242
9 High-Dimension Model Representation via Sparse GridTechniques
Fig. 9.29 Showing the Chebyshev distribution of points in various two-dimensional subsets of the
four-dimensional grid at level 8 (continued)
The five-dimensional parameter space ranges over: LO = [0, 9.99], Θ =
[0, 9.99], Ψ = [0, 90], L = [1.7, 2.3], D = [2.7, 3.3], which at level 4 will be
covered by 311 Chebyshev points. These are the values that are presented to VIC3D ® to generate the sparse interpolation table. The first NLSE full Cartesian table
consists of four points in each dimension, uniformly distributed over its range, which
yields 1024 nodes for NLSE. The results of the first inversion test are shown in
Table 9.7. The final column in the table gives the number of local minima generated
by the 500 random starting points that coalesce into the global minimum. In order to
improve the accuracy of the inversion of Ψ , we increase the number of NLSE nodes
in this parameter to 6, and get the result shown as Test 2 in Table 9.7. Clearly, this
parameter benefits from a denser nodal distribution because it covers a large range
of [0,90].
9 High-Dimension Model Representation via Sparse GridTechniques
Fig. 9.29 Showing the Chebyshev distribution of points in various two-dimensional subsets of the
four-dimensional grid at level 8 (continued)
The five-dimensional parameter space ranges over: LO = [0, 9.99], Θ =
[0, 9.99], Ψ = [0, 90], L = [1.7, 2.3], D = [2.7, 3.3], which at level 4 will be
covered by 311 Chebyshev points. These are the values that are presented to VIC3D ® to generate the sparse interpolation table. The first NLSE full Cartesian table
consists of four points in each dimension, uniformly distributed over its range, which
yields 1024 nodes for NLSE. The results of the first inversion test are shown in
Table 9.7. The final column in the table gives the number of local minima generated
by the 500 random starting points that coalesce into the global minimum. In order to
improve the accuracy of the inversion of Ψ , we increase the number of NLSE nodes
in this parameter to 6, and get the result shown as Test 2 in Table 9.7. Clearly, this
parameter benefits from a denser nodal distribution because it covers a large range
of [0,90].