Algorithm: Preconditioned NCG-MLS

$$\begin{gathered} \begin{array}{cc}\hfill & \mathrm{Inputs}:\hfill \\ \hfill & \mathit{y},\mathit{s},\mathit{t},\mathit{C},\beta \hfill \\ \hfill & \mathrm{Intialization}:\hfill \\ \hfill & {\omega }^0\text{by}(8)\text{or}(9)\hfill \\ \hfill & z^0=-\nabla \Psi ({\omega }^0)\hfill \\ \hfill & {\alpha }^0=0\hfill \\ \hfill & \text{precompute}{r}_{cdmnj}\text{by (4) and}{t}_m-t_n\hfill \\ \hfill & \mathbf{for}i=0,1,\dots ,N-1\mathbf{do}\hfill \\ \hfill & \text{compute gradient}{\mathit{g}}^i=\nabla \Psi ({\omega }^i)\text{with (10)}\hfill \\ \hfill & \text{precondition}{\mathit{p}}^i=({\mathit{P}}^i{)}^{-1}{\mathit{g}}^i\text{with (14)}\hfill \\ \hfill & \text{compute}{\mu }^i\text{with conjugacy}\hfill \\ \hfill & \text{search direction}{z}^{i+1}={\mathit{p}}^i+{\mu }^i{z}^i\in {\mathbb{R}}^{N_{\mathrm{v}}}\hfill \\ \hfill & \mathbf{for}k=0,1,\dots ,N_i-1\mathbf{do}\hfill \\ \hfill & \\ \text{update step size}{\alpha }^{(k+1)}\text{by (17)}\hfill \\ \hfill & \mathbf{end}\mathbf{for}\hfill \\ \hfill & \text{update}{\omega }^{i+1}={\omega }^i+{\alpha }^{(N_i)}{z}^{i+1}\hfill \\ \hfill & \mathbf{end}\mathbf{for}\hfill \\ \hfill & \mathrm{output}:{\omega }^N\hfill \end{array} \end{gathered}$$