Randomized progressive iterative approximation for B-spline curve and surface fittings

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA

The cyclicity of the period annulus of a reversible quadratic system

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system x˙ = y + ax2, y˙ = −x with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycle, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods can not be applied, except in [6] a computer-assisted method was used. In this paper we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by F. Dumortier and R. Roussarie in [5] for q ≤ 2. The method may be used in other problems

Preconditioned geometric iterative methods for cubic B-spline interpolation curves

The geometric iterative method (GIM) is widely used in data interpolation/fitting, but its slow convergence affects the computational efficiency. Recently, much work was done to guarantee the acceleration of GIM in the literature. In this work, we aim to further accelerate the rate of convergence by introducing a preconditioning technique. After constructing the preconditioner, we preprocess the progressive iterative approximation (PIA) and its variants, called the preconditioned GIMs. We show that the proposed preconditioned GIMs converge and the extra computation cost brought by the preconditioning technique is negligible. Several numerical experiments are given to demonstrate that our preconditioner can accelerate the convergence rate of PIA and its variants

On the number of limit cycles bifurcating from a non-global degenerated center

AbstractWe give an upper bound for the number of zeros of an Abelian integral. This integral controls the number of limit cycles that bifurcate, by a polynomial perturbation of arbitrary degree n, from the periodic orbits of the integrable system (1+x)dH=0, where H is the quasi-homogeneous Hamiltonian H(x,y)=x2k/(2k)+y2/2. The tools used in our proofs are the Argument Principle applied to a suitable complex extension of the Abelian integral and some techniques in real analysis

On the extended randomized multiple row method for solving linear least-squares problems

The randomized row method is a popular representative of the iterative algorithm because of its efficiency in solving the overdetermined and consistent systems of linear equations. In this paper, we present an extended randomized multiple row method to solve a given overdetermined and inconsistent linear system and analyze its computational complexities at each iteration. We prove that the proposed method can linearly converge in the mean square to the least-squares solution with a minimum Euclidean norm. Several numerical studies are presented to corroborate our theoretical findings. The real-world applications, such as image reconstruction and large noisy data fitting in computer-aided geometric design, are also presented for illustration purposes

Coagulation Behavior of Aluminum Salts in Eutrophic Water: Significance of Al13Species and pH Control

The coagulation behavior of aluminum salts in a eutrophic source water was investigated from the viewpoint of Al(III) hydrolysis species transformation. Particular emphasis was paid to the coagulation effect of Al-13 species on removing particles and organic matter. The coagulation behavior of Al coagulants with different basicities was examined through jar tests and hydrolyzed Al(III) speciation distribution characterization in the coagulation process. The results showed that the coagulation efficiency of Al coagulants positively correlated with the content of Al-13 in the coagulation process rather than in the initial coagulants. Aluminum chloride (AlCl3) was more effective than polyaluminum chloride (PACT) in removing turbidity and dissolved organic matter in eutrophic water because AlCl3 could not only generate Al-13 species but also function as a pH control agent in the coagulation process. The solid-state Al-27 NMR spectra revealed that the precipitates formed from AlCl3 and PACT were significantly different and proved that the preformed Al-13 polymer was more stable than the in situ formed one during the coagulation process. Through regulating Al speciation, pH control could improve the coagulation process especially in DOC removal, and AlCl3 benefited most from pH control