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A new iterative method for solving a class of two-by-two block complex linear systems Davod Khojasteh Salkuyeh∗ Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran Center of Excellence for Mathematical Modelling, Optimization and Combinational Computing (MMOCC), University of Guilan, Rasht, Iran

Abstract. We present an iterative method for solving the system arisen from finite element discretization of a distributed optimal control problem with time-periodic parabolic equations. We prove that the method is unconditionally convergent. Numerical results are presented to demonstrate the efficiency of the proposed method. Keywords: Iterative, finite element, PDE-constrained, optimization, convergence. AMS Subject Classification:

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49M25, 49K20, 65F10, 65F50.

Introduction

Consider the distributed control problems of the form: (see [6]) Z Z Z Z ν T 1 T 2 |y(x, t) − yd (x, t)| dxdt + |u(x, t)|2 dxdt, min y,u 2 0 Ω 2 0 Ω ∂ y(x, t) − ∆y(x, t) = u(x, t) in QT , s.t. ∂t y(x, t) = 0 on ΣT , y(x, 0) = y(x, T ) on ∂Ω, u(x, 0) = u(x, T ) in Ω, where Ω is an open and bounded domain in Rd (d ∈ {1, 2, 3}) and its boundary ∂Ω is Lipschitz-continuous. We introduce the space-time cylinder QT = Ω × (0, T ) and its lateral surface ΣT = ∂Q × (0, T ). Here, ν is a regularization parameter, yd (x, t) is a desired state and T > 0. We may assume that yd (x, t) is time-harmonic, i.e., yd (x, t) = yd (x)eiωt , with ∗

Corresponding author. Emails: [email protected]

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D.K. Salkuyeh

ω = 2πk/T for some k ∈ Z. Substituting yd (x, t) in the problem and using finite element discretization of the problem method we get the following system of linear equations      M 0 K − iωM y¯ M y¯d  0 νM −M  u ¯ =  0  , (1) K + iωM −M 0 p¯ 0

where M ∈ Rm×m and K ∈ Rm×m are the mass and stiffness matrices, respectively. Both of the matrices M and K are symmetric symmetric positive definite (SPD). From the second ¯ = p¯/ν and by substituting u ¯ in the third equation, we obtain the equation in (1) we have, u following system  M y¯ + (K − iωM )¯ p = M y¯d , 1 (K + iωM )¯ y − ν M p¯ = 0,

which is itself equivalent to      √ M y¯ yˆd ν(K − iωM ) Ax = √ = = b, (2) ν(K + iωM ) −M q¯ 0 √ where q¯ = p¯/ ν and yˆd = M y¯d . In [2], Krendl proposed the real block diagonal and the alternative indefinite preconditioners for the system (2). Zheng et al. in [6] proposed the block alternating splitting (BAS) iteration method which can be summarized as ( 1 (αV + H1 )xk+ 2 = (αV − S1 )xk + P1 b, 1 (αV + H2 )xk+1 = (αV − S2 )xk+ 2 + P2 b, √ √ where α > 0, V = blkdiag(M, M ), H1 = blkdiag(M, −M ), H2 = blkdiag( νK, νK),  √   √ 1 νK −M i νωM −iωνK √ √ S1 = and S2 = . M −i νωM 1 + ω 2 ν − νK iωνK Numerical results presented in [6] show that the BAS iteration method outperform the GMRES method [4]. They also showed that the parameter α = 1 + νω 2 often gives quite suitable results. Therefore, if νω 2 ≪ 1, then α = 1 is a good choice. In this paper we present a new iterative method for the system (2) and investigate its convergence properties.

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The new iterative method

Using the idea of [5], the system (2) can be written in the 4-by-4 block real system      √ √ νK ω νM M 0 ℜ(¯ y) ℜ(ˆ yd ) √ √    0 M −ω νM νK  y)  yd )  ˆ   ℑ(¯  =  ℑ(ˆ  ≡ b. √ √ Ax ≡       νK −ω νM −M 0 ℜ(¯ q) 0  √ √ ω νM νK 0 −M ℑ(¯ q) 0

(3)

A new iterative method for solving a class of two-by-two block complex linear systems We define the matrices G1 and G2 as following   √ I 0 0 ω νI √  0 I −ω νI 0  , √ G1 =   0 −I 0  −ω νI √ ω νI 0 0 −I



0 p  0 G2 = 1 + νω 2   I 0

where I ∈ Rm×m is the identity matrix. Then, the system (3) as √ ν ˆ ˆ = b. G2 K)x (G1 M + √ 1 + νω 2 where    M 0 0 0 K  0 M 0 0   0  ˆ  M=  0 0 M 0  and K =  0 0 0 0 M 0



0 ωνI  1 0  −ωνI √ G := G1−1 G2 = p  2 − νI 0 ν(1 + νω ) √ 0 − νI

I 0 0 0

 0 I  , 0  0

can be equivalently rewritten (4)  0 0 0 K 0 0  . 0 K 0  0 0 K

It is easy to see that G1 is nonsingular and  √ I 0 0 ω νI √  0 1 0 I −ω νI  √ G1−1 =  2 0 −ω νI −I 0 1 + νω √ 0 0 −I ω νI

and

0 0 0 I

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

 , 

√

 νI 0 √ νI  0 . 0 −ων  ων 0

Moreover, we have G 2 = −I with I being the identity matrix of order 4m, G −1 = −G and G T = −G. Premultiplying both sides of the system (4), gives the system Bx = (M + GK)x = b,

(5)

(αI + M)x = (αI − GK)x + b.

(6)

√ ˆ √ ˆ where K = ν K/ 1 + νω 2 and b = G1−1 b. Given α > 0, we rewrite Eq. (6) as

We also rewrite (6) as G(αI + K)x = (αG − M)x + b.

Premultiplying both sides of this equation by G −1 and having in mind that G −1 = −G, gives (αI + K)x = (αI + GM)x − Gb.

(7)

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D.K. Salkuyeh

Now, using Eqs. (6) and (7) we establish the Alternating SPD and Scaled symmetric positive semidefinite splitting (ASSS) method for solving the system (6) as ( 1 (αI + M)x(k+ 2 ) = (αI − GK)x(k) + b, (8) 1 (αI + K)x(k+1) = (αI + GM)x(k+ 2 ) − Gb, 1

where x(0) is an initial guess. Eliminating x(k+ 2 ) from Eq. (8) yields the following stationary iterative method x(k+1) = Tα x(k) + f , (9)

where

Tα = (αI + K)−1 (αI + GM)(αI + M )−1 (αI − GK),