On the convergence of hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we derive the bounds on the magnitude of l  th (l=2,3)$(l=2,3)$ order derivatives of rational Bézier curves, estimate the error, in the _L∞$L_{\infty }$ norm sense, for the hybrid polynomial approximation of the l  th (l=1,2,3)$(l=1,2,3)$ order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the l  th (l=1,2,3)$(l=1,2,3)$ derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.     

Necessary and sufficient conditions for rational quartic representation of conic sections

Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than π$\pi$. However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as being able to represent conics up to 2π$2\pi$ (but not including 2π$2\pi$) without resorting to negative weights and possessing better parameterization. Therefore, it is actually worth studying the necessary and sufficient conditions for the rational quartic Bézier representation of conics. This paper attributes the rational quartic conic sections to two special kinds, that is, degree-reducible and improperly parameterized; on this basis, the necessary and sufficient conditions for the rational quartic Bézier representation of conics are derived. They are divided into two parts: Bézier control points and weights. These conditions can be used to judge whether a rational quartic Bézier curve is a conic section; or for a given conic section, present positions of the control points and values of the weights of the conic section in form of a rational quartic Bézier curve. Many examples are given to show the use of our results.     

Circle approximation using LN Bézier curves of even degree and its application

We present an approximation method of circular arcs using linear-normal (LN) Bézier curves of even degree, four and higher. Our method achieves G^m$G^m$ continuity for endpoint interpolation of a circular arc by a LN Bézier curve of degree 2m  , for m=2,3$m=2,3$. We also present the exact Hausdorff distance between the circular arc and the approximating LN Bézier curve. We show that the LN curve has an approximation order of 2m+2$2m+2$, for m=2,3$m=2,3$. Our approximation method can be applied to offset approximation, so obtaining a rational Bézier curve as an offset approximant. We derive an algorithm for offset approximation based on the LN circle approximation and illustrate our method with some numerical examples.     

A note on constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

In the paper [H.S. Kim, Y.J. Ahn, Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain, J. Comput. Appl. Math. 216 (2008) 14–19], Kim and Ahn proved that the best constrained degree reduction of a polynomial over d$d$-dimensional simplex domain in _L2$L_2$-norm equals the best approximation of weighted Euclidean norm of the Bernstein–Bézier coefficients of the given polynomial. In this paper, we presented a counterexample to show that the approximating polynomial of lower degree to a polynomial is virtually non-existent when d≥2$d\ge 2$. Furthermore, we provide an assumption to guarantee the existence of solution for the constrained degree reduction.     

Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

In this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d  -dimensional simplex domain with respect to the _L2$L_2$-inner product and the weighted Euclidean inner product of BB (Bézier–Bernstein) coefficients are equal. Using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.     

On the approximation of smooth functions using generalized digital nets

In this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order α>1$\alpha >1$ in each variable. The approximation error is studied in the worst case setting in the _L2$L_2$ norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.     

Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions

In the present paper, we estimate the rate of pointwise convergence of the Bézier Variant of Chlodowsky operators _Cn,α$C_{n,\alpha }$ for functions, defined on the interval extending infinity, of bounded variation. To prove our main result, we have used some methods and techniques of probability theory.     

Finite approximation schemes for Lévy processes,and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T]$[0,T]$, and having a finite number of states, for a pure jump Lévy process _Lt$L_t$. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a “complexity” measure for the option pricing algorithm.     

Weak and quasi-polynomial tractability of approximation of infinitely differentiable functions

We comment on recent results in the field of information based complexity, which state (in a number of different settings), that the approximation of infinitely differentiable functions is intractable and suffers from the curse of dimensionality. We show that renorming the space of infinitely differentiable functions in a suitable way allows weakly tractable uniform approximation by using only function values. Moreover, the approximating algorithm is based on a simple application of Taylor’s expansion about the center of the unit cube. We discuss also the approximation on the Euclidean ball and the approximation in the _L1$L_1$-norm.     

The approximation of a Crank–Nicolson scheme for the stochastic Navier–Stokes equations

In this paper we prove the convergence of stochastic Navier–Stokes equations driven by white noise. A linearized version of the implicit Crank–Nicolson scheme is considered for the approximation of the solutions to the N–S equations. The noise is defined as the distributional derivative of a Wiener process and approximated by using the generalized _L2$L_2$-projection operator. Optimal strong convergence error estimates in the _L2$L_2$ norm are obtained.     

Approximate Bézier curves by cubic LN curves

In order to derive the offset curves by using cubic Bézier curves with a linear field of normal vectors (the so-called LN Bézier curves) more efficiently, three methods for approximating degree n Bézier curves by cubic LN Bézier curves are considered, which includes two traditional methods and one new method based on Hausdorff distance. The approximation based on shifting control points is equivalent to solving a quadratic equation, and the approximation based on L₂ norm is equivalent to solving a quartic equation. In addition, the sufficient and necessary condition of optimal approximation based on Hausdorff distance is presented, accordingly the algorithm for approximating the degree n Bézier curves based on Hausdorff distance is derived. Numerical examples show that the error of approximation based on Hausdorff distance is much smaller than that of approximation based on shifting control points and L₂ norm, furthermore, the algorithm based on Hausdorff distance is much simple and convenient.     

Poincaré and Opial inequalities for vector-valued convolution products

Various L^p$L^p$ form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of _C0$C_0$-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.     

Simultaneous classification and feature selection via convex quadratic programming with application to HIV-associated neurocognitive disorder assessment

Support vector machines (SVMs), that utilize a mixture of the _L1$L_1$-norm and the _L2$L_2$-norm penalties, are capable of performing simultaneous classification and selection of highly correlated features. These SVMs, typically set up as convex programming problems, are re-formulated here as simple convex quadratic minimization problems over non-negativity constraints, giving rise to a new formulation – the pq-SVM method. Solutions to our re-formulation are obtained efficiently by an extremely simple algorithm. Computational results on a range of publicly available datasets indicate that these methods allow greater classification accuracy in addition to selecting groups of highly correlated features. These methods were also compared on a new dataset assessing HIV-associated neurocognitive disorder in a group of 97 HIV-infected individuals.     

Nonparametric estimation for pure jump Lévy processes based on high frequency data

In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n$n$ discrete time observations with step Δ$\Delta$. The asymptotic framework is: n$n$ tends to infinity, Δ=_Δn$\Delta ={\Delta }_n$ tends to zero while n_Δn$n{\Delta }_n$ tends to infinity. First, we use a Fourier approach (“frequency domain”): this allows us to construct an adaptive nonparametric estimator and to provide a bound for the global L²${\mathbb{L}}^2$-risk. Second, we use a direct approach (“time domain”) which allows us to construct an estimator on a given compact interval. We provide a bound for L²${\mathbb{L}}^2$-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.     

Cauchy problem of Schrödinger-Improved Boussinesq Systems on the torus

In this paper, we will study the local well-posedness of Schrödinger-Improved Boussinesq System with additive noise in T^d${\mathbb{T}}^d$, d?1$d?1$, and we will also study the global well-posedness of dimension 1 case with the initial data (_u0,_v1,_v2)∈L²×L²×L²$(u_0,v_1,v_2)\in L^2\times L^2\times L^2$ almost surely, gaining some exponential growth of L²$L^2$ norm of v.     

Approximation rates for the hierarchical tensor format in periodic Sobolev spaces

In this note we estimate the asymptotic rates for the _L2$L_2$-error decay and the storage cost when approximating 2π$2\pi$-periodic, d$d$-variate functions from isotropic and mixed Sobolev classes by the recent hierarchical tensor format as introduced by Hackbusch and Kühn. To this end, we survey some results on bilinear approximation due to Temlyakov. The approach taken in this paper improves and generalizes recent results of Griebel and Harbrecht for the bi-variate case.     

Approximation of conic section by quartic Bzier curve with endpoints continuity condition

A new method for approximation of conic section by quartic B′ezier curve is presented, based on the quartic B′ezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G2 continuous spline approximation of conic section when using the subdivision scheme,and the effectiveness of this method is demonstrated by some numerical examples.     

Invariance principles for generalized domains of semistable attraction

Let X,_X1,_X2,…$X,X_1,X_2,\dots$ be independent and identically distributed R^d${\mathbb{R}}^d$-valued random vectors and assume X$X$ belongs to the generalized domain of attraction of some operator semistable law without normal component. Then without changing its distribution, one can redefine the sequence on a new probability space such that the properly affine normalized partial sums converge in probability and consequently even in L^p$L^p$ (for some p>0$p>0$) to the corresponding operator semistable Lévy motion.