Two-dimensional Function from Cheng and Sandu (2010)

Two-dimensional Function from Cheng and Sandu (2010)#

import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf

The two-dimensional test function from Cheng and Sandu (2002) (or Cheng2D for short) is used in a metamodeling exercise via polynomial chaos expansion in [CS10].

../_images/c1c2be5d5ca5c0b30a90cce788f2498a6b6eb4a3e2e6faae582cbda32a424380.png

Test function instance#

To create a default instance of the test function:

my_testfun = uqtf.Cheng2D()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : Cheng2D
Input Dimension  : 2 (fixed)
Output Dimension : 1
Parameterized    : False
Description      : Two-dimensional test function from Cheng and Sandu (2010)
Applications     : metamodeling

Description#

The test function is defined as follows[1]:

\[ \mathcal{M}(\boldsymbol{x}) = \cos{(x_1 + x_2)} \exp{(x_1 x_2)}, \]

where \(\boldsymbol{x} = \{ x_1, x_2 \}\) is the two-dimensional vector of input variables further defined below.

Probabilistic input#

The input consists of two uniformly distributed random variables as shown below.

Hide code cell source 
print(my_testfun.prob_input)

Function ID     : Cheng2D
Input ID        : Cheng2010
Input Dimension : 2
Description     : Probabilistic input model for the 2D test function from
                  Cheng and Sandu (2010)
Marginals       :

 No.    Name    Distribution    Parameters    Description
-----  ------  --------------  ------------  -------------
  1      X1       uniform        [0. 1.]           -
  2      X2       uniform        [0. 1.]           -

Copulas         : Independence

Reference results#

This section provides several reference results of typical UQ analyses involving the test function.

Sample histogram#

Shown below is the histogram of the output based on \(100'000\) random points:

Hide code cell source 
xx_test = my_testfun.prob_input.get_sample(100000)
yy_test = my_testfun(xx_test)

plt.hist(yy_test, bins="auto", color="#8da0cb");
plt.grid();
plt.ylabel("Counts [-]");
plt.xlabel("$\mathcal{M}(\mathbf{X})$");
plt.gcf().set_dpi(150);
../_images/87776b6de2c605eaf8010ec03275a679aa3609074e8f0a2f7e26461200f7ec4f.png

References#

[CS10] (1,2)

Haiyan Cheng and Adrian Sandu. Collocation least-squares polynomial chaos method. In Proceedings of the 2010 Spring Simulation Multiconference, SpringSim ’10, 1–6. Society for Computer Simulation International, 2010. doi:10.1145/1878537.1878621.