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A parallel multisplitting method with selfadaptive weightings for solving Hmatrix linear systems
Journal of Inequalities and Applications volume 2017, Article number: 95 (2017)
Abstract
In this paper, a parallel multisplitting iterative method with the selfadaptive weighting matrices is presented for the linear system of equations when the coefficient matrix is an Hmatrix. The zero pattern in weighting matrices is determined in advance, while the nonzero entries of weighting matrices are determined by finding the optimal solution in a hyperplane of α points generated by the parallel multisplitting iterations. Especially, the nonnegative restriction of weighting matrices is released. The convergence theory is established for the parallel multisplitting method with selfadaptive weightings. Finally, a numerical example shows that the parallel multisplitting iterative method with the selfadaptive weighting matrices is effective.
1 Introduction and preliminaries
To solve the large sparse linear system of equations
O’Leary and White [1] first proposed parallel methods based on multisplittings of matrices in 1985, where several basic convergence results may be found. A multisplitting of A is a collection of triples of \(n\times n\) matrices \((M_{i},N_{i},E_{i})_{i=1}^{\alpha}\) (\(\alpha\leq n\), a positive integer) with \(M_{i},N_{i},E_{i}\in\mathbf{R}^{n\times n}\), and then the following method for solving system (1.1) was given:
where \(M_{i}\), \(i=1,2,\ldots,\alpha\), are nonsingular and \(E_{i}=\operatorname{diag}(e_{1}^{(i)}, e_{2}^{(i)}, \ldots, e_{n}^{(i)})\), \(i=1,2,\ldots,\alpha\), satisfy \(\sum_{i=1}^{\alpha}E_{i}=I\) (\(I\in\mathbf{R}^{n\times n}\) is the identity matrix). By the way, \(x^{(0)}\) is an initial vector of \(x_{*}=A^{1}b\).
There has been a lot of study (see [1–13]) on the parallel iteration methods for solving the large sparse system of linear equations (1.1). In particular, when the coefficient matrix A is an Mmatrix or an Hmatrix, many parallel multisplitting iterative methods (see, e.g., [1–7, 10]) were presented, and the weighting matrices \(E_{i}\), \(i=1,2,\ldots,\alpha\), were generalized (see, e.g., [3, 4, 6–10])
but these weighting matrices were preset as multiparameter.
As we know, the weighting matrices play an important role in parallel multisplitting iterative methods, but the weighting matrices in all the abovementioned methods are determined in advance, they are not known to be good or bad, and this influences the efficiency of parallel methods. Recently, Wen and coauthors [13] discussed selfadaptive weighting matrices for a symmetric positive definite linear system of equations; Wang and coauthors [11] discussed selfadaptive weighting matrices for nonHermitian positive definite linear system of equations. In this paper, we focus on the Hmatrix, which originates from Ostrowski [14]. The Hmatrix is a class of the important matrices that has many applications. For example, numerical methods for solving PDEs are a source of many linear systems of equations whose coefficients form Hmatrices (see [15–18]). Are selfadaptive weighting matrices true for the linear system of equations when the coefficient matrix is an Hmatrix? We will discuss this problem in this paper.
Here, we generalize the weighting matrices \(E_{i}\) (\(i=1,2,\ldots,\alpha\)) to \(E_{i}^{(k)}\) (\(i=1,2,\ldots,\alpha\); \(k=1,2, \ldots\)), which can be divided into two steps: each splitting is first dealt with by a different processor, the weighting matrices \(E_{i}^{(k)}\) (\(i=1,2,\ldots,\alpha\); \(k=1,2,\ldots\)) will mask large portion of the matrix, so that each processor deals with a smaller matrix; later, the weighting matrices \(E_{i}^{(k)}\) (\(i=1,2,\ldots,\alpha\); \(k=1,2, \ldots\)) will be approximated to ‘the best’ choices well for kstep iteration in the hyperplane generated by \(\{x_{i}^{(k)}, i=1,2, \ldots, \alpha\}\), \(k=1,2, \ldots \) . In this paper, we determine the weighting matrices \(E_{i}^{(k)}\) (\(i=1,2,\ldots,\alpha\); \(k=1,2, \ldots\)) for above reasons. Firstly, decreasing execution times is completed by the zeroentry’s set in weighting matrices. Secondly, the selfadaptive weighting matrices are reached by finding the ‘good’ point in the hyperplane generated by \(\{ x_{i}^{(k)}, i=1,2, \ldots, \alpha\}\), \(k=1,2, \ldots \) . Thus, the scheme of finding the selfadaptive weighting matrices is established for the parallel multisplitting iterative method, it has two advantages as follows:

The weighting matrices are not necessarily nonnegative;

Only one splitting of α splittings is required to be convergent.
In the rest of this study, we first give some notations and preliminaries in Section 1, and then a parallel multisplitting iterative method with the selfadaptive weighting matrices is put forward in Section 2. The convergence of the parallel multisplitting iterative method is established in Section 3. Moreover, we give the computational results of the parallel multisplitting iterative method by a problem in Section 4. We end the paper with a conclusion in Section 5.
Here are some essential notations and preliminaries. \(\mathbf{R}^{n\times n}\) is used to denote the \(n\times n\) real matrix set, and \(\mathbf{R}^{n}\) is the ndimensional real vector set. \(A^{T}\) represents the transpose of the matrix A, and \(x^{T}\) denotes the transpose of the vector x. \(\langle A\rangle\) stands for the comparison (Ostrowski) matrix of the matrix A, and \(A\) is the absolute matrix of the matrix A.
In what follows, when A is a strictly diagonally dominant matrix in column, A is called a strictly diagonally dominant matrix.
Definition 1.1
[17]
The matrix A is an Hmatrix if there exists a positive diagonal matrix D such that the matrix DA is a strictly diagonally dominant matrix.
Property 1.2
[14]
The matrix A is an Hmatrix if and only if \(\langle A\rangle\) is an Mmatrix.
Definition 1.3
Suppose that A is an Hmatrix. Let \(A=MN\), which is called an Hcompatible splitting if \(\langle A\rangle =\langle M\rangleN\).
Property 1.4
[15]
Let A be an Hmatrix, then \(A^{1}\le\langle A\rangle^{1}\).
2 Description of the method
Here, we present a parallel multisplitting iterative method with the selfadaptive weighting matrices.
Let
By introducing
The iteration matrix of kth step
The preset nonzero set
Method 2.1
 Step 0.:

Given a tolerance \(\epsilon>0\) and an initial vector \(x^{(0)}\). For \(i=1,2, \ldots, \alpha\); \(k=1,2, \ldots \) , give also the set \(S(i,k)\) and for some \(i_{0}\), \(S(i_{0},k)=\{1,2,\ldots,n\}\). The sequence of \(x^{(k)}\), \(k=1,2,\ldots\) , until converges.
 Step 1.:

Solution in parallel. For the ith processor, \(i=1,2,\ldots,\alpha\): compute \(x_{j}^{(i,k)}\), \(j\in S(i,k)\) by
$$ M_{i}x^{(i,k)}=N_{i}x^{(i,k1)}+b, $$(2.6)where \(x^{(i,k)}=(x_{1}^{(i,k)},\ldots,x_{n}^{(i,k)})^{T}\).
 Step 2.:

For the \(i_{0}\)th processor, set
$$x=\sum_{i=1}^{\alpha}E_{i}^{(k)}x_{i}^{(k)}= \left ( \textstyle\begin{array}{@{}c@{}} \sum_{i=1}^{\alpha}e_{1}^{(i,k)}x_{1}^{(i,k)} \\ \sum_{i=1}^{\alpha}e_{2}^{(i,k)}x_{2}^{(i,k)}\\ \vdots\\ \sum_{i=1}^{\alpha}e_{n}^{(i,k)}x_{n}^{(i,k)} \end{array}\displaystyle \right ). $$Solve the following optimization problem:
$$\begin{aligned}& \min_{e_{j}^{(i,k)}\in S(k)}\Axb\_{1} \\& \quad \mbox{s.t. } \sum_{i=1}^{\alpha}E_{i}^{(k)}=I. \end{aligned}$$(2.7)Or
$$\begin{aligned}& \Axb\_{1}\le\bigl\ Ax_{i_{0}}^{(k)}b\bigr\ _{1} \\& \quad \mbox{s.t. } \sum_{i=1}^{\alpha}E_{i}^{(k)}=I. \end{aligned}$$(2.8)  Step 3.:

Compute
$$ x^{(k)}=\sum_{i=1}^{\alpha}E_{i}^{(k)}x_{i}^{(k)}. $$(2.9)  Step 4.:

If \(\Ax^{(k)}b\_{1}\leq\epsilon\), stop; otherwise, \(k\Leftarrow k+1\), go to Step 1.
Remark 2.1
The implementation of this method is that at each iteration there are α independent problems of the kind (2.6) with \(x_{j}^{(i,k)}\), \(j\in S(i,k)\) represents the solution to the local problem. The work for each equation in (2.6) is assigned to one processor, and communication is required only to produce the update given in (2.9). In general, some (most) of the diagonal elements in \(E_{i}\) are zero and therefore the corresponding components of \(x_{j}^{(i,k)}\), \(j\in S(i,k)\) need not be calculated.
Remark 2.2
We may use some optimization methods such as the simplex method (see [19]) to solve an approximated solution satisfying inequality (2.7). Usually, we can compute the optimal weighting at every two or three iterations to replace at each iteration step. Hence, the computational complexity of (2.7) is about \(4n^{2}\) or \(6n^{2}\) flops.
On the other hand, if \(S(i_{0},k)=n\), \(S(i,k)=1\), \(i\neq i_{0}\), let \(x=(1t)x^{(i_{0},k)}+t\bar{x}^{(k)}\), (2.7) becomes
Now, we consider the following programming:
Assumptions

(a)
\(b_{j}\geq0\), \(j=1,2,\ldots,n\).

(b)
\(\frac{b_{1}}{a_{1}}\leq \frac{b_{2}}{a_{2}}\leq\cdots\leq\frac{b_{n}}{a_{n}}\).
Lemma 2.1
Let programming (2.10) satisfy Assumptions and \(x_{j}=\frac{b_{j}}{a_{j}}\), \(j=1,2,\ldots,n\). Then there exists some \(j_{0}\) such that \(x_{j_{0}}\) is the solution of programming (2.10).
Proof
As we know, the solution \(x^{*}\in[\frac{b_{1}}{a_{1}},\frac{b_{n}}{a_{n}}]\). Let \(P=\{\frac{b_{1}}{a_{1}}, \frac{b_{2}}{a_{2}},\ldots,\frac{b_{k}}{a_{k}},\ldots,\frac{b_{n}}{a_{n}}\}\) be a partition of \([\frac{b_{1}}{a_{1}},\frac{b_{n}}{a_{n}}]\), it implies that \(\frac{b_{1}}{a_{1}}< \frac{b_{2}}{a_{2}}<\cdots<\frac{b_{k}}{a_{k}}<\cdots<\frac{b_{n}}{a_{n}}\). Thus, we obtain a set of subinterval induced by the partition P.
In every subinterval, the function \(\sum_{j=1}^{n}b_{j}a_{j}x\) is a linear function, then the minimization point of the linear function is just a partition point. Hence, the lemma has been proved. □
Corollary 2.2
Let programming (2.10) satisfy Assumptions. Then
with \(\frac{b_{k}}{a_{k}}\leq 0\leq\frac{b_{k+1}}{a_{k+1}}\).
From Lemma 2.1 we obtain an approximated solution, its complexity is about \(4n^{2}\) flops.
3 Convergence analysis
In this section, we discuss the convergence of Method 2.1 under the reasonable assumptions.
Lemma 3.1
Let \(A=MN\) be Hcompatible splitting of the strictly diagonally dominant matrix A, then
Proof
From the definition of Hcompatible splitting, we know that \(\langle A\rangle=\langle M\rangleN\).
Let \(N=(N_{1}^{T} , N_{2}^{T},\ldots,N_{n}^{T} )^{T}\), \(N_{j}=(n_{j1},n_{j2},\ldots ,n_{jn})\). From Property 1.4, it holds that
Let \(e^{T}N\langle M\rangle^{1}=x^{T}\), where \(x^{T}=(x_{1},x_{2},\ldots,x_{n})\). We have
Let \(x_{j_{0}}=\max_{1\leq j\leq n}x_{j}\), which implies
From the Hcompatible splitting \(\langle A\rangle=\langle M\rangle N\) and the strict diagonal dominance of \(\langle A\rangle\), it holds that
Hence, \(\NM^{1}\_{1}<1\). □
Lemma 3.2
Let \(A=MN\) be Hcompatible splitting of the Hmatrix A. Then there exists a positive diagonal matrix D such that
Proof
There is a positive diagonal matrix D such that DA is a strictly diagonally dominant matrix since A is an Hmatrix. Thus
Let
Then
is a regular splitting of the strictly diagonally dominant matrix Ā. From Lemma 3.1, we know that
Hence,
□
Theorem 3.3
Let \(A=M_{i}N_{i}\), \(i=1,2,\ldots,\alpha\), be α splittings of the Hmatrix A, and for some \(i_{0}\), let \(A=M_{i_{0}}N_{i_{0}}\) be Hcompatible splitting. Assume that \(E_{i}^{(k)}\), \(i=1,2,\ldots, \alpha\); \(k=1,2,\ldots \) are yielded by (2.7) or (2.8) in Method 2.1. Then \(\{x^{(k)}\} \) generated by Method 2.1 converges to the unique solution \(x_{*}\) of (1.1).
Proof
Let \(\varepsilon^{(k)}=x^{(k)}x_{*}\). We have
On the other hand,
Let D be a positive diagonal matrix such that DA is a strictly diagonally dominant matrix. Thus, from (2.8) (or (2.9)) and Lemma 3.2 we know that
where \(r=\DN_{i_{0}}\langle M_{i_{0}}\rangle^{1}D^{1}\_{1}<1\).
Thus, \(\lim_{k\rightarrow\infty}\DA\varepsilon^{(k+1)}\_{1} =0\), which implies that
We have completed the proof of the theorem. □
4 Numerical experiments
In this section, a test problem to assess the feasibility and effectiveness of Method 2.1 in terms of both iteration number (denoted by IT) and computing time (in seconds, denoted by CPU) is given. All our tests are started from zero vector and terminated when the current iterate satisfied \(\r^{(k)}\_{1}<10^{6}\), where \(r^{(k)}\) is the residual of the current, say kth iteration or the number of iterations is up to 20,000. For the latter the iteration is failing. We solve (2.7) or (2.8) in the optimization step by the simplex method (see [19]).
Problem
Consider the generalized convectiondiffusion equations in a twodimensional case. The equation is
with the homogeneous Dirichlet boundary condition. We use the standard RitzGalerkin finite element method by \(P_{1}\) conforming triangular element to approximate the following continuous solutions \(u=x\cdot y\cdot(1x)\cdot(1y)\) in the domain \(\Omega=[0,1]\times[0,1]\), the stepsizes along both x and y directions are the same, that is, \(h=\frac{1}{2^{m}}\), \(m=5,6,7\). Let \(q=1\).
After discretization, the matrix A of this equation is given by
where \(A_{i,i}\), \(i=1,\ldots, p\), are sbys nonsymmetric matrices and \(B^{T}_{i,i+1}\neq C_{i+1,i}\). Thus, \(n=s\cdot p\) and \(s=p=2^{m}=32,64,128\) from the above.
Let
where D is a block diagonal matrix, L is the strictly block lower triangle matrix, U is the strictly block upper triangle matrix.
We construct the multisplitting as follows:

(a)
The block Jacobi splitting (denoted by BJ)
$${A}={M}_{1}{N}_{1}, \qquad {M}_{1}={D}. $$ 
(b)
The block GaussSeidel splitting I (denoted by BGSI)
$${A}={M}_{2}{N}_{2},\qquad {M}_{2}= {D}{L}. $$ 
(c)
The block GaussSeidel splitting II (denoted by BGSII)
$${A}={M}_{2}{N}_{2},\qquad {M}_{2}= {D}{U}. $$
The three weighting matrices are chosen as follows:
where I is the \(\frac{n}{2} \times\frac{n}{2}\) identity matrix.
In the first results presented in Table 1 we show the spectral radius of the corresponding iteration matrix for the classical iterative methods (a)(c). It is well known that an iterative method is convergent when the spectral radius of the corresponding iteration matrix is less than one. We can see that the numbers listed in Table 1 approximate to 1 such that these iterative methods converge to the unique solution of (4.1) slowly. However, the behavior of the multisplitting method based on three splittings (a)(c) can be improved by selfadaptive weightings, and the results are shown in Tables 2 and 3.
Here, we denote the basic parallel multisplitting iterative method with fixed weighting matrix by BMeth (see [1]). In Basic Methods, we propose three groups of weighting matrices, which are generated by random selection. Thus, the corresponding parallel multisplitting iterative methods are denoted by BMeth 1, BMeth 2, BMeth 3, respectively.
The speedup is defined in the following:
From the above numerical experiments, it is obtained that the average speedup of the new parallel multisplitting iterative method (Method 2.1) is about 2.3 (the average value of all computational results by Basic Methods).
5 Conclusion
The parallel multisplitting iterative method with the selfadaptive weighting matrices has been proposed for the linear system of equations (1.1) when the coefficient matrix is an Hmatrix. The convergence theory is established for the parallel multisplitting method with selfadaptive weightings. The numerical results show that the new parallel multisplitting iterative method with the selfadaptive weightings is effective.
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Acknowledgements
The authors are very much indebted to the anonymous referees for their helpful comments and suggestions. And the authors are so thankful for the support from the NSF of Shanxi Province. This work is supported by grants from the NSF of Shanxi Province (201601D011004).
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Authors’ contributions
RW presented the optimization model of weighting matrices, carried out the convergence studies and drafted the manuscript. HD implemented the algorithm and helped to draft the manuscript. All authors read and approved the final manuscript.
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Wen, R., Duan, H. A parallel multisplitting method with selfadaptive weightings for solving Hmatrix linear systems. J Inequal Appl 2017, 95 (2017). https://doi.org/10.1186/s1366001713707
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DOI: https://doi.org/10.1186/s1366001713707