222
9 High-Dimension Model Representation via Sparse GridTechniques
SURPLUS(0, 20, 0, 10)S(x 1 )R(x 2 )S(x 3 )φ 1,10 (x 4 )
+SURPLUS(20, 20, 0, 10)R(x 1 )R(x 2 )S(x 3 )φ 1,10 (x 4 ) +
SURPLUS(0, 0, 20, 10)S(x 1 )S(x 2 )R(x 3 )φ 1,10 (x 4 )
+SURPLUS(20, 0, 20, 10)R(x 1 )S(x 2 )R(x 3 )φ 1,10 (x 4 ) +
SURPLUS(0, 20, 20, 10)S(x 1 )R(x 2 )R(x 3 )φ 1,10 (x 4 )
+SURPLUS(20, 20, 20, 10)R(x 1 )R(x 2 )R(x 3 )φ 1,10 (x 4 ) .
(9.12)
The arguments of the BF and SURPLUS functions are the depth parameters of
the corresponding slabs in Fig. 6.3. Thus, BF (0, 20, 20, 20) is the blending function
computed by VIC-3D ® when slab 1 has a depth of 0, and the other three slabs are
at a full depth of 20. In (9.9)–(9.12), we have
SURPLUS(10, a, b, c) = BF (10, a, b, c) − 1/2(BF (20, a, b, c) + BF (0, a, b, c))
SURPLUS(a, 10, b, c) = BF (a, 10, b, c) − 1/2(BF (a, 20, b, c) + BF (a, 0, b, c))
SURPLUS(a, b, 10, c) = BF (a, b, 10, c) − 1/2(BF (a, b, 20, c) + BF (a, b, 0, c))
SURPLUS(a, b, c, 10) = BF (a, b, c, 10) − 1/2(BF (a, b, c, 20) + BF (a, b, c, 0))
(9.13)
9.3 Clenshaw-Curtis Grids
The Clenshaw-Curtis family of grids [57, 58] are, for the most part, superior to
others of the genre that we have just described, in the sense that the number of points
in C-C grids increases more slowly with the level of refinement, while retaining the
same asymptotic error decay rate. Our presentation follows [58].
The points, x i
j , of the C-C grid in one dimension are defined as
m i =
1,
if i = 1,
2 i−1 + 1, if i > 1,
x
i
j =
(j − 1)/(m i − 1) for j = 1, . . . , m i if m i > 1,
0.5
f o r j = 1 if m i = 1.
(9.14)
The index, i, is used to indicate a level of refinement of the grid in the appropriate
dimension. It is clear from (9.14) that the set of points, X i , generated at the ith level,
is a subset of X i+1 : X i ⊂ X i+1 . Furthermore, this implies that the multidimensional
sparse grid generated by the tensor product of the one-dimensional grids
9 High-Dimension Model Representation via Sparse GridTechniques
SURPLUS(0, 20, 0, 10)S(x 1 )R(x 2 )S(x 3 )φ 1,10 (x 4 )
+SURPLUS(20, 20, 0, 10)R(x 1 )R(x 2 )S(x 3 )φ 1,10 (x 4 ) +
SURPLUS(0, 0, 20, 10)S(x 1 )S(x 2 )R(x 3 )φ 1,10 (x 4 )
+SURPLUS(20, 0, 20, 10)R(x 1 )S(x 2 )R(x 3 )φ 1,10 (x 4 ) +
SURPLUS(0, 20, 20, 10)S(x 1 )R(x 2 )R(x 3 )φ 1,10 (x 4 )
+SURPLUS(20, 20, 20, 10)R(x 1 )R(x 2 )R(x 3 )φ 1,10 (x 4 ) .
(9.12)
The arguments of the BF and SURPLUS functions are the depth parameters of
the corresponding slabs in Fig. 6.3. Thus, BF (0, 20, 20, 20) is the blending function
computed by VIC-3D ® when slab 1 has a depth of 0, and the other three slabs are
at a full depth of 20. In (9.9)–(9.12), we have
SURPLUS(10, a, b, c) = BF (10, a, b, c) − 1/2(BF (20, a, b, c) + BF (0, a, b, c))
SURPLUS(a, 10, b, c) = BF (a, 10, b, c) − 1/2(BF (a, 20, b, c) + BF (a, 0, b, c))
SURPLUS(a, b, 10, c) = BF (a, b, 10, c) − 1/2(BF (a, b, 20, c) + BF (a, b, 0, c))
SURPLUS(a, b, c, 10) = BF (a, b, c, 10) − 1/2(BF (a, b, c, 20) + BF (a, b, c, 0))
(9.13)
9.3 Clenshaw-Curtis Grids
The Clenshaw-Curtis family of grids [57, 58] are, for the most part, superior to
others of the genre that we have just described, in the sense that the number of points
in C-C grids increases more slowly with the level of refinement, while retaining the
same asymptotic error decay rate. Our presentation follows [58].
The points, x i
j , of the C-C grid in one dimension are defined as
m i =
1,
if i = 1,
2 i−1 + 1, if i > 1,
x
i
j =
(j − 1)/(m i − 1) for j = 1, . . . , m i if m i > 1,
0.5
f o r j = 1 if m i = 1.
(9.14)
The index, i, is used to indicate a level of refinement of the grid in the appropriate
dimension. It is clear from (9.14) that the set of points, X i , generated at the ith level,
is a subset of X i+1 : X i ⊂ X i+1 . Furthermore, this implies that the multidimensional
sparse grid generated by the tensor product of the one-dimensional grids