9.2 Mathematical Structure of the Problem
219
cube, resulting in eight nodes at the boundaries of the hypercube. Similarly, the
third through fifth entries correspond to the intersection of hyperplanes with the
hypercube, resulting in eight nodes for each entry, yielding a total of 48 nodes. Each
node carries a blending function that VIC-3D ® must compute using the appropriate
parameters of Fig. 6.3. For example, the blending function corresponding to the
second entry in the table would have the first slab of Fig. 6.3 fixed at 10 mils depth,
and the other three cycling through 0 and 20 mils, each, giving a VIC-3D ® problem
with 8 range values.
We turn our attention, now, to the hierarchical structure of the algorithm, which
lies at the heart of (9.5). Using the format of Table 9.2, we expand (9.5) as follows:
f (x) =
|l| 1 ≤1
j∈B l
α l,j φ l,j (x)
=
j∈B 0000
α 0000,j φ 0000,j (x) +
j∈B 1000
α 1000,j φ 1000,j (x) +
j∈B 0100
α 0100,j φ 0100,j (x)
+
j∈B 0010
α 0010,j φ 0010,j (x) +
j∈B 0001
α 0001,j φ 0001,j (x) .
(9.6)
Because the expansion functions, {φ l,j (x)}, are nonoverlapping for a given level, l,
and have a unit amplitude, the expansion coefficients, {α l,j }, are simply equal to the
blending function associated with the node of the appropriate function at level l.
Figure 9.2 illustrates the situation in one dimension at levels 0 and 1. In this
example, we have α 0,0 = BF (0), α 0,20 = BF (20), α 1,10 = SURPLUS, where
SURPLUS = BF (10) − 1/2(BF (20) + BF (0)). Hence, the expansion shown in
Fig. 9.2 is given by
f (x) = BF (0)φ 0,0 (x) + BF (20)φ 0,20 (x) + SURPLUSφ 1,10 (x) ,
(9.7)
where φ 0,0 (x) is the slide function, S(x), and φ 0,20 (x) is the ramp function, R(x),
in Fig. 9.2.
It is clear that the name ‘SURPLUS’ (called hierarchical surplus in [43])
denotes the excess in function value that the higher-order levels are supposed
to accommodate. It’s evaluation at level l requires only one additional blending
function to be computed, while using two previously computed at level l − 1. This is
an advantage of the hierarchical structure of the algorithm. If SURPLUS=0, then the
interpolator would treat this function as being linear, instead of piecewise linear, in
this dimension. Thus, the expansion, (9.6), is reminiscent of a Taylor series, in which
the terms corresponding to higher levels of the hierarchy correspond to higher-order
polynomial terms in the Taylor series.
The full four-dimensional expansion of (9.6) is given next. The first term is:
BF (0, 0, 0, 0)S(x 1 )S(x 2 )S(x 3 )S(x 4 ) + BF (20, 0, 0, 0)R(x 1 )S(x 2 )S(x 3 )S(x 4 ) +
BF (0, 20, 0, 0)S(x 1 )R(x 2 )S(x 3 )S(x 4 ) + BF (20, 20, 0, 0)R(x 1 )R(x 2 )S(x 3 )S(x 4 ) +
219
cube, resulting in eight nodes at the boundaries of the hypercube. Similarly, the
third through fifth entries correspond to the intersection of hyperplanes with the
hypercube, resulting in eight nodes for each entry, yielding a total of 48 nodes. Each
node carries a blending function that VIC-3D ® must compute using the appropriate
parameters of Fig. 6.3. For example, the blending function corresponding to the
second entry in the table would have the first slab of Fig. 6.3 fixed at 10 mils depth,
and the other three cycling through 0 and 20 mils, each, giving a VIC-3D ® problem
with 8 range values.
We turn our attention, now, to the hierarchical structure of the algorithm, which
lies at the heart of (9.5). Using the format of Table 9.2, we expand (9.5) as follows:
f (x) =
|l| 1 ≤1
j∈B l
α l,j φ l,j (x)
=
j∈B 0000
α 0000,j φ 0000,j (x) +
j∈B 1000
α 1000,j φ 1000,j (x) +
j∈B 0100
α 0100,j φ 0100,j (x)
+
j∈B 0010
α 0010,j φ 0010,j (x) +
j∈B 0001
α 0001,j φ 0001,j (x) .
(9.6)
Because the expansion functions, {φ l,j (x)}, are nonoverlapping for a given level, l,
and have a unit amplitude, the expansion coefficients, {α l,j }, are simply equal to the
blending function associated with the node of the appropriate function at level l.
Figure 9.2 illustrates the situation in one dimension at levels 0 and 1. In this
example, we have α 0,0 = BF (0), α 0,20 = BF (20), α 1,10 = SURPLUS, where
SURPLUS = BF (10) − 1/2(BF (20) + BF (0)). Hence, the expansion shown in
Fig. 9.2 is given by
f (x) = BF (0)φ 0,0 (x) + BF (20)φ 0,20 (x) + SURPLUSφ 1,10 (x) ,
(9.7)
where φ 0,0 (x) is the slide function, S(x), and φ 0,20 (x) is the ramp function, R(x),
in Fig. 9.2.
It is clear that the name ‘SURPLUS’ (called hierarchical surplus in [43])
denotes the excess in function value that the higher-order levels are supposed
to accommodate. It’s evaluation at level l requires only one additional blending
function to be computed, while using two previously computed at level l − 1. This is
an advantage of the hierarchical structure of the algorithm. If SURPLUS=0, then the
interpolator would treat this function as being linear, instead of piecewise linear, in
this dimension. Thus, the expansion, (9.6), is reminiscent of a Taylor series, in which
the terms corresponding to higher levels of the hierarchy correspond to higher-order
polynomial terms in the Taylor series.
The full four-dimensional expansion of (9.6) is given next. The first term is:
BF (0, 0, 0, 0)S(x 1 )S(x 2 )S(x 3 )S(x 4 ) + BF (20, 0, 0, 0)R(x 1 )S(x 2 )S(x 3 )S(x 4 ) +
BF (0, 20, 0, 0)S(x 1 )R(x 2 )S(x 3 )S(x 4 ) + BF (20, 20, 0, 0)R(x 1 )R(x 2 )S(x 3 )S(x 4 ) +