Implicit fidelity schemes for quadratic regularization

$$\text{gradient\hspace{0.17em}descent:\hspace{0.17em}}\Delta u_{\alpha }^n=-\frac{\Delta t}{1+\lambda \Delta t}\left(\lambda \left(u_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{u_{\alpha +e_k}^n-2u_{\alpha }^n+u_{\alpha -e_k}^n}{\Delta x^2}\right)$$ (46)

$$\begin{array}{l}\text{\hspace{0.17em}1-order\hspace{0.17em}accelerated:\hspace{0.17em}}\Delta u_{\alpha }^n=\frac{\Delta u_{\alpha }^{n-1}-\Delta t^2\left(\lambda \left(u_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{u_{\alpha +e_k}^n-2u_{\alpha }^n+u_{\alpha -e_k}^n}{\Delta x^2}\right)}{1+(a+\lambda \Delta t)\Delta t}\hfill \end{array}$$ (47)

$$\text{2-order\hspace{0.17em}accelerated:\hspace{0.17em}}{u}_{\alpha }^n=\frac{(2-a\Delta t)\Delta u_{\alpha }^{n-1}-2\Delta t^2\left(\lambda \left(u_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{u_{\alpha +e_k}^n-2u_{\alpha }^n+u_{\alpha -e_k}^n}{\Delta x^2}\right)}{2+a\Delta t+2\lambda \Delta t^2}$$ (48)

$$\text{semi-implicit:\hspace{0.17em}}\{\begin{array}{l}{v}_{\alpha }^n=u_{\alpha }^n+\frac{2-a\Delta t}{2+a\Delta t}\Delta u^{n-1}\hfill \\ u_{\alpha }^{n+1}=v_{\alpha }^n-\frac{2\Delta t^2}{2+a\Delta t+2\lambda \Delta t^2}\left(\lambda \left(v_{\alpha }^n-g_{\alpha }\right)-c\sum _{k=1}^N\frac{v_{\alpha +e_k}^n-2v_{\alpha }^n+v_{\alpha -e_k}^n}{\Delta x^2}\right)\hfill \end{array}$$ (49)