The three-dimensional test function from Xiu (2007)1 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).
The three-dimensional test function from Xiu (2007)1 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).
Footnotes
D. Xiu, “Efficient Collocational Approach for Parametric Uncertainty Analysis,” Communication in Computational Physics, vol. 2, no. 2, pp. 293–309, 2007. ↩