Two-dimensional Function from Alemazkoor and Meidani (2018)

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

The test function from [AM18] (or Alemazkoor2D for short) is a two-dimensional polynomial function. It was used as a test function for a metamodeling exercise (i.e., sparse polynomial chaos expansion). The function features a low-dimensional polynomial function (two-dimensional) with a high degree (a total degree of \(20\)); in other words, the function is low in dimension but of high-degree.

The surface and contour plots of the Alemazkoor2D function are shown below.

Test function instance

To create a default instance of the Alemazkoor2D function:

my_testfun = uqtf.Alemazkoor2D()

Check if it has been correctly instantiated:

print(my_testfun)

Name              : Alemazkoor2D
Spatial dimension : 2
Description       : Low-dimensional high-degree polynomial from Alemazkoor & Meidani (2018)

Description

The Alemazkoor2D function is defined as follows:

\[ \mathcal{M}(\boldsymbol{x}) = \sum_{i = 1}^{5} x_1^{2i} x_2^{2i} \]

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

Probabilistic input

Based on [AM18], the probabilistic input model for the function consists of two independent random variables as shown below.

my_testfun.prob_input

Name: 2D-Alemazkoor2018

Spatial Dimension: 2

Description: Input specification for the 2D test function from Alemazkoor & Meidani (2018)

Marginals:

No.	Name	Distribution	Parameters	Description
1	X1	uniform	[-1 1]	None
2	X2	uniform	[-1 1]	None

Copulas: None

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:

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);

References

AM18(1,2)

Negin Alemazkoor and Hadi Meidani. A near-optimal sampling strategy for sparse recovery of polynomial chaos expansions. Journal of Computational Physics, 371:137–151, 2018. doi:10.1016/j.jcp.2018.05.025.