COMSOL 6.4 - Uncertainty Quantification of the Ishigami Function

Uncertainty Quantification of the Ishigami Function

This example demonstrates how to perform uncertainty quantification analysis of the Ishigami function. This random function of three variables is a well-known benchmark used to test global sensitivity analysis and uncertainty quantification algorithms. The mean, standard deviation, maximum, and minimum values as well as Sobol indices of the Ishigami function can be calculated analytically for the input distributions used here.

For this test problem, the Ishigami function is

where X1, X2, and X3 are independent uniformly distributed random variables in [−π,+π] with a = 7 and b = 0.1.

The function can be visualized in 3D by using, for example, a slice plot as in Figure 1.

Figure 1: Slice plot of the Ishigami function.

The analytically computed values are according to Table 1.

Table 1: Analytical benchmark values.
Quantity	Expression	Numerical Value (Rounded)
Mean value	a/2	3.5
Variance (V)	(a^2)/8+b(pi^4)/5+b^2(pi^8)/18+1/2	13.845
Maximum	8+(pi^4)/10	17.741
Minimum	-1-(pi^4)/10	-10.741
Standard deviation	sqrt(V)	3.7208
First-order Sobol index X1	(0.5(1+b(pi^4)/5)^2)/V	0.31391
First-order Sobol index X2	((a^2)/8)/V	0.44241
First-order Sobol index X3	0	0
Total Sobol index X1	((1/2)(1+b(pi^4)/5)^2+(8b^2pi^8)/225)/V	0.55759
Total Sobol index X2	((a^2)/8/V	0.44241
Total Sobol index X3	((8b^2pi^8)/225/V	0.24368

For reference, these values are entered as global parameters in the model.

Model Definition

The model runs through 3 uncertainty quantification studies: , , and using the Ishigami function as the quantity of interest. In order to perform the uncertainty quantification analysis, the three random variables need to be defined as global parameters using arbitrary values. The actual values for these variables will, during the simulation, be randomized by the uncertainty quantification algorithms. All the global parameters in the model are shown in Figure 2.

Figure 2: The model parameters.

The Ishigami function is defined as an analytic function with three input arguments as shown in Figure 3.

Figure 3: The Ishigami function entered as an Analytic function, ishigami.

Results and Discussion

The sensitivity analysis shows that the computed Sobol indices are consistent with the true analytical values, as shown in Figure 4 below.

Figure 4: The computed Sobol indices.

Figure 5: The computed values for mean, standard deviation, minimum, maximum, and confidence intervals.

The accuracy of the results can be increased by lowering tolerances or increasing the number of sample input points.

The computed kernel density estimation is displayed in Figure 6.

Figure 6: The KDE plot for the Ishigami function.

These uncertainty quantification results can be compared not only with the analytical values but also with that of the direct Monte Carlo simulation performed in the model Direct Monte Carlo Simulation of the Ishigami Function.

Reference

1. T. Ishigami and T. Homma, “An importance quantification technique in uncertainty analysis for computer models,” Proc. First Int’l Symp. Uncertainty Modeling and Analysis, IEEE, pp. 398-403, 1990.

Uncertainty_Quantification_Module/Tutorials/ishigami_function_uncertainty_quantification

Modeling Instructions

From the menu, choose .

New

In the window, click

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Global Definitions

Parameters 1

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
X1	1	1	Random variable 1
X2	1	1	Random variable 2
X3	1	1	Random variable 3
a	7	7	Ishigami parameter 1
b	0.1	0.1	Ishigami parameter 2
M	a/2	3.5	Mean
V	(a^2)/8+b(pi^4)/5+b^2(pi^8)/18+1/2	13.845	Variance
STD	sqrt(V)	3.7208	Standard deviation
V1	0.5(1+b(pi^4)/5)^2	4.3459	First-order variance 1
V2	(a^2)/8	6.125	First-order variance 2
V3	0	0	First-order variance 3
S1	V1/V	0.31391	First-order Sobol index 1
S2	V2/V	0.44241	First-order Sobol index 2
S3	V3/V	0	First-order Sobol index 3
VT1	(1/2)(1+b(pi^4)/5)^2+(8b^2pi^8)/225	7.7196	Total variance 1
VT2	(a^2)/8	6.125	Total variance 2
VT3	(8b^2pi^8)/225	3.3737	Total variance 3
ST1	VT1/V	0.55759	Total Sobol index 1
ST2	VT2/V	0.44241	Total Sobol index 2
ST3	VT3/V	0.24368	Total Sobol index 3
imax	8+(pi^4)/10	17.741	Function max
imin	-1-(pi^4)/10	-10.741	Function min

Ishigami Function

In the toolbar, click

and choose > .

In the window for , type ishigami in the text field.

Locate the section. In the text field, type sin(x1)+a*(sin(x2))^2+b*x3^4*sin(x1).

In the text field, type x1,x2,x3.

In the text field, type Ishigami Function.

Locate the section. In the table, enter the following settings:


Plot	Argument	Lower limit	Upper limit	Fixed value	Unit
√	x1	-pi	pi	0
√	x2	-pi	pi	0
√	x3	-pi	pi	0

Uncertainty Quantification

In the toolbar, click

and choose .

In the window for , locate the section.

Click

In the table, enter the following settings:


Expression	Include study-dependent input
ishigami(X1,X2,X3)	Reduce to single global output

Locate the section. Click

three times.

In the table, click to select the cell at row number 1 and column number 1.

In the text field, type -pi.

In the text field, type pi.

In the table, click to select the cell at row number 2 and column number 1.

In the text field, type -pi.

In the text field, type pi.

In the table, click to select the cell at row number 3 and column number 1.

In the text field, type -pi.

In the text field, type pi.

In the toolbar, click

Results

MOAT, ishigami(X1,X2,X3)

The screening study indicates that, while all parameters are influential, X3 and X2 stand out as the most critical ones.

Study 1

Uncertainty Quantification

In the window, under right-click and choose > .

Study 2

To achieve a high level of accuracy, configure the surrogate model to use a lower relative tolerance.

In the window, under click .

In the window for , locate the section.

Find the subsection. In the text field, type 0.0005.

Uncertainty Quantification 2

In the toolbar, click

Results

Sobol Index, QoI1

In the window, under > click .

The Sensitivity analysis study computes Sobol indices that are consistent with the analytical values.

Study 2

Uncertainty Quantification

In the window, under right-click and choose > .

Study 3

Now, change the to . For the Ishigami function, the polynomial chaos expansion surrogate model turns out to be much more efficient than the default option.

In the window, under click .

In the window for , locate the section.

Find the subsection. From the list, choose .

Since the surrogate model has already been built from the Sensitivity analysis study, we can reuse the surrogate function, change to .

From the list, choose .

Uncertainty Quantification 3

In the toolbar, click

Results

Kernel Density Estimation, pce1_1

In the window, under > click .