Application of least square method to arbitrary-order problems with separated boundary conditions

In this paper, differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. In fact it will be shown that for every ε>0$\epsilon >0$, there exists an approximate solution _vε$v_{\epsilon }$ of B-spline functions such that the corresponding least square error is less than ε>0$\epsilon >0$, and also _vε$v_{\epsilon }$ satisfies the exact boundary conditions. Some examples are given and the optimal errors are obtained for the sake of comparison.     

用B-splines技术计算超强平行外磁场中氢分子离子的能级

考虑两核之间的相互作用,采用单中心展开方法和B-splines技术构建氢分子离子的径向和角向波函数,计算超强平行外磁场中氢分子离子的能级E_b(m=0,-1,-2,-3,-4,-5)和两核间的平衡距离2R.计算中,磁场强度从2.35×10⁶T变到2.35×10⁸T.对比绝热变分近似计算结果,计算精度提高的范围在0.00015 Ry~0.0488 Ry之间;对比变分近似计算结果,计算精度提高的范围在0.0029 Ry~0.0257 Ry之间.这些计算结果表明,用二重B-splines基函数展开方法使氢分子离子的波函数的完备性更好,并能得到比绝热变分近似和绝热近似方法更高的计算精度.     

Energy Spectra of the Confined Atoms Obtained by Using B-Splines

We have calculated the energy spectra of one- and two-electron atoms (ions) centered in an impenetrable spherical box by variational method with B-splines as basis functions. Accurate results are obtained for both large and small radii of confinement. The critical box radius of confined hydrogen atom is also calculated to show the usefulness of our method. A partial energy degeneracy in confined hydrogen atom is found when the radius of spherical box is equal to the distance at which a node of single-node wavefunctions of free hydrogen atom is located.     

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In this paper an efficient estimation methodology for the partially linear models with random effects is proposed. For this, we use the generalized least square estimate (GLSE) and the B-splines methods to estimate the unknowns, and employ the penalized least square method to obtain the estimators of the random effects item. Further, we also consider the estimation for the variance components. Compared with the existing methods, our proposed methodology performs well. The asymptotic properties of the estimators are obtained. A simulation study is carried out to assess the performance of our proposed methodology.     

H2^＋-Like Impurities Confined by Spherical Quantum Dots： a Candidate for Charge Qubits

We have calculated the electron energy of the ground and lower excited states for H2^＋-like impurity states confined in finite spherical quantum dots in GaAs. Based on the characteristics of energy levels, we have proposed a scheme for realizing charge qubit composed by the the ground and the first excited states of this confined double donor system for the first time. In the proposed scheme the charge qubit is coded in terms of the located electronic states.     

A few remarks on recurrence relations for geometrically continuous piecewise Chebyshevian B-splines

This works complements a recent article (Mazure, J. Comp. Appl. Math. 219(2):457–470, 2008) in which we showed that T. Lyche’s recurrence relations for Chebyshevian B-splines (Lyche, Constr. Approx. 1:155–178, 1985) naturally emerged from blossoms and their properties via de Boor type algorithms. Based on Chebyshevian divided differences, T. Lyche’s approach concerned splines with all sections in the same Chebyshev space and with ordinary connections at the knots. Here, we consider geometrically continuous piecewise Chebyshevian splines, namely, splines with sections in different Chebyshev spaces, and with geometric connections at the knots. In this general framework, we proved in (Mazure, Constr. Approx. 20:603–624, 2004) that existence of B-spline bases could not be separated from existence of blossoms. Actually, the present paper enhances the powerfulness of blossoms in which not only B-splines are inherent, but also their recurrence relations. We compare this fact with the work by G. Mühlbach and Y. Tang (Mühlbach and Tang, Num. Alg. 41:35–78, 2006) who obtained the same recurrence relations via generalised Chebyshevian divided differences, but only under some total positivity assumption on the connexion matrices. We illustrate this comparison with splines with four-dimensional sections. The general situation addressed here also enhances the differences of behaviour between B-splines and the functions of smaller and smaller supports involved in the recurrence relations.     

Statistical Encounters with Complex B-Splines

Complex B-splines as introduced in Forster et al. (Appl. Comput. Harmon. Anal. 20:281–282, 2006) are an extension of Schoenberg’s cardinal splines to include complex orders. We exhibit relationships between these complex B-splines and the complex analogues of the classical difference and divided difference operators and prove a generalization of the Hermite–Genocchi formula. This generalized Hermite–Genocchi formula then gives rise to a more general class of complex B-splines that allows for some interesting stochastic interpretations.      

B-spline solver for one-electron Schrödinger equation

A numerical algorithm for solving the one-electron Schrödinger equation is presented. The algorithm is based on the Finite Element method, and the basis functions are tensor products of univariate B-splines. The application of cubic or higher order B-splines guarantees that the searched solution belongs to a continuous and one time differentiable function space, which is a desirable property in the Kohn–Sham equation context from the Density Functional Theory with pseudopotential approximation. The theoretical background of the numerical algorithm is presented, and additionally, the implementation on parallel computers with distributed memory is described. The current implementation of the algorithm uses the MPI, HYPRE and ParMETIS libraries to distribute matrices on processing units. Additionally, the LOBPCG algorithm from HYPRE library is used to solve the algebraic generalized eigenvalue problem. The proposed algorithm works for any smooth interaction potential, where the domain of the problem is a finite subspace of the ?³ space. The accuracy of the algorithm is demonstrated for a selected interaction potential. In the current stage, the algorithm can be applied to solve the linearized Kohn–Sham equation for molecular systems.     

Spline R-Function and Applications in FEM

R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives. However, there is square root operation in the representation. Considering that the use of splines will facilitate the calculations within the CAD system, in this paper, we propose a system of R-functions represented in spline form called Spline R-function (SR). After transforming the function ranges of two base primitives to a new coordinate system, a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise function. Representation of SR in both Bézier form and B-spline form have been given. Among which the Bézier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of R-functions. The construction processes for both Boolean intersection and union operations with different smoothness are discussed in detail. Numerical experiments are conducted to show the potential of the proposed spline R-function.     

Polyharmonic splines: An approximation method for noisy scattered data of extra-large size

The aim of this paper is to provide a fast method, with a good quality of reproduction, to recover functions from very large and irregularly scattered samples of noisy data, which may present outliers. To the given sample of size N, we associate a uniform grid and, around each grid point, we condense the local information given by the noisy data by a suitable estimator. The recovering is then performed by a stable interpolation based on isotropic polyharmonic B-splines. Due to the good approximation rate, we need only M?N degrees of freedom to recover the phenomenon faithfully.