220
9 High-Dimension Model Representation via Sparse GridTechniques
0
2 0
S
R
BF(0)
BF(20)
BF(10)
BF(0)
BF(20)
10
20
0
(a)
(b)
Surplus
Surplus
Fig. 9.2 Illustrating a one-dimensional example of the hierarchical expansion at levels 0 (a) and
1 (b). The blending functions, BF (), are labeled at the nodes of the one-dimensional simplex. The
‘Surplus’ value defines the magnitude of the level-1 spline
BF (0, 0, 20, 0)S(x 1 )S(x 2 )R(x 3 )S(x 4 ) + BF (20, 0, 20, 0)R(x 1 )S(x 2 )R(x 3 )S(x 4 ) +
BF (0, 20, 20, 0)S(x 1 )R(x 2 )R(x 3 )S(x 4 ) + BF (20, 20, 20, 0)R(x 1 )R(x 2 )R(x 3 )S(x 4 ) +
BF (0, 0, 0, 20)S(x 1 )S(x 2 )S(x 3 )R(x 4 ) + BF (20, 0, 0, 20)R(x 1 )S(x 2 )S(x 3 )R(x 4 ) +
BF (0, 20, 0, 20)S(x 1 )R(x 2 )S(x 3 )R(x 4 ) + BF (20, 20, 0, 20)R(x 1 )R(x 2 )S(x 3 )R(x 4 ) +
BF (0, 0, 20, 20)S(x 1 )S(x 2 )R(x 3 )R(x 4 ) + BF (20, 0, 20, 20)R(x 1 )S(x 2 )R(x 3 )R(x 4 ) +
BF (0, 20, 20, 20)S(x 1 )R(x 2 )R(x 3 )R(x 4 ) + BF (20, 20, 20, 20)R(x 1 )R(x 2 )R(x 3 )R(x 4 ) ;
(9.8)
the second:
SURPLUS(10, 0, 0, 0)φ 1,10 (x 1 )S(x 2 )S(x 3 )S(x 4 )
+SURPLUS(10, 20, 0, 0)φ 1,10 (x 1 )R(x 2 )S(x 3 )S(x 4 ) +
SURPLUS(10, 0, 20, 0)φ 1,10 (x 1 )S(x 2 )R(x 3 )S(x 4 )
+SURPLUS(10, 20, 20, 0)φ 1,10 (x 1 )R(x 2 )R(x 3 )S(x 4 ) +