Parallel Hybrid Algorithms for a Finite Family of G-Nonexpansive Mappings and Its Application in a Novel Signal Recovery

Abstract

This article considers a parallel monotone hybrid algorithm for a finite family of G-nonexpansive mapping in Hilbert spaces endowed with graphs and suggests iterative schemes for finding a common fixed point by the two different hybrid projection methods. Moreover, we show the computational performance of our algorithm in comparison to some methods. Strong convergence theorems are proved under suitable conditions. Finally, we give some numerical experiments of our algorithms to show the efficiency and implementation of the LASSO problems in signal recovery with different types of blurred matrices and noise.

1. Introduction

Let C be a nonempty subset of a real Banach space X. Let $\Delta$ denotes the diagonal of the cartesian product $C\times C$, i.e., $\Delta =\left\{\right(s,s):s\in C\}$. Assume that G is a directed graph such that $V\left(G\right)$ is the set of its vertices that coincides with C, and $E\left(G\right)$ is the set of its edges. We assume that G has no parallel edge and $\Delta \subseteq E\left(G\right)$. A graph of G is defined by $\left(V\right(G),E(G\left)\right)$. A mapping $\mathsf{\Theta }:C\to C$ is said to be G-nonexpansive if $\mathsf{\Theta }$ satisfies the conditions: $\mathsf{\Theta }$ preserves edges of G, i.e.,

$$(s,t)\in E\left(G\right)\Rightarrow (\mathsf{\Theta }s,\mathsf{\Theta }t)\in E\left(G\right),\forall (s,t)\in E\left(G\right);$$

and $\mathsf{\Theta }$ non-indecreases the weights of edges of G in the following way:

$$(s,t)\in E\left(G\right)\Rightarrow \parallel \mathsf{\Theta }s-\mathsf{\Theta }t\parallel \le \parallel s-t\parallel ,\forall (s,t)\in E\left(G\right).$$

It’s easy to see that G-nonexpansive mapping generalizes nonexpansive mapping. Many problems in mathematical sciences have been solved by finding a fixed-point approximation of a nonexpansive mapping in many metric spaces. Iterative sequences have been proposed for finding fixed points and their applications by many mathematicians, see [1,2,3]. One of the most famous is the S-iteration method introduced by Agarwal et al. [4] for some operators in norm linear spaces. Recently, Suparatulatorn et al. [2] used the S-iteration method for finding a fixed point of three different G-nonexpansive mappings ${\mathsf{\Theta }}_1$, ${\mathsf{\Theta }}_2$, ${\mathsf{\Theta }}_3$ with directed graphs. The weak convergence was proved under some conditions on the parameters in Hilbert spaces endowed with graphs. This modified S-iteration method is defined as Algorithm 1:

Algorithm 1: Choose $s_1\in C$, and $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$ are real sequences in $[0,1]$.

(STEP 1) Compute $$u_n=(1-{\beta }_n)s_n+{\beta }_n{\mathsf{\Theta }}_1{s}_n.$$ (STEP 2) Construct $s_{n+1}$ by         $$s_{n+1}=(1-{\alpha }_n){\mathsf{\Theta }}_1{s}_n+{\alpha }_n{\mathsf{\Theta }}_2{u}_n,\forall n\ge 1.$$ (STEP 3) Set $n:=n+1$, and go to (Step 1).

In 2017, Sridarat et al. [5] studied the convergence analysis of SP-iteration in Hilbert spaces endowed with graphs. A weak convergence theorem was proved. The following iteration in Algorithm 2 is known as SP-iteration:

Algorithm 2: Choose $s_1\in C$, and $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$, $\left\{{\gamma }_n\right\}$ are real sequences in $[0,1]$.

(STEP 1) Compute $$u_n=(1-{\gamma }_n)s_n+{\gamma }_n{\mathsf{\Theta }}_3{s}_n.$$ (STEP 2) Compute $$t_n=(1-{\beta }_n)u_n+{\beta }_n{\mathsf{\Theta }}_2{u}_n.$$ (STEP 3) Construct $s_{n+1}$ by $$s_{n+1}=(1-{\alpha }_n)t_n+{\alpha }_n{\mathsf{\Theta }}_1{t}_n,\forall n\ge 1.$$ (STEP 4) Set $n:=n+1$, and go to (Step 1).

In studying fixed-point algorithms, the rate of convergence is very important to show the efficiency of an algorithm. Recently, Yambangwai et al. [6] introduced a new modified three-step iteration method for three different G-nonexpansive mappings in Banach spaces with a graph and showed a better rate of convergence compared with Sridarat et al. [5]. The algorithm also gets a weak convergence theorem under some suitable conditions. This algorithm is defined as Algorithm 3:

Algorithm 3: Choose $s_1\in C$, and $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$, $\left\{{\gamma }_n\right\}$ are real sequences in $[0,1]$.

(STEP 1) Compute $$u_n=(1-{\gamma }_n)s_n+{\gamma }_n{\mathsf{\Theta }}_3{s}_n.$$ (STEP 2) Compute $$t_n=(1-{\beta }_n)u_n+{\beta }_n{\mathsf{\Theta }}_2{u}_n.$$ (STEP 3) Construct $s_{n+1}$ by $$s_{n+1}=(1-{\alpha }_n){\mathsf{\Theta }}_2{u}_n+{\alpha }_n{\mathsf{\Theta }}_1{t}_n,\forall n\ge 1.$$ (STEP 4) Set $n:=n+1$, and go to (Step 1).

For obtaining strong convergence theorems, Nakajo and Takahashi [7] proposed the following hybrid projection algorithm which is well-known as the CQ projection algorithm for finding a fixed point of a nonexpansive mapping in a real Hilbert space H. For each $t\in H$, ${\mathrm{argmax}}_{s\in H}\parallel s-t\parallel =\{x\in H:\parallel s-t\parallel \le \parallel x-t\parallel$ for all $s\in H\}$ They investigated the sequence $\left\{s_n\right\}$ generated by Algorithm 4 as follows:

Algorithm 4: Choose $s_1\in C$, $Q_1=C=C_1$ and $\left\{{\alpha }_n\right\}\subset [0,a]$ for some $a\in [0,1)$.

(STEP 1) Set $$t_n={\alpha }_n{s}_n+(1-{\alpha }_n){\mathsf{\Theta }}_n{s}_n.$$ (STEP 2) Compute $${\overline{t}}_n=\mathrm{argmax}\{\parallel t_n^i-s_n\parallel :i=1,2,\dots ,N\}.$$ (STEP 3) Compute $$C_n=\{z\in C:\parallel {\overline{t}}_n-z\parallel \le \parallel s_n-z\parallel \}$$ and $$Q_n=\{z\in C:\langle z-s_n,s_1-s_n\rangle \le 0\}.$$ (STEP 4) Construct $s_{n+1}$ by $$s_{n+1}=P_{C_n\cap Q_n}\left(s_1\right),\forall n\ge 1.$$ (STEP 5) Set $n:=n+1$, and go to (Step 1).

In 2005, Anh and Hieu [8,9] introduced a projection method which is called the parallel monotone hybrid algorithm for solving common fixed point problems of a finite family of quasi $\varphi$-nonexpansive mappings ${\left\{{\mathsf{\Theta }}_i\right\}}_{i=1}^N$ in a Banach space. This algorithm is presented in a real Hilbert space as Algorithm 5:

Algorithm 5: Choose $s_1\in C$, $C_1=C$ and $\left\{{\alpha }_n\right\}$ is a sequence in $[0,1]$.

(STEP 1) Set $$t_n^i={\alpha }_n{s}_n+(1-{\alpha }_n){\mathsf{\Theta }}_i{s}_n,i=1,2,\dots ,N.$$ (STEP 2) Compute $${\overline{t}}_n=\mathrm{argmax}\{\parallel t_n^i-s_n\parallel :i=1,2,\dots ,N\}.$$ (STEP 3) Compute $$C_{n+1}=\{v\in C_n:\parallel v-{\overline{t}}_n\parallel \le \parallel v-s_n\parallel \}.$$ (STEP 4) Construct $s_{n+1}$ by $$s_{n+1}=P_{C_{n+1}}\left(s_1\right),n\ge 1.$$ (STEP 5) Set $n:=n+1$, and go to (Step 1).

A strong convergence theorem has been proved under a condition on the parameter ${\alpha }_n$ with ${\displaystyle \underset{n\to \infty }{lim\; sup}{\alpha }_n<1}$. Recently, there have been some works involving the parallel method for solving the fixed point problem (see [2,10,11]).

In this work, we wish to study the parallel monotone hybrid algorithm for a finite family of G-nonexpansive mappings in Hilbert spaces and introduce algorithms, based on the hybrid projection method. We then prove the strong convergence theorems of the proposed methods using the parallel monotone hybrid algorithm. Finally, some numerical experiments in signal recovery are provided to show the efficiency and implementation of our algorithms.

2. Main Results

In this section, we prove strong convergence theorems for a finite family of G- nonexpansive mappings and propose a projection Algorithm 6 for finding a common fixed point.

Assume that six below conditions hold.

(1)

The mappings ${\mathsf{\Theta }}_i:C\to C$ are $G$-nonexpansive for all $i=1,2,\dots ,N$.

(2)

The solution set $F:={\bigcap }_{i=1}^{N}F\left({\mathsf{\Theta }}_i\right)\ne \mathsf{\varnothing }$.

(3)

F is closed and $F\left({\mathsf{\Theta }}_i\right)\times F\left({\mathsf{\Theta }}_i\right)\subseteq E\left(G\right)$ for all $i=1,2,\dots ,N$.

(4)

$\left\{s_n\right\}$ dominates p for all $p\in F$ and if there exists a subsequence $\left\{s_{n_k}\right\}$ of $\left\{s_n\right\}$ such that $s_{n_k}\rightharpoonup w\in C$, then $(s_{n_k},w)\in E\left(G\right)$.

(5)

The sequence $\left\{{\alpha }_n^i\right\}\subset [0,1]$ and ${\displaystyle \underset{n\to \infty }{lim\; inf}{\alpha }_n^i(1-{\alpha }_n^i)>0}$ for all $i=1,2,\dots ,N$.

(6)

${\displaystyle \underset{n\to \infty }{lim\; inf}{\alpha }_n^0{\alpha }_n^i>0}$ for all $i=1,2,\dots ,N$.

Algorithm 6: Choose $s_1\in C$, $Q_1=C$ and $\left\{{\alpha }_n^i\right\}$ is a sequence in $[0,1]$.

(STEP 1) Set $$t_n^i=(1-{\alpha }_n^i)s_n+{\alpha }_n^i{\mathsf{\Theta }}_i{s}_n,i=1,2,\dots ,N.$$ (STEP 2) Compute $${\overline{t}}_n=\mathrm{argmax}\{\parallel t_n^i-s_n\parallel :i=1,2,\dots ,N\}.$$ (STEP 3) Compute $$C_n=\{v\in C:\parallel {\overline{t}}_n{-v\parallel }^2\le \parallel s_n-v{\parallel }^2\}$$ and $$Q_n=\{v\in Q_{n-1}:\langle s_1-s_n,s_n-v\rangle \ge 0\}.$$ (STEP 4) Construct $s_{n+1}$ by $$s_{n+1}=P_{C_n\cap Q_n}\left(s_1\right).$$ (STEP 5) Set $n:=n+1$, and go to (Step 1).

Theorem 1.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let $G=\left(V\right(G),E(G\left)\right)$ be a directed graph such that $V\left(G\right)=C$ and $E\left(G\right)$ be convex. Assume that $\left\{s_n\right\}$ is generated by Algorithm 6. Then under conditions (1)–(5), $\left\{s_n\right\}$ strongly converges to $w=P_F\left(s_1\right).$

Proof. 

We divide the proof in five steps.

Claim 1:$P_{C_n\cap Q_n}$ is well-defined for every $s_1\in H$. By Theorem 3.2 of Tiammee et al. [12], we obtain that F is closed and convex, and $F\left({\mathsf{\Theta }}_i\right)$ is convex for all $i=1,2,\dots ,N$. From definitions of $C_n$ and $Q_n$, and from Lemma 2.2 in [13] that $C_n\cap Q_n$ is closed and convex. Let $p\in F$. Since $\left\{s_n\right\}$ dominates p and ${\mathsf{\Theta }}_i$ is edge-preserving, we have $({\mathsf{\Theta }}_i{s}_n,p)\in E\left(G\right)$ for all $i=1,2,\dots ,N$. This implies that $(t_n^i,p)=((1-{\alpha }_n^i)s_n+{\alpha }_n^i{\mathsf{\Theta }}_i{s}_n,p)\in E\left(G\right)$ ans as $E\left(G\right)$ is convex, we have

$$\begin{array}{ccc}\hfill \parallel t_n^i-p\parallel & =& \parallel (1-{\alpha }_n^i)s_n+{\alpha }_n^i{\mathsf{\Theta }}_i{s}_n-p\parallel \hfill \\ & \le & (1-{\alpha }_n^i)\parallel s_n-p\parallel +{\alpha }_n^i\parallel {\mathsf{\Theta }}_i{s}_n-p\parallel \hfill \\ & \le & \parallel s_n-p\parallel .\hfill \end{array}$$

(1)

This implies that $\parallel {\overline{t}}_n-p\parallel \le \parallel s_n-p\parallel$. Thus, we have $p\in C_n.$ Therefore $F\subset C_n\cap Q_n$. This implies that $P_{C_n\cap Q_n}\left(s_1\right)$ is well-defined.

Claim 2:${\displaystyle \underset{n\to \infty }{lim}\parallel s_n-s_1\parallel }$ exists. By the definition of the metric projection $P_F$, and since F is a nonempty, closed and convex subset of H, we know that there exists a unique $v\in F$ such that $v=P_F\left(s_1\right).$ From $s_{n+1}=P_{C_n\cap Q_n}\left(s_1\right)$, we get

$$\parallel s_{n+1}-s_1\parallel \le \parallel z-s_1\parallel ,\mathrm{for}\mathrm{all}z\in C_n\cap Q_n\mathrm{and}n\in \mathbb{N}.$$

(2)

On the other hand, as $F\subset C_n\cap Q_n$, we obtain

$$\parallel s_{n+1}-s_1\parallel \le \parallel v-s_1\parallel ,\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(3)

Thus, $\left\{s_n\right\}$ is bounded. Since $s_n=P_{Q_n}\left(s_1\right)$ and $s_{n+1}\in Q_n$, we get

$$\parallel s_n-s_1\parallel \le \parallel s_{n+1}-s_1\parallel ,\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

It follows that the sequence $\{\parallel s_n-s_1\parallel \}$ is bounded and non-decreasing. Therefore ${\displaystyle \underset{n\to \infty }{lim}\parallel s_n-s_1\parallel }$ exists.

Claim 3:${\displaystyle \underset{n\to \infty }{lim}{s}_n=w\in C.}$ For $m>n$, by the definition of $C_n$, since $s_m=P_{C_m}\left(s_1\right)\in C_m\subset C_n$, it follows from the property of the metric projection $P_{C_m}$ that

$$\parallel s_m-s_n{\parallel }^2\le \parallel s_m-s_1{\parallel }^2-{\parallel s_n-s_1\parallel }^2.$$

Since $\parallel s_n-s_1\parallel$ exists, therefore by Step 2, $s_m\to s_n$ as $n\to \infty$. From the completeness of a Hilbert space, $\left\{s_n\right\}$ is a Cauchy sequence. Hence, there exists $w\in C$ such that $s_n\to w$ as $n\to \infty$. In particular, we get

$$\underset{n\to \infty }{lim}\parallel s_{n+1}-s_n\parallel =0.$$

(4)

Claim 4:$w\in F$. Since $s_{n+1}\in C_n$, therefore from (4), we have

$$\parallel {\overline{t}}_n-s_{n+1}\parallel \le \parallel s_n-s_{n+1}\parallel \to 0\mathrm{as}n\to \infty .$$

Now, we obtain

$$\parallel {\overline{t}}_n-s_n\parallel \le \parallel {\overline{t}}_n-s_{n+1}\parallel +\parallel s_{n+1}-s_n\parallel .$$

That is

$${\displaystyle \underset{n\to \infty }{lim}\parallel {\overline{t}}_n-s_n\parallel =0.}$$

(5)

We know that $\parallel t_n^i-s_n\parallel \le \parallel {\overline{t}}_n-s_n\parallel$. By (5), we obtain

$${\displaystyle \underset{n\to \infty }{lim}\parallel t_n^i-s_n\parallel =0,}$$

(6)

for all $i=1,2,\dots ,N$. It follows from the properties in a real Hilbert space, that

$$\begin{array}{ccc}\hfill \parallel t_n^i{-p\parallel }^2& =& \parallel (1-{\alpha }_n^i)s_n+{\alpha }_n^i{\mathsf{\Theta }}_i{s}_n{-p\parallel }^2\hfill \\ & =& (1-{\alpha }_n^i)\parallel s_n{-p\parallel }^2+{\alpha }_n^i\parallel {\mathsf{\Theta }}_i{s}_n{-p\parallel }^2-{\alpha }_n^i(1-{\alpha }_n^i){\parallel {\mathsf{\Theta }}_i{s}_n-s_n\parallel }^2\hfill \\ & \le & (1-{\alpha }_n^i)\parallel s_n{-p\parallel }^2+{\alpha }_n^i\parallel s_n{-p\parallel }^2-{\alpha }_n^i(1-{\alpha }_n^i){\parallel {\mathsf{\Theta }}_i{s}_n-s_n\parallel }^2\hfill \\ & =& \parallel s_n{-p\parallel }^2-{\alpha }_n^i(1-{\alpha }_n^i){\parallel {\mathsf{\Theta }}_i{s}_n-s_n\parallel }^2.\hfill \end{array}$$

(7)

By (7), we obtain

$${\alpha }_n^i(1-{\alpha }_n^i)\parallel {\mathsf{\Theta }}_i{s}_n-s_n{\parallel }^2\le \parallel s_n{-p\parallel }^2-{\parallel t_n^i-p\parallel }^2.$$

(8)

By our Assumption (5) and (8), we obtain

$${\displaystyle \underset{n\to \infty }{lim}\parallel {\mathsf{\Theta }}_i{s}_n-s_n\parallel =0.}$$

From $s_n\to w$ as $n\to \infty$, the assumption (1) and Lemma 6 in [2], we have $w\in F$.

Claim 5:$w=P_F\left(s_1\right)$. Since $s_{n+1}=P_{C_n\cap Q_n}\left(s_1\right)$ and $w\in F\subset C_n\cap Q_n$, we have

$$\langle s_1-s_n,s_n-p\rangle \ge 0,\forall p\in C_n\cap Q_n.$$

(9)

By taking the limit in (9), we obtain

$$\langle s_1-w,w-p\rangle \ge 0,\forall p\in C_n\cap Q_n.$$

Since $F\subset C_n\cap Q_n$, so we have $\langle s_1-w,w-p\rangle \ge 0,$ for each $p\in F$, which gives $w=P_F\left(s_1\right)$. This completes the proof.   □

We know that if $\mathsf{\Theta }$ is $G$-nonexpansive, that $\mathsf{\Theta }$ is nonexpansive. From direct consequences of Theorem 1, we have the following corollary.

Corollary 1.

Assume that $\left\{s_n\right\}$ is a sequence generated by Algorithm 6. Let C be a nonempty closed and convex subset of a real Hilbert space H, and let ${\mathsf{\Theta }}_i:C\to C$ be a nonexpansive mapping for all $i=1,2,\dots ,N$ such that ${\displaystyle F:=\bigcap _{i=1}^{N}F\left({\mathsf{\Theta }}_i\right)\ne \varnothing }$. Then under conditions (3)–(5), $\left\{s_n\right\}$ strongly convergence to $w=P_F\left(s_1\right)$.

Next, we propose the following Algorithm 7:

Algorithm 7: Choose $s_1\in C$, $Q_1=C$ and $\left\{{\alpha }_n^i\right\}$ is a sequence in $[0,1]$ for all $i=0,1,\dots ,N$ such that ${\sum }_{i=0}^N{\alpha }_n^i=1$ for all $n\ge 1$.

(STEP 1) Set $${\displaystyle t_n={\alpha }_n^0{s}_n+\sum _{i=1}^N{\alpha }_n^i{\mathsf{\Theta }}_i{s}_n,}$$ (STEP 2) Compute $$C_n=\{v\in C_n:\parallel v-t_n{\parallel }^2\le \parallel v-s_n{\parallel }^2\}$$ and $$Q_n=\{v\in Q_{n-1}:\langle s_1-s_n,s_n-v\rangle \ge 0\}.$$ (STEP 3) Compute $$s_{n+1}=P_{C_n\cap Q_n}\left(s_1\right).$$ (STEP 4) Set $n:=n+1$, and go to (Step 1).

Theorem 2.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let $G=\left(V\right(G),E(G\left)\right)$ be a directed graph such that $V\left(G\right)=C$ and $E\left(G\right)$ be convex. Assume that $\left\{s_n\right\}$ is generated by Algorithm 7. Then under conditions (1)–(4) and (6), $\left\{s_n\right\}$ strongly converges to $w=P_F\left(s_1\right).$

Proof. 

We shell show that $P_{C_n\cap Q_n}$ is well-defined and $F\subseteq C_n\cap Q_n,\forall n\ge 0$. Similar to Step 1 in Theorem 1, we can show that $C_n\cap Q_n$ is closed and convex, $\forall n\ge 0$. Also, we can show that

$$\begin{array}{ccc}\hfill \parallel t_n-p\parallel & =& {\displaystyle \parallel {\alpha }_n^0{s}_n+\sum _{i=1}^N{\alpha }_n^i{\mathsf{\Theta }}_i{s}_n-p\parallel }\hfill \\ & \le & {\displaystyle {\alpha }_n^0\parallel s_n-p\parallel +\sum _{i=1}^N{\alpha }_n^i\parallel {\mathsf{\Theta }}_i{s}_n-p\parallel }\hfill \\ & \le & \parallel s_n-p\parallel .\hfill \end{array}$$

(10)

Thus, we have $p\in C_n.$ Therefore $F\subset C_n\cap Q_n$. This implies that $P_{C_n\cap Q_n}\left(s_1\right)$ is well-defined. By the same proof of Step 2–3 in Theorem 1, we obtain ${\displaystyle \underset{n\to \infty }{lim}\parallel s_n-s_1\parallel }$ exists. Hence, there exists $w\in C$ such that $s_n\to w$ as $n\to \infty$. In particular, we have

$$\underset{n\to \infty }{lim}\parallel s_{n+1}-s_n\parallel =0.$$

(11)

Next, we show that $w\in F$. By $s_{n+1}\in C_n$, it follows from (11) that

$$\parallel t_n-s_n\parallel \le \parallel t_n-s_{n+1}\parallel +\parallel s_{n+1}-s_n\parallel \le 2\parallel s_{n+1}-s_n\parallel \to 0$$

(12)

as $n\to \infty$. For $p\in \mathsf{\Omega }$, it follows from Lemma 2.1 in [14] and since $\left\{s_n\right\}$ dominates p that

$$\begin{array}{ccc}\hfill \parallel t_n{-p\parallel }^2& =& {\displaystyle \parallel {\alpha }_n^0{s}_n+\sum _{i=1}^N{\alpha }_n^i{\mathsf{\Theta }}_i{s}_n{-p\parallel }^2}\hfill \\ & \le & {\displaystyle {\alpha }_n^0\parallel s_n{-p\parallel }^2+\sum _{i=1}^N{\alpha }_n^i\parallel {\mathsf{\Theta }}_i{s}_n{-p\parallel }^2-\sum _{i=1}^N{\alpha }_n^0{\alpha }_n^i{\parallel {\mathsf{\Theta }}_i{s}_n-s_n\parallel }^2}\hfill \\ & \le & \parallel s_n{-p\parallel }^2-\sum _{i=1}^N{\alpha }_n^0{\alpha }_n^i{\parallel {\mathsf{\Theta }}_i{s}_n-s_n\parallel }^2.\hfill \end{array}$$

This implies that

$${\displaystyle \sum _{i=1}^N{\alpha }_n^0{\alpha }_n^i\parallel {\mathsf{\Theta }}_i{s}_n-s_n{\parallel }^2\le \parallel s_n{-p\parallel }^2-{\parallel t_n-p\parallel }^2.}$$

By our assumption (3) and (12), we obtain

$${\displaystyle \underset{n\to \infty }{lim}\parallel {\mathsf{\Theta }}_i{s}_n-s_n\parallel =0}$$

for all $i=1,2,\dots ,N$. From the fact that $s_n\to w$ as $n\to \infty$, the assumption (1) and Lemma 6 in [2], we have $w\in F$. By the same proof of Step 5 in Theorem 1, we obtain $w\in P_F\left(s_1\right)$. This completes the proof.   □

Corollary 2.

Assume that $\left\{s_n\right\}$ is a sequence generated by Algorithm 7. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let ${\mathsf{\Theta }}_i:C\to C$ be a nonexpansive mapping for all $i=1,2,\dots ,N$ such that ${\displaystyle F:=\bigcap _{i=1}^{N}F\left({\mathsf{\Theta }}_i\right)\ne \varnothing }$. Then under conditions (3), (4) and (6), $\left\{s_n\right\}$ convergence strongly to $w=P_F\left(s_1\right)$.

3. Numerical Experiments

In this section, we give numerical results to support our main theorem. We now give an example in a Euclidean space ${\mathbb{R}}^3$ with a numerical experiment to support our main results.

Example 1.

Let $H={\mathbb{R}}^3$ and $C=[0,\infty )\times [-10,5]\times [0,\infty )$. Assume that $(s,t)\in E\left(G\right)$ if and only if $1\le s_1,t_1$, $-9\le s_2,t_2\le 1.5$ and $0\le s_3,t_3\le 1.25$ or $s=t$ for all $s=(s_1,s_2,s_3)$, $t=(t_1,t_2,t_3)\in C$. Define mappings ${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2,{\mathsf{\Theta }}_3:C\to C$ by

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_{1}s& =& (log\frac{s_1}{2}+2,\frac{arctan{s}_2}{4},1);\hfill \\ \hfill {\mathsf{\Theta }}_{2}s& =& (2,0,\frac{tan(s_3-1)}{4}+1);\hfill \\ \hfill {\mathsf{\Theta }}_{3}s& =& (2,\frac{e^{s_2}-1}{2},1)\hfill \end{array}$$

for all $s=(s_1,s_2,s_3)\in C$. It is easy to check that ${\mathsf{\Theta }}_1$ and ${\mathsf{\Theta }}_2$ are G-nonexpansive such that $F\left({\mathsf{\Theta }}_1\right)\cap F\left({\mathsf{\Theta }}_2\right)=\left\{(2,0,1)\right\}$. On the other hand, ${\mathsf{\Theta }}_1$ is not nonexpansive since for $s=(0.31,1,7)$ and $t=(0.22,1,7)$, this implies that $\parallel {\mathsf{\Theta }}_{1}s-{\mathsf{\Theta }}_{1}t\parallel >0.1>\parallel s-t\parallel$. ${\mathsf{\Theta }}_2$ is not nonexpansive since for $s=(5,-0.5,2.11)$ and $t=(5,-0.5,2.28)$, we have $\parallel {\mathsf{\Theta }}_{2}s-{\mathsf{\Theta }}_{2}t\parallel >0.3>\parallel s-t\parallel$. Moreover, ${\mathsf{\Theta }}_3$ is not nonexpansive since for $s=(1,1.19,0.2)$ and $t=(1,1.02,0.2)$, we have $\parallel {\mathsf{\Theta }}_{3}s-{\mathsf{\Theta }}_{3}t\parallel >0.2>\parallel s-t\parallel$. In this section, CPU and Iter are denoted by the time of CPU and the number of iterations, respectively. All numerical experiments presented were obtained from MATLAB R2019b running on the same laptop computer. In our experiment, we give three cases as follows:

Case 1: Algorithm 6 with ${\alpha }_n=0.1$ of the initial point (1.01, −0.02, 1.26).

Case 2: Algorithm 6 with ${\alpha }_n=0.5$ of the initial point (1.01, −0.02, 1.26).

Case 3: Algorithm 6 with ${\alpha }_n=0.9$ of the initial point (1.01, −0.02, 1.26).

The numerical results are reported as follows:

From

Table 1. The convergence behavior of inputting ${\mathsf{\Theta }}_i$, $i=1,2,3$, stop condition (Cauchy error) $<{10}^{-9}$.

		${\mathsf{\Theta }}_1$	${\mathsf{\Theta }}_2$	${\mathsf{\Theta }}_3$	${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2$	${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_3$	${\mathsf{\Theta }}_2,{\mathsf{\Theta }}_3$	${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2,{\mathsf{\Theta }}_3$
Case 1	Iter	369	348	418	313	339	311	301
	CPU	0.0192	0.0130	0.0113	0.0161	0.0192	0.0192	0.0166
Case 2	Iter	78	72	89	65	71	64	61
	CPU	0.0194	0.0122	0.0101	0.0166	0.0166	0.0164	0.0209
Case 3	Iter	52	54	50	43	37	36	35
	CPU	0.0223	0.0121	0.0109	0.0127	0.0181	0.0191	0.0199

and Figure 1, we see that in the case of two or more inputting ${\mathsf{\Theta }}_i(i\ge 2)$ of the proposed parallel monotone hybrid Algorithm 6 achieves fewer iterations than the inputted one. In the case of three inputting, a little more CPU time is required than in some cases of one or two inputting.

Next, we give an experiment of Algorithm 7. We give 9 cases for parameter α as follows:

Case 1: ${\alpha }_0=0.1,$ ${\alpha }_1=\frac{0.9}{3},$ ${\alpha }_2=\frac{0.9}{3}$ and ${\alpha }_3=\frac{0.9}{3}$.

Case 2: ${\alpha }_0=0.5,$ ${\alpha }_1=\frac{0.5}{3},$ ${\alpha }_2=\frac{0.5}{3}$ and ${\alpha }_3=\frac{0.5}{3}$.

Case 3: ${\alpha }_0=0.9,$ ${\alpha }_1=\frac{0.1}{3},$ ${\alpha }_2=\frac{0.1}{3}$ and ${\alpha }_3=\frac{0.1}{3}$.

Case 4: ${\alpha }_0=0.1,$ ${\alpha }_1=\frac{0.9}{6},$ ${\alpha }_2=2\left(\frac{0.9}{6}\right)$ and ${\alpha }_3=3\left(\frac{0.9}{6}\right)$.

Case 5: ${\alpha }_0=0.1,$ ${\alpha }_1=\frac{0.9}{6},$ ${\alpha }_2=3\left(\frac{0.9}{6}\right)$ and ${\alpha }_3=2\left(\frac{0.9}{6}\right)$.

Case 6: ${\alpha }_0=0.1,$ ${\alpha }_1=2\left(\frac{0.9}{6}\right),$ ${\alpha }_2=\frac{0.9}{6}$ and ${\alpha }_3=3\left(\frac{0.9}{6}\right)$.

Case 7: ${\alpha }_0=0.1,$ ${\alpha }_1=2\left(\frac{0.9}{6}\right),$ ${\alpha }_2=3\left(\frac{0.9}{6}\right)$ and ${\alpha }_3=\frac{0.9}{6}$.

Case 8: ${\alpha }_0=0.1,$ ${\alpha }_1=3\left(\frac{0.9}{6}\right),$ ${\alpha }_2=\frac{0.9}{6}$ and ${\alpha }_3=2\left(\frac{0.9}{6}\right)$.

Case 9: ${\alpha }_0=0.1,$ ${\alpha }_1=3\left(\frac{0.9}{6}\right),$ ${\alpha }_2=2\left(\frac{0.9}{6}\right)$ and ${\alpha }_3=\frac{0.9}{6}$.

For all cases, we choose the initial point (1.48, −0.06, 2.72).

From

Table 2. The convergence behavior of inputting ${\mathsf{\Theta }}_i$, $i=1,2,3$, stop condition (Cauchy error) $<{10}^{-9}$.

		${\mathsf{\Theta }}_1$	${\mathsf{\Theta }}_2$	${\mathsf{\Theta }}_3$	${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2$	${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_3$	${\mathsf{\Theta }}_2,{\mathsf{\Theta }}_3$	${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2,{\mathsf{\Theta }}_3$
Case 1	Iter	43	42	60	37	44	44	37
	CPU	0.01127	0.0081	0.0066	0.0061	0.0097	0.0087	0.0075
Case 2	Iter	73	81	103	68	104	70	68
	CPU	0.0118	0.0063	0.0053	0.0075	0.0099	0.0073	0.0076
Case 3	Iter	357	419	448	335	365	340	324
	CPU	0.0125	0.0085	0.0053	0.0075	0.0099	0.0073	0.0076
Case 4	Iter	43	42	60	35	51	53	40
	CPU	0.0123	0.0061	0.0048	0.0084	0.0109	0.0060	0.0109
Case 5	Iter	43	42	60	35	51	37	36
	CPU	0.0120	0.0062	0.0061	0.0061	0.0086	0.0087	0.0102
Case 6	Iter	43	42	60	40	47	38	41
	CPU	0.0143	0.0089	0.0056	0.0070	0.0082	0.0078	0.0098
Case 7	Iter	43	42	60	35	47	37	49
	CPU	0.0122	0.0067	0.0051	0.0053	0.0105	0.0085	0.0094
Case 8	Iter	43	42	60	40	43	38	41
	CPU	0.0124	0.0071	0.0056	0.0062	0.0079	0.0088	0.0103
Case 9	Iter	43	42	60	35	43	53	34
	CPU	0.00126	0.0057	0.0048	0.0065	0.0085	0.0075	0.0142

, we see that the CPU time and the number of iterations of Algorithm 7 decrease when the parameter ${\alpha }_0$ approaches 0 and the rate of ${\alpha }_n$, by $3:2:1$ this has an effect on the CPU time and the number of iterations for input many mappings ${\mathsf{\Theta }}_i$.

Next, we present some numerical examples of signal recovery. We provide a comparison between Algorithms 2, 3, 6 and 7. In this case, we set $\mathsf{\Theta }\left(s_n\right)={\mathrm{prox}}_{\lambda g}(s_n-\lambda \nabla f\left(s_n\right))$. It is known that $\mathsf{\Theta }$ is a nonexpansive mapping when $\lambda \in (0,2/L)$ and L is the Lipschitz constant of $\nabla f$. Compressed sensing can be modeled as the following underdeterminated linear equation system:

$$t=As+ϵ,$$

(13)

where $s\in {\mathbb{R}}^N$ is a original signal vector, $t\in {\mathbb{R}}^M$ is the observed signal which disturbed by filer operator $A:{\mathbb{R}}^N\to {\mathbb{R}}^M(M<N)$ and noisy $ϵ$. It is known that the solution of (13) can be seen as solving the LASSO problem:

$${\displaystyle \underset{s\in {\mathbb{R}}^N}{min}\frac{1}{2}{\parallel t-As\parallel }_2^2+\lambda {\parallel s\parallel }_1,}$$

(14)

where $\lambda >0$. So we can apply our method for solving (14) in the case that $f\left(s\right)=\frac{1}{2}{\parallel t-As\parallel }_2^2$ and $g\left(s\right)={\lambda \parallel s\parallel }_1.$ It is noted that $\nabla f\left(s\right)=A^T(As-t)$.

The goal of this paper is to remove noise without knowing the type of it. Thus, we focus on the following problem:

$$\begin{array}{ccc}\hfill {\displaystyle \underset{s\in {\mathbb{R}}^N}{min}\frac{1}{2}{\parallel A_{1}s-t_1\parallel }_2^2}& +& {\lambda }_1{\parallel s\parallel }_1,\hfill \\ \hfill {\displaystyle \underset{s\in {\mathbb{R}}^N}{min}\frac{1}{2}{\parallel A_{2}s-t_2\parallel }_2^2}& +& {\lambda }_2{\parallel s\parallel }_1,\hfill \\ & ⋮& \hfill \\ \hfill {\displaystyle \underset{s\in {\mathbb{R}}^N}{min}\frac{1}{2}{\parallel A_{N}s-t_N\parallel }_2^2}& +& {\lambda }_N{\parallel s\parallel }_1,\hfill \end{array}$$

(15)

where s is the original signal, $A_i$ is a bounded linear operator and $t_i$ is observed signal with noisy for all $i=1,2,\dots ,N.$ We can apply the Algorithms 6 and 7 to solve the problem (15) by setting ${\mathsf{\Theta }}_{i}s={\mathrm{prox}}_{{\lambda }_i{g}_i}(s_n-{\lambda }_i\nabla f_i\left(s_n\right))$.

In our experiment, the observations $t_1,t_2,t_3$ are generated by different Gaussian noise white signal-to-noise ratio SNR and normal distribution with zero mean and one invariance matrix $A_1,A_2,A_3\in {\mathbb{R}}^{M\times N}$, respectively. The initial point $s_1$ is picked randomly. We use the mean squared error (MSE) for showing the restoration accuracy. This MSE is defined by

$$\mathrm{MSE}=\frac{1}{N}{\parallel s_n-s^{*}\parallel }_2^2<{10}^{-5},$$

where $s^{*}$ is an estimated signal of s.

In what follows, let ${\alpha }_n^i=0.5$ for all $i=1,2,3$ and let the step sizes ${\lambda }_1=\frac{1}{\parallel A_1{\parallel }^2}$, ${\lambda }_2=\frac{1}{\parallel A_2{\parallel }^2}$ and ${\lambda }_3=\frac{1}{\parallel A_3{\parallel }^2}$. The numerical results are shown as follows:

The performance of the studied proposed Algorithm 6 with the following original signal (Figure 2) is tested.

The different types of blurred matrices $A_1$, $A_2$ and $A_3$ are shown in Figure 3.

The results of the Algorithm 6 with $(N=1)$ by inputting $A_i,i=1,2,3$ for the following three cases:

Case 1.1: Inputting $A_1$, SNR = 40 on the proposed algorithm;

Case 1.2: Inputting $A_2$, SNR = 50 on the proposed algorithm;

Case 1.3: Inputting $A_3$, SNR = 60 on the proposed algorithm;

Are shown in Figure 4 which are composed of the recovered signal.

Next, we present finding the common solutions to signal recovery problem (15) with $(N\ge 2)$ by using the Algorithm 6. So, we can consider the results algorithm in the following three cases:

Case 2.1: Inputting $A_1$, SNR = 40 and $A_2$, SNR = 50;

Case 2.2: Inputting $A_1$, SNR = 40 and $A_3$, SNR = 60;

Case 2.3: Inputting $A_2$, SNR = 50 and $A_3$, SNR = 60;

Case 3.1: Inputting $A_1$, SNR = 40, $A_2$, SNR = 50 and $A_3$, SNR = 60;

Are shown in Figure 5 which are composed of the recovered signal.

From Figure 4 and Figure 5, we see that Case 2.1–2.3 has less number of iterations and CPU time than Case 1.1–1.3 in all of the cases.

Finally, we present the common solution the signal recovery problem (15) with $(N=3)$ of Case 3.1 and a comparison among Algorithms 2 and 3 shown in Figure 6.

From Figure 5 and Figure 6, we see that Case 3.1 has a lower number of iterations and CPU time than Case 2.1–2.3 all of the cases. This means that the efficiency of the proposed Algorithm 6 is better when the number of subproblems is increasing. Moreover, we see that our proposed Algorithm 6 gets less CPU time and fewer iterations than the other two algorithms. The following Figure 7 shows the efficiency of our parallel Algorithm 6 in all cases and a comparison with the other two algorithms by MSE versus the number of iterations.

Next, we analyze the convergence and the effects of $\left\{{\alpha }_n\right\}$ in Algorithm 6 in each case.

From

Table 3. The convergence of Algorithm 6 with each $\left\{{\alpha }_n\right\}.$

Given: Random Initial Point, Stop Condition (Cauchy Error) $<{10}^{-9}$.
$\left\{{\mathit{\alpha }}_{\mathit{n}}\right\}$	SNR	N = 512, M = 256		N = 1024, M = 512
		m = 20		m = 20
		Iter	CPU	Iter	CPU
${\displaystyle \frac{4n^2+12}{20n^2+10}}$	40	21,189	206.1261	32,906	1001.1
	50	22,454	232.5960	34,356	1084.8
	40, 50	5529	16.6651	8145	48,778
${\displaystyle \frac{25n^5+15}{50n^5+10}}$	40	7360	27.2077	14,285	201.2996
	50	7560	28.3236	13,531	182.0590
	40, 50	2210	3.4878	3610	20.8226
${\displaystyle \frac{81n^9+19}{90n^9+10}}$	40	5190	14.0915	8534	77.5054
	50	4316	9.8401	7622	64.8060
	40, 50	1301	1.5145	1660	6.7987

, we observe that the CPU time and the number of iterations of Algorithm 6 small reduction when the parameter $\left\{{\alpha }_n\right\}$ approaches 1. The following Figure 8 and Figure 9 show numerical results for each $\left\{{\alpha }_n\right\}$ in Table 3.

Next, we provide a comparison between Algorithms 2, 3 and 7. For convenience, we set all conditions as in the previous example.

The performance of the studied proposed Algorithm 7 with the following original signal (Figure 10) is tested.

The different types of blurred matrices $A_1$, $A_2$ and $A_3$ are shown in Figure 11.

The results of the Algorithm 7 with $(N=1)$ by inputting $A_i,i=1,2,3$ for the following three cases:

Case 1.1: Inputting $A_1$, SNR = 40 on the proposed algorithm;

Case 1.2: Inputting $A_2$, SNR = 50 on the proposed algorithm;

Case 1.3: Inputting $A_3$, SNR = 60 on the proposed algorithm.

Figure 12 which are composed of the recovered signal.

Next, we present the finding of the common solutions to the signal recovery problem (15) with $(N\ge 2)$ by using the Algorithm 7. So, we can consider the results algorithm in the following three cases:

Case 2.1: Inputting $A_1$, SNR = 40 and $A_2$, SNR = 50;

Case 2.2: Inputting $A_1$, SNR = 40 and $A_3$, SNR = 60;

Case 2.3: Inputting $A_2$, SNR = 50 and $A_3$, SNR = 60;

Case 3.1: Inputting $A_1$, SNR = 40, $A_2$, SNR = 50 and $A_3$, SNR = 60;

Are shown in Figure 13 which are composed of the recovered signal.

From Figure 12 and Figure 13, we see that Case 2.1–2.3 has a lower number of iterations and CPU time than Case 1.1–1.3 in all of the cases. Finally, we present the common solution of signal recovery problem (15) with $(N=3)$ of Case 3.1 and a comparison between Algorithms 2 and 3 which has been shown in Figure 14.

Figure 15 shows the efficiency of our parallel Algorithm 7 in all cases and a comparison with the other two algorithms by MSE versus the number of iterations.

From Figure 14 and Figure 15, we see that Case 3.1 has a lower number of iterations and CPU time than Case 2.1–2.3 all of the cases. This means that the efficiency of the proposed Algorithm 7 is also better when the number of subproblems is increasing. Moreover, we see that our proposed Algorithm 7 requires a lower CPU time and number of iterations than the other two algorithms.

4. Conclusions

This paper proposes two parallel hybrid projection algorithms for finding a common fixed point of a finite family of G-nonexpansive mappings in Hilbert spaces with a directed graph. Under some suitable conditions on the update parameters generated in two different algorithms, we obtain strong convergence theorems. We also give examples of numerical experiments to support our main results and compare the rate between the proposed and the existing two methods. It is found that our algorithms have a better convergence behavior than these methods through experiments.