*5.2. Multilevel Approximation*

The multilevel scattered approximation was implemented first in [8] and then studied by a number of other researchers [9–13]. In the multilevel algorithm, the residual can be formed on the coarsest level first and then be approximated on the later finer level by the compactly supported radial basis functions with gradually smaller support. This process can be repeated and be stopped on the finest level. An advantage of this multilevel interpolation algorithm is its recursive property (i.e., the same routine can be applied recursively at each level in the programming language), of course the disadvantage being the allocation that memory needs.

In this experiment, suppose a 3D target surface is an explicit function *f*(*<sup>x</sup>*, *y*, *z*) = <sup>64</sup>*x*(<sup>1</sup> − *x*)*y*(<sup>1</sup> − *y*)*z*(<sup>1</sup> − *<sup>z</sup>*). We generate a uniform points set in the domain [0, 1] 3, with levels *k* = 1, 2, 3, 4 and *N* = 27, 125, 729, 4913. The scale parameter *ε* = 0.07 × <sup>2</sup>[0:3], and *l* = 3. The corresponding slice plots, the iso-surfaces, and slice plots of the absolute error are shown in Figures 4–7. Both the iso-surfaces and the slice plots are color coded according to the absolute error. At each level, the trial space is constructed by a series of truncated exponential radial basis functions with varying support radii. Hence, the multilevel approximation algorithm can produce a well conditioned sparse discrete algebraic system in each recursion and keep ideal approximation accuracy at the same time. Numerical experiments show that TERBF multilevel interpolation is very effective for 3D explicit surface approximation. These observations can be found from Figures 4–7. Similar experiments and observations are reported in detail in Fasshauer's book [3], where Wendland's function *C*<sup>4</sup> has been used for approximation. However, to improve the allocation memory needs of the multilevel algorithm, we can make use of the hierarchical collocation method developed in [13].

**Figure 4.** Fits and errors at Level 1.

**Figure 6.** Fits and errors at Level 3.

**Figure 7.** Fits and errors at Level 4.
