Algorithm 1 Multi Fidelity Gaussian Process

Require: Low-fidelity input, ${\mathbf{x}}_1$; Low-fidelity output, ${\mathbf{y}}_{\mathbf{1}}$; Hyperparameter of the low fidelity kernel, ${\theta }_1$; High-fidelity input, ${\mathbf{x}}_{\mathbf{2}}$; High-fidelity output, ${\mathbf{y}}_{\mathbf{2}}$; Hyperparameter of the high fidelity kernel, ${\theta }_2$; Kernel function, k (for simplicity, here $k_1$ = $k_2$); Test input, ${\mathbf{x}}^{tst}$; Noise-level, ${\sigma }_{ϵ}^2$ (for simplicity, here ${\sigma }_{{ϵ}_1}^2$ = ${\sigma }_{{ϵ}_2}^2$) 1:L = cholesky(K +${\sigma }_{ϵ}^2$I) ▹ K as calculated from Equation (9) 2:Y = [${\mathbf{y}}_{\mathbf{1}}{\mathbf{y}}_{\mathbf{2}}$] 3:$\alpha$ = ${\mathbf{L}}^T\setminus (\mathbf{L}\setminus \mathbf{Y})$ 4:${\psi }_1=\rho k({\mathbf{x}}^{tst},{\mathbf{x}}_1,{\theta }_1)$ 5:${\psi }_2={\rho }^{2}k({\mathbf{x}}^{tst},{\mathbf{x}}_2;{\theta }_1)+k({\mathbf{x}}^{tst},{\mathbf{x}}_2;{\theta }_2)$ 6:$\Psi =\left[{\psi }_1{\psi }_2\right]$ 7:${\widehat{f}}_{x^{tst}}=\Psi .\alpha$ ▹ predictive mean 8:$\beta ={\mathbf{L}}^T\setminus (\mathbf{L}\setminus {\Psi }^T)$ 9:$\mathbb{V}\left[f_{x^{tst}}\right]={\rho }^{2}k({\mathbf{x}}^{tst},{\mathbf{x}}^{tst},{\theta }_1)+k({\mathbf{x}}^{tst},{\mathbf{x}}^{tst};{\theta }_2)-\Psi \beta$ ▹ predictive variance