On a splitting for a class of block two- by-two linear systems 1Salisu Lukunti, 2Aminu Haruna, and 3Muhammad Mubarak Muhammad

3Department of Mathematics, Lovely Professional University, Punjab. India. 1,2& 3 Department of Mathematics and Statistics, Federal Polytechnic Bauchi, P.M.B. 0231, Bauchi State, Nigeria.

Abstract:

Recently, Bai et al. (2013) proposed an effective and efficient matrix splitting , called preconditioned modified Hermitian/Skew-Hermitian splitting (PMHSS) iteration method, for two-by-two block linear systems of equations. The eigenvalue distribution of the iterative matrix suggests that the splitting matrix could be advantageously used as a . In this study, the Conjugate gradient on the normal equations(CGNR) method is utilized for solving the PMHSS preconditioned linear systems, and the performance of the method is considered by estimating the condition number of the normal equations. Furthermore, the proposed method is compared with other PMHSS preconditioned methods by solving linear systems arising in complex partial differential equations and a distributed control problem. The numerical results demonstrate the difference in the performance of the methods under consideration.

Keywords: PMHSS preconditioner, Krylov subspace method, Two-by-two block matrix splitting

1. Introduction

The following two-by-two block systems of linear equations are considered:

(1.1) where are real, symmetric, and positive semidefinite matrices with at least one of them, e.g., W, positive definite. In this case, the coefficient matrix is nonsingular. Thus the linear system has a unique solution. This class of linear systems can be formally regarded as a special case of generalized saddle-point problems [1]. Moreover, they possess the special structure and properties so that particular numerical solutions are of considerable interest. The linear system (1.1) arises in a variety of applications, such as complex symmetric linear systems in electromagnetism, electrical power system model structural dynamics and quantum mechanics. Another important application of (1.1) is the finite-element discretization of elliptic PDE-constrained optimization problems such as distributed control problems [2]. (1.1) also arises in finite-element discretization and first-order linearization of two- phase flow problems based on the Cahn– Hilliard equation [3].For large-scale problems, iterative solutions,

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particularly Krylov subspace methods with suitable preconditioners, are the preferred numerical methods. Recently, several effective preconditioners have been developed for (1.1). For problems arising in PDE-constrained optimization, the induced iterative methods converge fast, and the convergence is independent of the mesh size and the regularized parameters. For instance, in [4–6], and [7], the preconditioners are proved to cluster the eigen values in a small region, and the preconditioned iterative methods primarily consist in solving two linear systems of equations, in which the coefficient matrices are linear combinations of W and T. An effective preconditioner, namely, the PMHSS preconditioner, is presented in [8]. It is a nonsymmetrical preconditioner and is utilized to improve the performance of a solver of nonsymmetrical linear systems. In [7,9], two different symmetric positive definite preconditioners are derived for two symmetric linear systems equivalent to (1.1). The MINRES method, which performs three-term recurrence, is more efficient than the GMRES method when their iterative counters are comparable. PMHSS preconditioners are also generalized to the case of indefinite W and T in [10]. In the present study, the CGNR method is considered for solving the linear system (1.1). Generally, the condition number of the normal equations should be the square of the condition number of the original equations. However, according to the eigenvalue distribution of the PMHSS preconditioned matrix, the condition number is quite small. It enlightens us to study CGNR method for solving the PMHSS preconditioned linear system. The numerical behavior of the proposed method is compared with that of other PMHSS-type preconditioned Krylov subspace methods using two problems arising from different applications.

The remainder of the paper is organized as follows. The PMHSS preconditioner is reviewed in Section 2. Section 3 contains the description of PMHSS preconditioned CGNR and its features. Two problems are introduced in Section 4. Then, the PMHSS preconditioned Krylov subspace methods are implemented on the linear systems arising from these problems in Section 5. Section 6 concludes the paper.

2. PMHSS preconditioners and preconditioning methods

In [8], a matrix splitting for the following two-by-two block matrix is presented:

Where

and (2.1)

It is verified that F is a good preconditioner of A. According to Remark 2.2 and Theorem 2.3 in [8], the preconditioned matrix has the following eigenvalue decomposition:

(2.2) Where 2 | P a g e

, .

The eigenvalues are , , where are the eigenvalues of . The

modulus of is not greater than , as . The columns of are the corresponding eigenvectors. It is

clear that eigenvector matrices are not unique and hardly to find the infimum of their condition number. In [8], the

authors construct a matrix such that . Moreover, there is an eigenvector matrix whose

condition number is equal to 1 when is normal. In [9], by introducing

, with , a symmetric linear system equivalent to (1.1) is obtained, namely,

, with .

This system is solved by the preconditioned MINRES method with preconditioner

.

In the present study, this is called the PMHSS–MINRES method. Another PMHSS-type preconditioner

,

is defined in [7], which is called additive block diagonal preconditioner, and is used for preconditioning the MINRES method for solving the linear system

. (2.3)

It is proved that the eigenvalues of , are located in the union of the two intervals

where . In the present study, this is called the ABD-MINRES method.

3. PMHSS preconditioned CGNR method

In this section, the CGNR method is considered for solving the PMHSS preconditioned linear system

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The CGNR method adopts the conjugate gradient method for solving the normal equations

, where A is nonsingular and nonsymmetrical. CGNR yields the approximate solution that minimizes in the following Krylov subspace:

Unlike the GMRES method, which yields the approximate solution in the Krylov subspace

CGNR performs three-term recurrence and does not store all the orthogonal vectors generated by the Lanczos process [11]. A potential weakness of CGNR is the ill condition number of the coefficient matrix, which indicates slow convergence. To ascertain the feasibility of CGNR, the condition number of is checked as follows.

Theorem 3.1. Let A be a two-by-two matrix in (1.1) and F be the preconditioner of A in (2.1). W and T are positive definite and positive semi definite block matrices in A. The maximum and minimum singular values of F−1A are denoted by and , respectively. Then, the condition number of is bounded as

.

Proof. The maximum eigenvalue of is the square of the maximum singular value of F−1A, which satisfies

Moreover, the minimum eigenvalue of is the square of the minimum singular value of F−1A, which satisfies

.

Therefore, .

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According to Theorem 3.1, if the condition number of is acceptable, the efficiency of the PMHSS preconditioned CGNR method can be asserted. In particular, when A is normal, there is a nonsingular matrix satisfying . The eigenvalues of (F−1A)TF−1A are located in . Subsequently, we have . In every iterative step, there are two linear sub-systems with to be solved. The computing cost is nearly twice as much as that of PMHSS preconditioned MINRES. It is expected to be more efficient than other PMHSS preconditioned methods if the iteration counters are smaller

4. Experimental examples

Herein, two linear systems (1.1) from different applications are briefly introduced.

Problem 1. The following linear saddle point problem is considered:

, (4.1)

which arises in distributed control problems, where is the mass matrix, is the stiffness matrix, is the regularization parameter, and and are vectors containing the termscorrespondingtotheboundaryvaluesofthediscretesolutionandthedesiredstateintheoptimization problem [7,8,12]. Problem 2. The following complex linear system is considered:

,

Table 1

IT and CPU (in parentheses) of the methods for problem 1. h 2−4 2−5 2−6

PMHSS–GMRES 15 (2.31e−1) 15 (1.77e−1) 15 (5.75e−1)

β = 10−2 ABD-MINRES 13 (3.91e−2) 13 (7.30e−2) 13 (2.96e−1) PMHSS–MINRES 13 (1.27e−1) 13 (6.28e−2) 13 (3.28e−1)

PMHSS–CGNR 7 (2.69e−2) 7 (5.81e−2) 7 (2.49e−1)

PMHSS–GMRES 20 (4.14e−2) 22 (1.56e−1) 22 (6.68e−1)

β = 10−4 ABD-MINRES 17 (2.81e−2) 18 (9.86e−2) 19 (4.33e−1) PMHSS–MINRES 22 (3.15e−2) 22 (1.00e−1) 23 (4.94e−1)

PMHSS–CGNR 9 (1.90e−2) 9 (7.00e−2) 10 (3.53e−1)

PMHSS–GMRES 20 (3.40e−2) 18 (1.04e−1) 20 (6.07e−1)

β = 10−6 ABD-MINRES 16 (2.66e−2) 17 (9.49e−2) 17 (3.87e−1) PMHSS–MINRES 23 (3.31e−2) 23 (1.03e−1) 25 (5.35e−1)

PMHSS–CGNR 9 (1.90e−2) 9 (7.48e−2) 9 (3.19e−1)

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where ω, τ, and ν are positive parameters, and with being the mass matrix and

the discrete negative Laplace operator. The right-hand side vector h is given by . This complex system

appears in the time horizon of the discretized form of the following parabolic partial differential equation

, , .

5. Numerical results

In this section, the numerical behavior of PMHSS–GMRES, PMHSS–CGNR, ABD-MINRES, and PMHSS– MINRES is compared in terms of the number of iteration steps (denoted by IT) and CPU time in seconds (denoted by CPU). All tests are performed in Mat lab 8.0.0 (64 bit). In the implementations, all iteration processes are terminated once the relative residuals satisfy

, where the initial vector x0 = 0. The subsystems with W + T are solved directly. In Table 1, the number of iteration steps and CPU time for Problem 1are listed with respect to the differential mesh size h and the regularization parameter β. It is shown that all methods are independent of not only the mesh size but also the parameter β. The CGNR method is the most efficient with respect to both iteration counters and CPU time. In Table 2, the number of iteration steps and CPU time for solving Problem 2are listed for different values of ω and m. Compared with other PMHSS preconditioned Krylov subspace methods, CGNR uses nearly half the number of iteration steps to obtain a satisfactory solution. Its CPU time is less than that of the other methods with respect to most values of ω and m.

6. Conclusions

The CGNR method was presented for the linear system of equations (1.1). Owing to the clustering eigenvalue distribution, the normal equation is well conditioned. Theorem 3.1provides an upper bound for the condition number, as it is not easy to determine the infimum of the condition number of the eigenvector matrices. Compared with other methods, CGNR requires an efficient inner solver for W+T more than the other methods, as there are twice as many sublinear systems to be solved in each iterative step.

Table 2 IT and CPU (in parentheses) of the methods for Problem 2.

Method ω m × m 900 × 900 1764 × 1764 1936 × 1936 2025 × 2025 PMHSS–GMRES 1 9 (1.17e−1) 9 (7.10e−2) 9 (7.46e−2) 9 (8.02e−2)

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102 18 (7.03e−2) 18 (1.34e−1) 18 (1.47e−1) 18 (1.55e−1) 104 17 (6.56e−2) 17 (1.26e−1) 17 (1.44e−1) 17 (1.47e−1) ABD-MINRES 1 9 (4.34e−2) 7 (4.64e−2) 7 (5.09e−2) 7 (5.39e−2) 102 17 (6.02e−2) 17 (1.13e−1) 17 (1.25e−1) 17 (1.32e−1) 104 16 (5.84e−2) 17 (1.13e−1) 17 (1.25e−1) 17 (1.32e−1) PMHSS–MINRES 1 8 (8.44e−2) 7 (8.30e−2) 7 (9.14e−2) 7 (9.27e−2) 102 17 (5.98e−2) 17 (1.12e−1) 17 (1.26e−1) 17 (1.32e−1) 104 17 (6.00e−2) 17 (1.13e−1) 17 (1.24e−1) 17 (1.32e−1) PMHSS–CGNR 1 5 (3.71e−2) 5 (6.16e−2) 5 (6.75e−2) 5 (7.21e−2) 102 10 (5.95e−2) 10 (1.23e−1) 10 (1.36e−1) 10 (1.45e−1) 104 9 (5.38e−2) 9 (1.11e−1) 9 (1.22e−1) 9 (1.31e−1) References [1] M. Benzi, G. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005) 1–137. [2] J. Lions, Optimal Control of Systems Governed By Partial Differential Equations, Springer-Verlag, Berlin and Heidelberg, 1968. [3] J. Cahn, J. Hilliard, Free energy of a nonuniform system. i. interfacial free energy, J. Chem. Phys. 23 (1958) 258–267. [4] O. Axelsson, M. Neytcheva, B. Ahmad, A comparision of iterative methods to solve complex valued linear algebraic systems, Numer. Algorithms 66 (2014) 811–841. [5] W. Zulehner, Efficient Solvers for Saddle Point Problems with Applications to PDE–Constrained Optimization, Springer, 2013, pp. 197–216. [6] J. Pearson, A. Wathen, A new approximation of the schur complement in preconditioners for PDE- constrained optimization, Numer. Linear Algebra Appl. 19 (2012) 816–829. [7] Z.-Z. Bai, F. Chen, Z.-Q. Wang, Additive block diagonal preconditioning for block two-by-two linear systems of skewhamiltonian coefficient matrices, Numer. Algorithms 62 (2013) 655–675. [8] Z.-Z. Bai, M. Benzi, F. Chen, Z.-Q. Wang, Preconditioned MHSS iteration methods for a class of block two- by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal. 33 (2013) 343– 369. [9] Z.-Z. Bai, On preconditioned iteration methods for complex linear systems, J. Engrg. Math. 93 (2015) 41– 60. [10] Z.-R. Ren, Y. Cao, L.-L. Zhang, On preconditioned MHSS real-valued iteration methods for a class of complex symmetric indefinite linear systems, East Asian J. Appl. Math. 6 (2016) 192–210. [11] Y. Saad, Iterative Methods for Sparse Linear Systems, second ed., SIAM, Philadelphia, 2003. [12] T. Rees, H. Dollar, A. Wathen, Optimal solvers for PDE-constrained optimization, SIAM J. Sci. Comput. 32 (2010) 271–298.

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