7.9 Two-Dimensional Functions
183
0
5e-05
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
0.00045
-4
-3
-2
-1
0
1
2
3
4
Scan position (mm)
Total variance in R and X due to six random variables
R
X
Fig. 7.13 Total variances in resistance and reactance due to the one-dimensional functions defined
in (7.27) and (7.34) associated with the six random variables, ξ 1 , · · · , ξ 5 , ξ 32
Fig. 7.14 Two-dimensional
interpolation nodes
ξ
ξ
1
2
−0.5,0.5
−0.5,−0.5
0.0,0.5
0.0,0.0
0.5,0.5
0.5,0.0
−0.5,0.0
0.0,−0.5 0.5,−0.5
The interpolation nodes for the two-dimensional functions are the tensor product
of the one-dimensional nodes used above (see Fig. 7.14). With this in mind, we can
determine the expansion coefficients in (7.51) as the solution of the vector-matrix
equation:
183
0
5e-05
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
0.00045
-4
-3
-2
-1
0
1
2
3
4
Scan position (mm)
Total variance in R and X due to six random variables
R
X
Fig. 7.13 Total variances in resistance and reactance due to the one-dimensional functions defined
in (7.27) and (7.34) associated with the six random variables, ξ 1 , · · · , ξ 5 , ξ 32
Fig. 7.14 Two-dimensional
interpolation nodes
ξ
ξ
1
2
−0.5,0.5
−0.5,−0.5
0.0,0.5
0.0,0.0
0.5,0.5
0.5,0.0
−0.5,0.0
0.0,−0.5 0.5,−0.5
The interpolation nodes for the two-dimensional functions are the tensor product
of the one-dimensional nodes used above (see Fig. 7.14). With this in mind, we can
determine the expansion coefficients in (7.51) as the solution of the vector-matrix
equation: