13.3 Smolyak sparse grid applied
13.3.1 The 2-variate Test Equation
The stochastic Test equation with uniformαandβ has been investigated using the two sparse grids. The two variables are distributed with α∼ U(−1,1)and β ∼ U(0,2).
The SGL grid and CC grid with level 3 have been used to estimate the statistics for the stochastic Test equation and the results can be seen in the following sections. Since the estimates of the statistics are very similar to the ones ob-tained with the full tensor grids the numerical tests in this section will be an investigation of the errors. Furthermore the accuracy of the the grids have been compared with the accuracy obtained with the full tensor grid in section 13.3.1.3.
13.3.1.1 Sparse Gauss Legendre grid
In this section the SGL grid is applied and the error of the obtained estimates have been computed. The errors have been plotted as functions of the total number of grid points, M. The correspondence between number of points and the levels can be seen in table 13.1. The grids have been computed for the levels l= 0,1, . . . ,6and the errors for the corresponding grid sizes have been plotted in gure 13.3.
128 Tests with Smolyak sparse grids
100 101 102 103 10−13
10−9 10−5 10−1
M
Error
Mean Variance
Figure 13.3: Error of the estimated mean and variance for increasing M at timet= 1.
It is seen that the error drops rather quickly and the error reaches a minimum which is equivalent to the minimum error obtained in the previous tests of the multivariate Test equation at level l = 3 where the grid contains 21 unique points.
13.3.1.2 Sparse Clenshaw-Curtis grid
The sparse Clenshaw-Curtis grid has been use to estimate the statistics for the multivariate Test equation. The errors using the CC grid have been plotted in gure 13.4 as function of the total number of grid points,M.
13.3 Smolyak sparse grid applied 129
100 101 102
10−13 10−9 10−5 10−1
M
Error
Mean Variance
Figure 13.4: Error of the estimated mean and variance for increasing M at timet= 1.
It is seen that the convergence of the errors when using the CC grid is not impressive compared to the convergence obtained previously using full tensor grids. This means that the CC grid might not be suited for estimating the statistics of this problem.
13.3.1.3 Comparison of the errors on the estimated statistics
As illustrated in gure 13.3 and gure 13.4 the two sparse grids can be used as collocation points for the stochastic Collocation method which leads to fair results when the stochastic variables are uniformly distributed. In this section the accuracy of the SCM used with SGL and CC grids is investigated further. As a comparison the SCM has been used with full tensor grid on the same problem and the errors are included in Table 13.3.
The full tensor grids are computed from 1D Gauss Legendre quadrature and the same number of quadrature nodes has been used in each dimension, i.e.
the total number of nodes is M =m1×m2 =m2 where m is the number of quadrature nodes in each dimension.
130 Tests with Smolyak sparse grids
M Error in the mean Error in the variance
1 1.75·10−1 1.04
4 3.85·10−3 8.34·10−2 9 3.28·10−5 2.98·10−3 16 1.48·10−6 5.39·10−5 25 4.13·10−10 5.98·10−7 36 1.63·10−12 4.52·10−9 49 8.43·10−13 2.91·10−10 64 8.41·10−13 4.50·10−12 81 8.42·10−13 4.49·10−12 100 8.41·10−13 4.49·10−12
Table 13.3: The errors in the estimated statistics and the corresponding num-ber of nodes in the full tensor grid.
From table 13.3 it is seen that the minimum error in the mean is obtained for 49 grid points and 64 grid points for the variance. This corresponds very well with the investigations made previously for the univariate Test equation with uniformly distributedα. In this case 7 quadrature nodes was needed to obtain the minimum error in the estimated mean and 8 quadrature nodes to obtain the minimum error in the estimated variance.
Furthermore the previous numerical tests with the 2-variate stochastic Test equation indicated that the α-parameter was more dominant than β in terms of the error in the estimated statistics. This corresponds very well with the optimal number of tensor grid points for the full tensor grid since it corresponds to chose the optimal choice of quadrature nodes forαin each dimension.
The SGL grid is used as collocation points for the SCM for the stochastic Test equation. The grids with level l = 0,1, . . . ,6 are used and the errors of the estimated statistics are outlined in table 13.4. A comparison of Table 13.4 eciency of the SCM is higher when using the SGL grids than when using full tensor grids in the sense that the same accuracy is obtained using SGL grids but with fewer grid points.
13.3 Smolyak sparse grid applied 131 Level M Error in the mean Error in the variance
0 1 1.75·10−1 1.037
1 5 3.27·10−5 2.73·10−1
2 22 8.42·10−13 7.65·10−4
3 75 8.42·10−13 1.06·10−11 4 224 8.40·10−13 4.39·10−12 5 613 8.40·10−13 4.39·10−12 6 1578 8.48·10−13 4.39·10−12
Table 13.4: The errors in the estimated statistics and the corresponding levels and total number of nodes in the SGL grid.
Numerical tests are conducted using the CC grids as well and this resulted in computation of the errors of the estimated mean and variance which can be seen in Table 13.5.
Level M Error in the mean Error in the variance
0 1 1.75·10−1 1.04
1 5 5.83·10−3 1.78·10−1
2 13 1.35·10−5 3.49·10−2
3 29 1.11·10−11 3.06·10−4 4 65 8.39·10−13 3.81·10−9 5 145 8.39·10−13 4.39·10−12 6 321 8.40·10−13 4.39·10−12
Table 13.5: The errors of the estimated statistics and the corresponding levels and number of nodes in the CC grid.
The accuracy of the CC grids is lower than the accuracy obtained using the SGL grids on this particular test problem. This is not necessarily true in general but it still yields an interesting result. It seems like fewer grid points are needed when using the full tensor grid than when using the CC grid to obtain the minimum error. But this is might just be due to the fact that the CC grid has 29 grid points in the level 3 grid and 65 grid points for level 4. This means that because the optimal choice of level is not level 3 at least 65 nodes are used to obtain the best estimates which is more than in the full grid.
It is furthermore to be noted that the CC grid might be much more useful in higher dimensions than 2 since the sparse grids are introduced to reduce the eects of the curse of dimensionality in high dimensions.
132 Tests with Smolyak sparse grids
13.3.1.4 Discussion
The sparse grids seems to be very useful and the use of the SGL grid denitely demonstrates that a high accuracy can be obtained with fewer nodes in com-parison to the full tensor grid.
It should be noted that for the stochastic Test equation it seems likeαis more dominant in terms of inuence on the error than β. Therefore a full tensor grid might involve fewer quadrature nodes to represent β and thereby lead to a smaller number of total nodes, while maintaining a high accuracy. Therefore the sparse grids are more ecient than the full tensor grids investigated in this section but smarter grids might be applied which are full tensor grids but still involves relatively few grid points.
This is one of the reasons why a method like ANOVA can be very useful since ANOVA can be used to investigate which of the stochastic variables that have a great inuence on the statistics and which of them are less important.