272-279 ◽  
Author(s):  
H. Lipkin ◽  
E. Pohl
Keyword(s):  
Joint Angle ◽  
Robotic Manipulators ◽  
Systematic Procedure ◽  
Singular Configurations ◽  
Special Cases ◽  
Angle Space ◽  
Six Degree Of Freedom ◽  
Kinematic Singularities ◽  
And Control ◽  
Space Approach

Kinematic singularities are important considerations in the design and control of robotic manipulators. For six degree-of-freedom manipulators, the vanishing of the determinant of the Jacobian yields the conditions for the primary singularities. Examining the vanishing of the minors of the Jacobian yields further singularities which are special cases of the primary ones. A systematic procedure is presented to efficiently enumerate all possible singular configurations. Special geometries of representative manipulators are exploited by expressing the Jacobian in terms of vector elements. In contrast to using a joint-angle space approach, the resulting expressions yield direct physical interpretations.

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