The three-dimensional test function from Xiu (2007)[^Xiu] features two-dimensional outputs that mimic the solution of a system of partial differential equations (without the spatial and time dependencies):
where $\boldsymbol{x} = { p_1, p_2 p_3 }$ is the vector of input variables modeled as three independent Gaussian random variables with zero mean and standard deviation $\sigma = 0.1$.
The function was used to demonstrate the efficiency and accuracy of stochastic collocation algorithms for metamodeling in Xiu (2007).
[^Xiu]: D. Xiu, “Efficient Collocational Approach for Parametric Uncertainty Analysis,” Communication in Computational Physics, vol. 2, no. 2, pp. 293–309, 2007.
The three-dimensional test function from Xiu (2007)[^Xiu] features two-dimensional outputs that mimic the solution of a system of partial differential equations (without the spatial and time dependencies):
$$ \begin{aligned} \mathcal{M}_1(\boldsymbol{x}) & = \frac{p_1 e^{p_2}}{1 + p_3^2} \ \mathcal{M}_2(\boldsymbol{x}) & = \cos{(p_1)} \ln{\left( \frac{1}{2} + p_2^2 + p_3^2 \right)}, \end{aligned} $$
where $\boldsymbol{x} = { p_1, p_2 p_3 }$ is the vector of input variables modeled as three independent Gaussian random variables with zero mean and standard deviation $\sigma = 0.1$.
The function was used to demonstrate the efficiency and accuracy of stochastic collocation algorithms for metamodeling in Xiu (2007).
[^Xiu]: D. Xiu, “Efficient Collocational Approach for Parametric Uncertainty Analysis,” Communication in Computational Physics, vol. 2, no. 2, pp. 293–309, 2007.