B-spline basis

We employ a B-spline basis representation for the distribution function ... 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

M. Venuti, P. Decleva, Convergent multichannel continuum states by a general configuration interaction expansion in a B-spline basis Application to H photodetachment, J. Phys. B At. Mol. Opt. Phys. 30 (1997) 4839. 

The matrix M is determined by integrals of the products of the second derivatives of the B-spline basis functions used in the representation of the unknown distribution. This is a quadratic least-squares minimization problem. 

The quantitative method in Section 2.2 is used to determine the intrinsic magnetization intensity for each voxel. Cubic B-spline basis functions with a partition of 60 interior knots logarithmically spaced between 1 x 10 5 and 10 s are used to represent the relaxation distribution within each voxel. The optimal regularization parameter, A, of each voxel is found within the range between 1 x 10 5 and 5 x 10"18 s by using the UBPR9 criterion. 

For shape design purposes it is usually far more convenient to use a basis where the basis functions sum to f. All the coefficients then transform as points, and we call them control points. For polynomials this can be achieved by using the Bezier basis, which is a special case of the B-spline basis which will be encountered shortly. 

If the pieces are all of equal length in abscissa, all the B-spline basis functions are just translates of the same basic function, which is typically called the basis function of the given degree. The resulting curves are then called equal interval B-splines or uniform B-splines. We shall use the shorter term B-splines here, despite the fact that it is not strictly accurate. 

The characterisation of B-splines that we shall use here is that the B-spline basis function of degree n has the following properties... 

Clearly the first derivative of the B-spline basis function must have the properties that... 

From these we see that it must be a linear combination of exactly two B-spline basis functions of degree n— 1, and in fact it is exactly the difference of two consecutive basis functions of degree n — 1. 

To evaluate the sums over virtual states in Elqs. (102) and (103), we make use of the B-spline basis functions described later in this section. 

Taking the radial wave functions and energies for states n from the B-spline basis set, we may easily carry out the double sums in (128). The partial-wave contributions to from terms in square bracket are listed in Table 2. These terms fall-off approximately as L for large L and may easily be extrapolated. We find E = —0.0373736 a.u., leading to a binding energy of -0.8990800 a.u., differing from experiment by 0.5%. 

To solve Eqs. (176-181), an angular momentum decomposition is first carried out and the equations are then reduced to coupled equations involving single-body radial wave functions only. The radial wave functions for states v, m, n, a, b, are taken from a B-spline basis set [36] and the resulting coupled radial equations are solved iteratively. The core equations (176-177) are solved first and the valence equations (179-181) are then solved for valence states of interest using the converged core amplitudes. 

The set % is further symmetrized accordingly to the point group symmetry, and real harmonics are employed in order to avoid complex arithmetic. In this implementation the electron density is generated from a previous LCAO conventional DPT calculation with the program ADF [20], and further expanded using a numerical integration scheme with the same basis set in order to build the hamiltonian matrix. The Poisson equation is solved to get the Hartree term, as in the atomic calculation, and the B-spline basis set has proven very flexible so that the particular multipolar boundary conditions are easily satisfied. 

This input is then used for the CASINO programme which is a Quantum Monte Carlo code adapted to solid state work. It is converted into a B-spline basis, which has negligible influence on the accuracy and speeds up the Quantum Monte Carlo calculations considerably. These are cubic splines limited to radius 2 (i.e., /( ) = 0 if r = u > 2) of the form ... 

The orbitals were initialised using DFT in a plane-wave basis, then re-represent them in a blip (B-spline) basis for the QMC calculations, in order to improve the system-size scaling of QMC [1]. The blip coefficients and plane-wave coefficients are related by a Fourier transform. The number of blip coefficients is usually somewhat larger than the number of non-zero plane-wave coefficients, in order to make the blip grid finer in real space. 

Keywords Electronic correlation Relativistic effects Isotopic effects Multiply excited states Hollow atoms B-spline basis set Charge transfer Photodissociation Radiative association... 

Using a truncated diagonalization method with these B-spline basis sets, we have calculated the decay properties of all doubly- excited AlAl and A151 states of and... 

The idea of using B-spline basis sets for the representation of vibrational molecular wave functions emerged rapidly. For a Morse potential and a two-dimensional Henon-Heiles potential, we have assessed the efficiency of the B-splines over the conventional DVR (discrete variable representation) with a sine or a Laguerre basis sets [50]. In addition, the discretization of the vibrational continuum of energy when using the Galerkin method allows the calculation of photodissociation cross-sections in a time-independent approach. 

N.(u) are the normalised B-spline basis functions defined on the knot set ... 

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