IN this paper, we investigate spectral properties of block

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1 On the Eigenvalues Distribution of Preconditioned Block wo-by-two Matrix Mu-Zheng Zhu and a-e Qi Abstract he spectral properties of a class of block matrix are studied, which arise in the numercial solutions of PDE-constrained optimization problems. Based on the Schur complement approximate approach and inexact Uzawa preconditioner, the eigenvalues distribution of preconditioned matrix is discussed by the similarity transformation. Moreover, he numerical experiments originated in PDE-constrained optimization problem are presented to show that the theoretical bound of the eigenvalues is in good agreement with its practical bound. ndex erms block two-by-two linear systems, schur complement approach, inexact Uzawa preconditioner, eigenvalues distribution, PDE-constrained optimization problems.. NRODUCON N this paper, we investigate spectral properties of block matrix in following linear systems: x x y f g b, () where, n n are symmetric positive semi-definite (SPSD) and one of them is symmetric positive definite (SPD). ithout loss of generality, we assume is SPD. he matrix is nonsingular if and only if null( ) null( ) {} []. hus, the linear system () has a unique solution when the (, )-block in matrix is nonsingular. he linear system () can be formally regard as a special case of the saddle point problems [5] [7]. hey frequently arise from finite element discretizations of PDEconstrained optimization problems [7], [], [3], [3], complex symmetric linear systems [], [8], [9], [], finite element discretizations of first-order linearization of the two-phase flow problems based on Cahn-Hilliard equation [4], [6], matrix completions problems [8], and so on [6], [], [4], [6]. A large variety of applications and numerical solution methods of linear system () have been comprehensively reviewed by Benzi, Golub, and Liesen []. n recent years, the eigenvalue distribution of block linear systems has been deeply studied [], [3], [4], [], [9]. On the one hand, the bounds for eigenvalues of in () can be used to analyze the spectral properties Manuscript received January, 6; his work was supported by the Research Foundation of the Higher Education nstitutions of Gansu Province with code 4A- and 4B-3 and the project of the Key Laboratory of Hexi Corridor of characteristic resources utilization in Gansu Province with code XZ4. Mu-Zheng Zhu is with the School of Mathematics and Statistics, Hexi University, Zhangye, Gansu, 734 P. R. China; zhumzh7@yahoo.com. a-e Qi is corresponding author with the College of Chemistry and Chemical Engineering, Hexi University, Zhangye, Gansu, 734 P. R. China. of preconditioners such as symmetric indefinite preconditioners, inexact constraint preconditioners and primalbased penalty preconditioners for linear systems (); see [3] [5], [], [8]. On the other hand, the estimates for eigenvalues of in () can give theoretical basis for the CG method solving linear system () in a nonstandard inner product; see [], [7], [8]. Our focus is on an important block lower triangular preconditioner, called inexact Uzawa preconditioner, which exploit the knowledge of a good approximation for the (negative) Schur complement. he aim of this paper is to investigate the spectral properties and provide the eigenvalues distribution of preconditioned matrix. he remainder of this paper is organized as follows. n Section, an new Schur approximation with parameter is presented and the approximate degree is studied. n Section, an inexact Uzawa preconditioner is introduced and the eigenvalues distribution of preconditioned matrix are analyzed by the use of a similarity transformation. n Section V, the numerical experiments are given to show that the theoretical bound of eigenvalues distribution is in good agreement with the practical bound though PDEconstrained optimization problems. Finally, in Section V we end this paper with some conclusions.. SCHUR COMPLEMEN APPROXMAE N this section, we consider the approximate degree between the Schur complement S : + of the matrix and its approximate S(α) ( + α ) ( + α ), where Re (α) > and αα. As is symmetric positive semi-definite, then an orthogonal matrix Q n n and a diagonal matrix Σ (σ i i ) n n, σ i i are exist to such that Q ΣQ, so the following expression is true: S(α) ( + α ) ( + α ) ( + Re (α)q ΣQ +Q Σ Q) (Q ) ( + Re (α)σ + Σ ) (Q ). t is obvious that S S(i), where i is the imaginary unit. hus, hen S Q ( + Σ )Q. S(α) S (Q ) H(Q ), ()

2 where H ( + Re (α)σ + Σ ) ( + Σ ) is a diagonal matrix and its diagonal elements As V, we have h i i + Re (α)σ i i + σ, i i and the equality hold up if and only if Re (α). t can be found from () that S(α) S and H have the same eigenvalues. According we easily have h i i + Re (α)σ i i + σ i i + Re (α) ( i n), + Re (α)σ i i λ(s(α) S) + Re (α),. (3) Similarly, an approximation can be found for, and λ( ) [µ, µ], where µ >, µ <.. PRECONDONER AND EGENVALUES DSRBUON HERE are many indefinite" preconditioner with block form are presented, whose indefiniteness is tailored to compensate for the indefiniteness of systems matrix, and in this sense that the preconditioned matrix has only eigenvalues with positive real part. he eigenvalue distribution for this type preconditioners with Schur complement are also widely discussed. heses include indefinite block diagonal preconditioners [3], [9], block triangular preconditioners [3], for inexact Uzawa preconditioners [7], and block approximate factorization preconditioners [3]. Our focus is on the inexact Uzawa preconditioner n M S(α), (4) S(α) which is considered in [4]. he eigenvalues distribution of the preconditioned matrix M are discussed in this section. hen and S(α) S, it is well known that the preconditioner (4) is such that the preconditioned matrix has all eigenvalues equal to and minimal polynomial of degree at most []. However, using these ideal" preconditioners requires exact solves with and S, which is often impractical, just the computation of S can be prohibitive [4]. Here we investigate the effect of using approximations S(α) instead Schur complement S. e analyze how the eigenvalue distributions are affected by providing bounds, where bounds for non-, have to be understood as combinations of inequalities proving their clustering in a confined region of the complex plane. Assume n n is an orthogonal matrix, V < (V diag(v, v,, v n ), v v v n < ) is a diagonal matrix such that V. hen M n S(α) n S(α) ( ) ( +. ) n S(α) S(α) S(α) S(α) ( ) ( + ) S(α) V S(α) ( ) S(α) ( + ) V S(α) G G ( V ) ( C +G VG : J, ) where G S(α), C S(α) S(α). t is evident that C +G VG S(α) SS(α), i.e., λ( C +G VG ) + Re (α),. According the assumption V <, V and ( V ) exist. aking a similar transformation with respect to blkdiag(( V ) V, ) on J, we have ( V ) V V ( V ) J V (V V ) G G (V V ) ( C +G VG ) A B : K B C where A V and C C +G VG are all SPD, B G (Σ Σ ). Such matrices are nonnegative definite in n. Hence, their eigenvalues have positive real part [4]. hus, if the preconditioned matrix is similar to a matrix of the above form (5), the indefiniteness of the original matrix () is lost. However, we must note that this is at the expense of the loss of the symmetry, meaning that a portion of the eigenvalues will be in general complex. According the above discuss, M is similar to the matrix of the form K, thus the eigenvalue analysis of the preconditioned matrix M can be reduced to that of the matrix of the form K. Nextly the eigenvalues distribution of the preconditioned matrix is discussed and the main results is given. Lemma. ( [4]) Define A K B where A n n is SPD, C m m is positive semi-definite, m n. Assume B has full rank or C is positive definite in null space of B. Let S C C +BA B, and S A A+B C B when C is positive definite, then the λ of K satisfies the following condition: min λ min (A), λ min (S C ) B C, λ max λ max (A), λ max (C ). he non- of K satisfies the following condition: λ min (A) + λ min (C ) Re (λ) λ max (A) + λ max (C ) m (λ) λ max (B B ), and λ ξ ξ, where λ max (S A ) λ max (S C ) ξ λ max (S A ) + λ max (S C ), λ max (S C ), C is positive definite, otherwise. (5)

3 heorem. he λ of the inexact Uzawa preconditioned matrix M satisfy min v, λ. + Re (α) he non- λ of the inexact Uzawa preconditioned matrix M satisfy v + Re (λ) vn +, + Re (α) m (λ) σ n + Re (α)σ n + σ n and λ., Here, v, v n is the minimum and maximum diagonal element of matrix V mentioned above respectively, σ n is the maximum diagonal element of matrix Σ mentioned above. Proof. As A and C are all positive definite matrices, the condition in Lemma is satisfied. Now we consider σ max (B) λ max (B B ). As B G (V V ), we have λ max (B B ) λ max G (V V )G 4 λ max(gg ) 4 σ n + Re (α)σ n + σ n hen we have m (λ) σ n + Re (α)σ n + σ n.. Because A V <, >, i.e., >. hus the following expression is true: C S(α) + S(α) S(α) SS(α) + Re (α). As C C +G VG < C +GG <, we have + Re (α) λ <. Next the eigenvalues distribution of S C, S A is discussed. o analyze the Schur complement S C, one firstly has to obtain it explicitly. One way is to consider the Schur complement, S C C + BA B C +G VG +G (V V )V G C +GG, then λ(s C ) λ S(α) S holds. As seen in formula (3), we have λ(s C ) + Re (α),. Because S A A + B C B V ( + ( V ) G ( C +G VG ) G ( V ) )V V RV, we obtain the following result by using the Sherman-Morrisson- oodbury formula (SM): G R ( V ) G C +G VG +G ( V )G ( V ) ( V ) G ( C +GG ) G ( V ). As G ( C + GG ) G and GG ( C + GG ) have the same set of nonzero eigenvalues and they are banded by max x x GG x x ( C +GG )x. One then finds R ( V ) ( V ) V, i.e., H V. Further, we have λ(s A ). hus, ξ λ max(s A )λ max (S C ) λ max (S A ) + λ max (S C ). ndeed, when is trivial, more special conclusion can be reached. Corollary 3. f then V, the matrix K C +GG has only, and the distribution of eigenvalues is λ(k ) + Re (α),. V. NUMERCAL RESULS N this section, it is illustrated by using numerical examples that the theoretical bound for eigenvalues of preconditioned matrix are agreement with its practical bound. All the tests are performed in MALAB R3a with machine precision 6. we consider the distributed control problem which consists of a cost functional (6) to be minimized subject to a partial differential equation (PDE) problem posed on a domain Ω [5], [3]: min u,f u u + β f, (6) subject to u f, in Ω [, ], (7) with u u, on Ω, (8) where the function (x ) (y ), (x, y ),, u, otherwise, β is a regularization parameter, Ω is the regions boundaries of Ω. Such problems is firstly introduced by Lions in [3]. here are two approaches to obtain the solution of the PDEconstrained optimization problems (6 8). he one is discretizethen-optimize and the other is optimize-then-discretize. By using the discretize-then-optimize approach and the Q finite element discretize in this paper, the following linear systems can be obtained: β M M f M K u b, M K λ d where M, K n n is the mass matrix and stiffness matrix (the discrete Laplacian) respectively. hey are all SPD matrices. d n is the terms coming from the boundary values, b n is the discrete Galerkin projection of u, λ is the Laplace operator vector. As M is a symmetric matrix, λ β f and the following linear systems of the form M β K µ z b, (9) β K M β f β d can be obtained. we note that this system of linear equations have saddle point structure. he eigenvalues distribution of the preconditioned matrix M with. and V < are listed in ables, he symbol " denotes that the case is not exist. From these ables, we can find that the practical eigenvalues just fall in the interval theoretical eigenvalues, the theoretical bound of image are good agreement with the practical bound of image. o further illustrate this, he eigenvalues distribution of the preconditioned matrix M is given in Fig.. V. CONCLUSONS

4 N this paper, our focus is directed at the spectral property of the block linear systems, which frequently arise from saddle point problems and PDE-constrained optimization problems. Based on the Schur complement approximate approach, an inexact Uzawa preconditioner is introduced and the eigenvalues distribution of preconditioned matrix is discussed by the similarity transformation. he results of our numerical experiments utilizing test matrices from PDE-constrained optimization problems demonstrate that the theoretical bound for eigenvalues of the preconditioned matrix is good agreement with its practical bound. ACKNOLEDGEMENS he author would like to thank the anonymous referees for his/her careful reading of the manuscript and useful comments and improvements. REFERENCES [] O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numerische Mathematik, vol. 48, no. 5, pp , 986. [] O. Axelsson and A. Kucherov, Real valued iterative methods for solving complex symmetric linear systems, Numerical Linear Algebra with Applications, vol. 7, no. 4, pp. 97 8,. [3] O. Axelsson and M. Neytcheva, Eigenvalue estimates for preconditioned saddle point matrices, Numerical Linear Algebra with Applications, vol. 3, no. 4, pp , 6. [4] V. E. Badalassi, H. D. Ceniceros and S. Banerjee, Computation of multiphase systems with phase field models, Journal of Computational Physics, vol. 9, no., pp , 3. [5] Z.-Z. Bai, Construction and analysis of structured preconditioners for block two-by-two matrices, Journal of Shanghai University (English Edition), vol. 8, no. 4, pp , 4. [6] Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Mathematics of Computation, vol. 75, no. 54, pp , 6. [7] Z.-Z. Bai, Block preconditioners for elliptic PDE-constrained optimization problems, Computing, vol. 9, no. 4, pp ,. [8] Z.-Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, vol. 87, no. 3 4, pp. 93,. [9] Z.-Z. Bai, M. Benzi and F. Chen. On preconditioned MHSS iteration methods for complex symmetric linear systems, Numerical Algorithms, vol. 56, no., 97 37,. [] Z.-Z. Bai, M. Benzi, F. Chen and Z.-Q. ang, Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, MA Journal of Numerical Analysis, vol. 33, no., , 3. [] M. Benzi and D. Bertaccini, Block preconditioning of real-valued iterative algorithms for complex linear systems, MA Journal of Numerical Analysis, vol. 8, no. 3, , 8. ABLE : Bound of Eigenvalues of M for., h 4. heoretical Practical heoretical real Practical real heoretical image Practical image /3 [.75, ] [.895,.9545] [.968,.9544] [-6, 6] [ ] /4 [.8, ] [.8545,.9545] [.957,.9544] [-6, 6] [-.98,.98] /5 [.8333, ] [.87,.9545] [.93,.9545] [-7, 7] [-.98,.98] /6 [.857, ] [.883,.9545] [.9349,.9545] [-73, 73] [-.98,.98] /3 [.75, ] [.895,.9545] [.86,.954] [-6, 6] [-.98,.98] /4 [.8, ] [.8545,.9545] [.88,.954] [-6, 6] [-.98,.98] /5 [.8333, ] [.87,.9545] [.894,.9543] [-7, 7] [-.98,.98] /6 [.857, ] [.883,.9545] [.93,.9543] [-73, 73] [-.98,.98] ABLE : Bound of Eigenvalues of M for., h 5. heoretical Practical heoretical real Practical real heoretical image Practical image /3 [.75, ] [.895,.9545] [.967,.9545] [-6, 6] [-.98,.98] /4 [.8, ] [.8545,.9545] [.956,.9545] [-6, 6] [-.98,.98] /5 [.8333, ] [.87,.9545] [.93,.9545] [-7, 7] [-.98,.98] /6 [.857, ] [.883,.9545] [.9349,.9545] [-73, 73] [-.98,.98] /3 [.75, ] [.895,.9545] [.86,.9544] [-6, 6] [-.98,.98] /4 [.8, ] [.8545,.9545] [.88,.9545] [-6, 6] [-.98,.98] /5 [.8333, ] [.87,.9545] [.8939,.9545] [-7, 7] [-.98,.98] /6 [.857, ] [.883,.9545] [.93,.9545] [-73, 73] [-.98,.98] ABLE : Bound of Eigenvalues of M for., h 6. heoretical Practical heoretical real Practical real heoretical image Practical image / [.6667, ] [.7879,.9545] [.8998,.9545] [-.887,.887] [-.98,.98] /3 [.75, ] [.895,.9545] [.967,.9545] [-6, 6] [-.98,.98] /4 [.8, ] [.8545,.9545] [.956,.9545] [-6, 6] [-.98,.98] /5 [.8333, ] [.87,.9545] [.93,.9545] [-7, 7] [-.98,.98] /6 [.857, ] [.883,.9545] [.9349,.9545] [-73, 73] [-.98,.98] / [.6667, ] [.7879,.9545] [.86,.9535] [-.887,.887] [-.98,.98] /3 [.75, ] [.895,.9545] [.86,.9545] [-6, 6] [-.98,.98] /4 [.8, ] [.8545,.9545] [.889,.9545] [-6, 6] [-.98,.98] /5 [.8333, ] [.87,.9545] [.8939,.9545] [-7, 7] [-.98,.98] /6 [.857, ] [.883,.9545] [.99,.9545] [-73, 73] [-.98,.98]

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