Summary of schemes for Total Variation regularization

$$\text{gradient\hspace{0.17em}descent\hspace{0.17em}}\{\begin{array}{l}\Delta u^n=-\Delta t\left(\lambda K^T*K*\left(u^n-g\right)-D_{\Delta x}^2{u}^n\right)\hfill \\ \Delta t\le Q\Delta x\left(\frac{2}{4\sqrt{N}+\lambda Q\Delta x}\right)\hfill \end{array}$$ (65)

$$\text{1-order\hspace{0.17em}accelerated}\{\begin{array}{l}\Delta u^n=\frac{1}{1+a\Delta t}\Delta u^{n-1}-\frac{\Delta t^2}{1+a\Delta t}\left(\lambda K^T*K*\left(u^n-g\right)-D_{\Delta x}^2{u}^n\right)\hfill \\ \Delta t\le \sqrt{Q\Delta x}\left(\sqrt{\frac{4}{4\sqrt{N}+\lambda Q\Delta x}+{\left(\frac{a\sqrt{Q\Delta x}}{4\sqrt{N}+\lambda Q\Delta x}\right)}^2}+\frac{a\sqrt{Q\Delta x}}{4\sqrt{N}+\lambda Q\Delta x}\right)\hfill \end{array}$$ (66)

$$\text{2-orderaccelerated\hspace{0.17em}}\{\begin{array}{l}\Delta u^n=\frac{2-a\Delta t}{2+a\Delta t}\Delta u^{n-1}-\frac{2\Delta t^2}{2+a\Delta t}\left(\lambda K^T*K*\left(u^n-g\right)-D_{\Delta x}^2{u}^n\right)\hfill \\ \Delta t\le \sqrt{Q\Delta x}\left(\sqrt{\frac{4}{4\sqrt{N}+\lambda Q\Delta x}}\right)\hfill \end{array}$$ (67)

$$\text{semi-implicit}\{\begin{array}{l}{v}^n=u^n+\frac{2-a\Delta t}{2+a\Delta t}\Delta u^{n-1}\hfill \\ u^{n+1}=v^n-\frac{2\Delta t^2}{2+a\Delta t}\left(\lambda K^T*K*\left(v^n-g\right)-D_{\Delta x}^2{v}^n\right)\hfill \\ \Delta t\le \sqrt{Q\Delta x}\left(\sqrt{\frac{4}{3(4\sqrt{N}+\lambda Q\Delta x)}}\right)\hfill \end{array}$$ (68)