Moreover, usually the underlying simulation gives only and , so must be numerically differentiated to obtain . As discussed by Thornburg [156, Section 6.1], it is somewhat more efficient to combine the numerical differentiation and interpolation operations, essentially doing the differentiation inside the interpolator27.
Thornburg [156, Section 6.1] argues that for an elliptic-PDE algorithm (Section 8.5), for best convergence of the nonlinear elliptic solver the interpolated geometry variables should be smooth (differentiable) functions of the trial horizon surface position. He argues that that the usual Lagrange polynomial interpolation does not suffice here (in some cases his Newton’s-method iteration failed to converge) because this interpolation gives results which are only piecewise differentiable28. He uses Hermite polynomial interpolation to avoid this problem. Cook and Abrahams , and Pfeiffer et al.  use bicubic spline interpolation; most other researchers either do not describe their interpolation scheme or use Lagrange polynomial interpolation.
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