# Convergence Gain in Compressive Deconvolution: Application to Medical Ultrasound Imaging
## 2 Method
### Global Convergence  
To make the analysis more elegant, we reformulate ICD Equations (25a)-(25d) into the form \[z_{1}^{k+1}=\arg\min_{z_{1}\in\mathbb{R}^{n}}\{\theta_{1}(z_{1})-(\lambda^{k})^{ \mathsf{T}}Az_{1}+\frac{\beta}{2}\|\ Az_{1}+Bz_{2}^{k}\ \|^{2}+\frac{1}{2}\ \|\ z_{1}-z_{1}^{k}\ \|_{R}^{2}\} \tag{26a}\] \[\lambda^{k+\frac{1}{2}}=\lambda^{k}-\rho\beta(Az_{1}^{k+1}+Bz_{2}^{k})\] (26b) \[z_{2}^{k+1}=\arg\min_{z_{2}\in\mathbb{R}^{n}}\left\{\theta_{2}(z_{2})-( \lambda^{k+\frac{1}{2}})^{\mathsf{T}}Bz_{2}+\frac{\beta}{2}\|\ Az_{1}^{k+1}+Bz_{ 2}\ \|^{2}\right\}\] (26c) \[\lambda^{k+1}=\lambda^{k+\frac{1}{2}}-\varrho\beta(Az_{1}^{k+1}+Bz_{2}^{k+1}). \tag{26d}\]  
Now we analyze the convergence for our proposed ICD method expressed by Equation (26). We prove its global convergence from the contraction perspective. In order to further alleviate the notation in our analysis, we define an auxiliary sequence \(\tilde{w}^{k}\) as  
\[\tilde{w}^{k}=\left( \begin{array}{c}\underline{z}_{1}^{k}\\ \underline{z}_{2}^{k}\\ \tilde{\lambda}^{k}\end{array} \right)=\left( \begin{array}{c}z_{1}^{k+1}\\ z_{2}^{k+1}\\ \lambda^{k}-\beta(Az_{1}^{k+1}+Bz_{2}^{k})\end{array} \right) \tag{27}\]  
where \((z_{1}^{k+1},z_{2}^{k+1})\) is produced by Equations (26a) and (26c), and we immediately have  
\[z_{1}^{k+1}=\underline{z}_{1}^{k},\ \ \ z_{2}^{k+1}=z_{2}^{k},\ \ \ \ \ \lambda^{k+\frac{1}{2}}=\lambda^{k}-\rho( \lambda^{k}-\tilde{\lambda}^{k}),\]  
and  
\[\lambda^{k+1}= \lambda^{k+\frac{1}{2}}-\varrho\beta(Az_{1}^{k}+Bz_{2}^{k})\] \[= \lambda^{k}-\rho(\lambda^{k}-\tilde{\lambda}^{k})-\varrho[\beta (Az_{1}^{k}+Bz_{2}^{k})-\beta B(z_{2}^{k}-\underline{z}_{2}^{k})]\] \[= \lambda^{k}-\rho(\lambda^{k}-\tilde{\lambda}^{k})-\varrho[ \lambda^{k}-\tilde{\lambda}^{k}-\beta B(z_{2}^{k}-\underline{z}_{2}^{k})]\] \[= \lambda^{k}-[(\varrho+\varrho)(\lambda^{k}-\tilde{\lambda}^{k})- \varrho\beta B(z_{2}^{k}-\underline{z}_{2}^{k})].\]  
Moreover, we have the following relationship:  
\[\left( \begin{array}{c}z_{1}^{k+1}\\ z_{2}^{k+1}\\ \lambda^{k+1}\end{array} \right)=\left( \begin{array}{c}z_{1}^{k}\\ z_{2}^{k}\\ \lambda^{k}\end{array} \right)-\left( \begin{array}{ccc}I_{N}&0&0\\ 0&I_{N}&0\\ 0&-\varrho B&(\rho+\varrho)I_{N}\end{array} \right)\left( \begin{array}{c}z_{ 1}^{k}-\underline{z}_{1}^{k}\\ z_{2}^{k}-\underline{z}_{2}^{k}\\ \lambda^{k}-\tilde{\lambda}^{k}\end{array} \right),\]  
which can be reformulated as a compact form under the notation of \(w^{k}\) and \(\tilde{w}^{k}\):  
\[w^{k+1}=w^{k}-M(w^{k}-\tilde{w}^{k}), \tag{28}\]  
where \(M\) is defined in Equation (15). Now we start to prove some properties for the sequence \(\{\tilde{w}^{k}\}\) defined in Equation (27). We are interested in estimating how accurate the point \(\tilde{w}^{k}\) is to a solution point \(w^{*}\) of \(\text{VI}(F,\Omega,\theta)\). The main result is proved in Theorem 1. Now, we try to find a lower bound in terms of the discrepancy between \(\|w-w^{k+1}\|_{H}^{2}\) and \(\|w-w^{k}\|_{H}^{2}\) for any \(w\in\Omega\). Theorem 1: _Let \(\{w^{k}\}\) be generated by Equation (26) and let \(\{\tilde{w}^{k}\}\) be defined in Equation (27). Let \(H\) and \(G\) be defined in Equations (19) and (18), respectively. Then, for any \(w\in\Omega\), we have_  
\[\theta(\tilde{u}^{k})-\theta(The next lemma demonstrates the contraction property of the sequence \(\{w^{k}\}\) generated by Equation (26).  
Let \(\{w^{k}\}\) be generated by Equation (26) with \(0<\rho<1\) and \(0<\varrho<1\), and let \(\{\tilde{w}^{k}\}\) be defined in Equation (27). Let \(H\) and \(G\) be defined in Equations (18) and (19), respectively. Then, for any \(w^{*}\in\Omega^{*}\), we have  
\[\|w^{k+1}-w^{*}\|_{H}^{2}\leq\|w^{k}-w^{*}\|_{H}^{2}-\|w^{k}-\tilde{w}^{k}\|_{G }^{2}. \tag{30}\]  
Let the sequence \(\{w^{k}\}\) be generated by Equation (26) with \(0<\rho<1\) and \(\varrho=1\).