B. Spline interpolation

Figure 7. B-spline interpolation between a set of reference points.

The PME method with Cardinal B-spline interpolation seheme is included as a standard routine in the AMBER [115] and DL POLY [114] simulation packages. 

Fig. 6. Cubic B-spline interpolation versus surface spline interpolation.

An explicit convergence proof for Sobolev norms can be found in [15). Let us fix the sampling nodes and increase the domain fi. What we observe is that, similar to the surface spline, the bicubic B-spline interpolant tends to a linear polynomial. 

Figure 4. (a) The electric potential (circles) as a function of the distance from the surface for the system described in the text. The continuous line represents a Spline interpolation, (b) The average polarization of a water molecule (squares) as a function of the distance from surface for the system described in text the macroscopic electric field (triangles) obtained through the numerical derivative of the potential is not proportional to the average polarization. 

Fig. 4. (A) Parametrization of a shaped pulse envelope with the help of a cubic spline interpolation between a small number of anchor points (circles). (B) Approximation of the smooth pulse envelope by rectangular pulses with piecewise constant rf amplitude. (Adapted from Ewing et al., 1990, p. 123, with kind permission from Elsevier Science—NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

Fig. 11. Introduction to application of splines to chromatograms30). a measured curve with digitized data b digitized data only c polygonal interpolation d spline interpolation...

This linear subspace can usefully be made more explicit by taking as the corners of our barycentric combination not the B-splines, but schemes satisfying the interpolation degree constraints. 

A nice example is the dual of the quadratic B-spline, which turns into a primal interpolating scheme (because each old line in the old polygon is retained in the new one). The limit curve (the envelope of the lines in the limit polygon) turns out to be a concatenation of conic section pieces. 

The end-conditions described above cover two distinct cases, those of interpolating schemes, which are likened to Lagrange interpolation, and those of B-splines, likened to the Bezier end-conditions. The schemes which interpolate when the data lies on a cubic or higher polynomial do not really fit either of these cases. They are almost interpolating (when the data is really smooth) but not quite. Somebody needs to play with these schemes to find out how they currently misbehave at the ends and what kinds of control are required to make them do what the curve designer wants. 

The tabulated values were interpolated with a B-spline routine to provide numerical values for any nuclear charge number Z. We display these values in Fig. 11 together with the nonrelativistic reduced mass correction which is included in AErec- For uranium, the total effect to the lsi/2-state is 0.51 eV, including the nonrelativistic correction of 0.30 eV. Compared to the radiative effects discussed so far this might be thought tiny, but an experimental precision of better than 1 eV demands also for the proper calculation of Lamb shift contributions of this size of magnitude. 

Fig. 9.31 Spline interpolation procedures for background corrections of PM IRRAS spectra, (a) A raw PM IRRAS spectrum of a DM PC bilayer on a Au(lll) surface at the electrode potential E=-0.2 V. (b) A spline (dashed line) for the first rough baseline correction, open circles are data points used...

Greiner, G., and Hermann, K. Interpolating and Approximating Scattered 3D Data with Hierarchical Tensor Product B-splines, Surface Fitting and Multiresolution Methods, A. Le Mehaute, C. Rabut, and L. L. Schumaker (eds),Vanderbilt University Press, Nashville, TN, (1997) pp. 163-172. 

Park, H. Choosing Nodes and Knots in Closed B-spline Curve Interpolation to Point Data, Computer-Aided Design, vol. 33, no. 13 (2001) pp. 967-974. 

A sculptured surface is obtained by interpolation with two-dimensional B-spline-functions. A B-spline surface is considered as a collection of surface patches and the whole surface is a mosaic of these patches linked together with proper continuity (Figure 9). Due to its computational efficiency a uniform bicubic B-spline surface has been implemented. The bicubic B-spline patch can be written in matrix form as... 

In Fig. 2, we summarize the detailed steps involved in developing the atomistic pore model from the lattice model. From the lattice configuration (Fig. 2a), we isolate a pore and model its surfece as a b-sphne [24], The b-spline approach serves as a reasonable interpolation to obtain a pore surfece with sub-atomistic resolution <0.1 nm. The control points in this b-spline model are the lattice surface coordinates, which allows us to retain the sur ce undulations and irregularities fiom the mimetic simulation. The atomistic pore is now... 

Figure 41. A schematic representation of the SOPHE partition scheme (a) SOPHE partition grid in which the three sets of curves are described hy Eq. (8) SOPHE partition number N equals 10 (b) subpartitioning into smaller triangles ean be performed by using either Eq. (8) or alternatively the points along the edge of the triangle are interpolated by the eubie spline interpolation method and each point inside the triangle is linearly interpolated three times and an average is taken.

Molecular dynamics was performed at constant temperature with AMBER 4.1 all-atom force field [121] and Particle Mesh Ewald method (PME) was used for the calculation of electrostatic interactions [122]. This is a fast implementation of the Ewald summation method for calculating the full electrostatic energy of a unit cell in a macroscopic lattice of repeating images. The PME grid spacing was 1.0A. It was interpolated on a cubic B-spline, with the direct set tolerance set to 0.000001. Periodic boundary conditions were imposed in all directions. All solute-solute non-bonded interactions were calculated without jmy cut-off distance, while a non-bonded residue based cutoff distance of 9A was used for the solvent-solvent and for the solute-solvent interactions. The non-bonded pair list was updated every 20 steps and the... 

---