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Gaussian distributed α -parameter and initial condition β 105

11.3 Smolyak Sparse Grid Collocation

12.1.2 Gaussian distributed α -parameter and initial condition β 105

The multivariate Test equation is solved for the case where αand β are both Gaussian distributed. The α-parameter is chosen to be standard normal dis-tributed, i.e. α∼ N(0,1), and the initial condition is distributed asβ ∼ N(1,1). Due to the distribution of the stochastic parameters the Gauss Hermite quadra-ture is chosen and six points are chosen in each dimension which leads to 36 deterministic solutions due to the tensor product of the grid points.

In gure 12.1 the 36 deterministic solutions are plotted.

0 0.5 1

−50 0 50 100

t

Solutions

Figure 12.1: TheN = 36deterministic solutions.

It is seen that there is big variations in the 36 solutions. A dierence from the univariate case is that now there are negative solutions which was not obtained previously and this means that a single deterministic solution can dier a lot from the computed mean since it could be dierent in sign and very dierent in size. The negative solutions are obtained since some of the computed initial

106 Numerical tests for multivariate stochastic PDEs

conditions are negative which leads to negative solutions.

The 36 deterministic solutions have been used to compute the statistics. The estimated meanu¯ can be seen in gure 12.2 with an error bound computed by

¯

u±σ¯ as well as the exact mean.

0 0.5 1

−2 0 2 4

t

Mean

MeanStd µ

Figure 12.2: The estimated mean with the computed standard deviation and the exact mean.

From gure 12.2 it is seen that the mean has not changed a lot in comparison with the mean in gure 7.8.

In gure 12.2 it is seen that the stochastic initial condition introduces a lot more uncertainty in the solution in the rst half of the time domain compared to 7.8.

With time the eects of the stochastic initial condition becomes less dominant since the variance introduced by the α-parameter increases exponential with time.

The errors in mean and variance have been computed for dierent M and in gure 12.3 the errors have been plotted as function ofM, wheremα=mβ=m which means thatM =m2.

12.1 Test Equation with two random variables 107

100 101 102 103 104 10−10

10−6 10−2

M

Error

Mean Variance

Figure 12.3: Error on the mean and variance for increasingM at time t= 1. From the errors in gure 12.3 it is seen that the errors converge like it did in the univariate case except that now it is a function ofM instead of justm1. Unless the stochastic β reduces the eects of the stochastic α it is expected that at least M = 81 is needed for achieving the smallest error in the mean sinceM =m2 andmα= 9was needed to minimize the error inα. Apparently M =m2= 81is the optimal choice with regard to the error in the mean. This indicates that either is the eects of the parameter αreduced or else it holds that the best approximation in the mean can be obtained withmβ ≤mα. A simple test of this assumption is to maintainmα= 2while increasingmβand vice versa. This results in the error-plots in gure 12.4.

101 102 10−1

100

M

Error

mβ= 2

101 102 10−10

10−5 100

M

Error

mα= 2 Mean Variance

Figure 12.4: Error on the mean and variance at time t= 1for increasingM. Left: mβ= 2 and increasingmα. Right: mα= 2and increasing mβ.

108 Numerical tests for multivariate stochastic PDEs

From the two plots it is seen that it is the number of quadrature points for representingαthat makes a dierence with regard to the convergence of the es-timated mean and variance. The number of quadrature points used to represent theβ-parameter does not have a great impact on the quality of the estimates.

This means that instead of using M = 9·9 = 81deterministic solutions it is sucient to computeM = 9·2 = 18deterministic solutions as long asmα= 9 andmβ= 2 and not the other way around.

This means that the α-parameter has much more inuence on accuracy of the approximations of the statistics than theβ-parameter. This is a good example on how a smart choice of quadrature nodes in each stochastic dimension can lead to much more ecient computations without compromising the quality of the approximations.

12.1.3 Gaussian distributed α and uniformly distributed initial condition β .

In this section it has been investigated how it aects the statistics of the solution to have two stochastic variables with dierent distributions namelyα∼ N(0,1) andβ ∼ U(0,2). Sinceβ is uniformly distributed the quadrature chosen for this variable is Gauss Legendre quadrature. There are again 6 quadrature nodes and weights for bothαandβwhich by use of tensor product leads to 36 deterministic solutions. Like in the previous test case the deterministic solutions are plotted and can be seen in gure 12.5.

0 0.2 0.4 0.6 0.8 1 0

20 40

t

Solutions

Figure 12.5: 12 of theN= 36deterministic solutions.

12.1 Test Equation with two random variables 109

The deterministic solutions in gure 12.5 are dierent from the ones in gure 12.1 since none of them are negative and they do not grow as much as when β was Gaussian distributed. This is because the quadrature nodes are strictly limited to be in the interval[0,2]in contrast to the other test case whereβ was not limited in this way.

A comparison of the deterministic solutions in gure 12.5 with the ones in gure 12.1 yields that it can have a great impact on the individual deterministic solution whether β is Gaussian or uniformly distributed. This dierence is however not so visible when looking at the statistics as illustrated in gure 12.6 where the estimated mean, the std-bound and the exact mean are plotted.

0 0.2 0.4 0.6 0.8 1 0

2 4

t Std Meanµ

Figure 12.6: The estimated mean with the computed standard deviation as well as the exact mean.

It is seen that the mean is not greatly aected by the distribution of β but the standard deviation is dierent than the one seen in gure 12.2. It is smaller whenβis uniformly distributed than whenβis Gaussian distributed. Especially in the rst half of the time domain the dierence in the standard deviation is visible.

For this choice of parameters the error has been plotted in gure 12.7 as function ofM =m2 wherem=mα=mβ.

110 Numerical tests for multivariate stochastic PDEs

100 101 102 103 104 10−10

10−6 10−2

M

Error

Mean Variance

Figure 12.7: Error on the mean and variance for increasingM at timet= 1. The tendency in the error in gure 12.7 is the same as was seen in gure 12.3.

Again the optimal choice seems to be M = 81 which could indicate that the error is not greatly aected by the choice of distribution of β. But it worth to remember that the β-parameter apparently has much less inuence on the accuracy than the α-parameter which means that with 9 quadrature points to representβ the shift in distribution should not change a lot.

To investigate this further the same simple test as was conducted in the previous section is conducted here. This means that mα= 2is maintained whilemβ is increased and vice versa. This resulted in the error-plots visualized in gure 12.8.

101 102 10−1

100

M

Error

mβ= 2

101 102 10−10

10−5 100

M

Error

mα= 2

Mean Variance

Figure 12.8: Error on the mean and variance at time t= 1 for increasingM. Left: mβ= 2 and increasingmα. Right: mα= 2and increasing mβ.