Algorithm 1: Edge-enhancing FGK-based methods.

1:Input: initial guess $x_{0,★}$, $r_{0,★}=b-A{x}_{0,★}$, thresholds $\tau ,\eta ,\xi >0$, 2:Take $u_1=r_{0,★}/\parallel r_{0,★}\parallel$, $U_1=\left[u_1\right]$, $V_0=\left[\right]$, $Z_0=\left[\right]$ 3:Compute $x_0$ and E as in (14)–(16) 4:for$i=1,2,\dots$ until (31) is satisfied do 5:  Expand the approximation subspace, by updating the FGK factorization (21) $$\begin{array}{ccc}{\overline{v}}_i=A^T{u}_i,\hfill & v=(I-V_{i-1}{V}_{i-1}^T){\overline{v}}_i,v_i=v/{\parallel v\parallel }_2,\hfill & V_i=[V_{i-1},v_i]\hfill \\ z_i={\left(W_i{D}_{2d}\right)}^{†}{v}_i& \left(using\right(\left)\right),& Z_i=[Z_{i-1},z_i]\\ {\overline{u}}_{i+1}=A{z}_i,& u=(I-U_i{U}_i^T){\overline{u}}_{i+1},u_{i+1}=u/{\parallel u\parallel }_2,& U_{i+1}=[U_i,u_{i+1}]\end{array}$$ 6: Choose $\lambda ={\lambda }_i$ such that (30) holds 7: Solve problem (24) with $\lambda ={\lambda }_i$ 8: Update $W_{i+1}$ 9:end for