Preconditioners for reduced saddle point systems arising in elliptic PDE-constrained optimization problems
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1 Zeng et al. Journal of Inequalities and Applications :355 DOI 0.86/s x RESEARCH Open Access Preconditioners for reduced saddle point systems arising in elliptic PDE-constrained optimization problems Yuping Zeng*, Siqing Wang, Hongru Xu and Shuilian Xie * Correspondence: zeng_yuping@63.com School of Mathematics, Jiaying University, Meizhou, 5405, China Abstract In this paper, we propose some preconditioning techniques for reduced saddle point systems arising from linear elliptic distributed optimal control problems. The eigenvalues of preconditioned matrices are analyzed. Moreover, the bounds of these eigenvalues with respect to the mesh size h are also obtained. Some numerical tests are presented to validate the theoretical analysis. MSC: 65F0; 65F50 Keywords: PDE-constrained optimization; saddle point problem; preconditioning; eigenvalue estimate Introduction We consider the preconditioning techniques for solving the saddle point system arising from linear elliptic distributed optimal control problems: min u u L + β f L u,f subject to u = f u=g in, on, where R is a simply connected polygonal domain with a connected boundary. The data u is the target function, and the parameter β is the Tikhonov regular parameter. u is the state variable, and f is called the control variable. Such problems are simple models, which were originally introduced by Lions in [ ]. Some more complex formulations, which include control constraints or state constraints, can be found in [ ]. For the elliptic PDE-constrained optimization problem, we take the method based on discretize-then-optimize [ ] approach. More concretely, we use the P conforming finite elements for the approximations of the state variable u and the control variable f, which yields the following corresponding minimization problem [, ]: min ut Mu ut b + α + βf T Mf u,f subject to Ku = Mf + d, 205 Zeng et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Zeng et al. Journal of Inequalities and Applications :355 Page 2 of 4 where K R m m is the stiffness matrix, M R m m is the mass matrix. Both M and K are symmetric and positive definite. b R m is the discrete Galerkin projection of the target function û. α = 2 û 2 L 2,andd Rm contains the terms arising from the boundaries of finite element approximation of the state u. Then, using the Lagrange multiplier technique for the minimization problem 2, we can find that f, u and λ satisfy the following linear saddle point system [5]: 2βM 0 M f 0 Ax = 0 M K u = b = g, 3 M K 0 λ d where λ is a vector of Lagrange multipliers. To obtain the above system, we have used the fact that K = K T. Obviously, from the first equation of 3, we can obtain that f = λ. 4 2β Substituting 4 into3, we can easily obtain the following reduced system: Ay = M K u K 2β M λ = = g. 5 b d This paper is devoted to constructing efficient preconditioners for the reduced linear saddle point system 5. Comparing with 3, 5 is a smaller linear system so that it can help us to reduce computational cost. As we know, previous work has been mainly devoted to the development of efficient solution techniques for the original linear saddle point system 3, such as the block diagonal preconditioner and the constraint preconditioner [5], block triangular diagonal preconditioner [7], block-counter-diagonal and block-countertridiagonal preconditioning techniques [6]. Other recent work in this direction can also be found in [8 ]. For the reduced saddle point problem 5, there have existed several preconditioning algorithms, we refer the reader to [2 8]for details. In our work,we extend similar preconditioning techniques, which were used in [5, 6] for system3, to the saddle point system 5. We also refer the reader to [9] for analogous discussions, where the optimal control of Stokes equations was addressed. The major contribution of our work is to establish the bounds of eigenvalues of the preconditioned matrices with respect to the discretization mesh size h, though we also provide the analysis of eigenvalues of the preconditioned matrices using the usual linear algebraic techniques developed in [5, 6, 9]. In particular, for the case that the Tikhonov parameter β is not very small, the bounds of eigenvalues of preconditioned matrices are proved to remain bounded away from 0 as h 0 with a bound independent of h. This suggests that they are optimal preconditioners in the sense that their performances are independent of the mesh size h. The rest of our paper is organized as follows. In Section 2, we provide preconditioning techniques for the situation that the Tikhonov parameter β is not very small, then we discuss the corresponding spectral analysis. In Section 3, we propose preconditioning techniques for the case that the Tikhonov parameter β is sufficiently small, and then the corresponding spectral analysis is presented. In Section 4, some numerical results are presented to demonstrate our theoretical analysis.
3 Zeng et al. Journal of Inequalities and Applications :355 Page 3 of 4 2 Preconditioners for the Tikhonov parameter β not very small When the Tikhonov parameter β is not very small, we may provide block-counter diagonal preconditioner P bcd and block-counter-triangular diagonal preconditioner P bctd as the following form: 0 K P bcd = K 0 6 and M K P bctd =. 7 K 0 For the block-counter diagonal preconditioner P bcd, we have the following results. Theorem 2. Let A R 2m 2m be the coefficient matrix of the linear system 5 and P bcd R 2m 2m be defined in 6. Denote by ν l the eigenvalue of K M R m m, where ν l >0,l =,2,...,m, and by x l C m the corresponding eigenvector for l =,2,...,m. Then i the eigenvalues of the preconditioned matrix P bcd A are λ l k = 2β ν le 2k+πi 2, k =0,,l =,2,...,m, where i denotes the imaginary unit; ii the eigenvectors of the preconditioned matrix P bcd A areoftheform x l 2β e 2k+πi 2 K Mx l μ l Proof i By a direct calculation, we get, l =,2,...,m, k =0,. P 2 A = 0 K M K I K 0 K 2β 2β M = K M. 8 K M I To proceed, let 0 E = I P 2 A = 2β K M 0 2β = D, 9 K M 0 D 0 where D = K M. Therefore, E 2 = 2β D β D2 Then from 0 we can know that if ν l is the eigenvalue of K M, then the eigenvalues of E are 2β ν le 2k+πi 2, k = 0,. Thus, the eigenvalues of the preconditioned matrix P bcd A are 2β ν le 2k+πi 2, k =0,.Thisyieldsi.
4 Zeng et al. Journal of Inequalities and Applications :355 Page 4 of 4 ii By similar techniques used in the proof of ii in Theorem 3., we can obtain the corresponding conclusions. For the block-counter-triangular diagonal preconditioner P bctd,wehavethefollowing results. Theorem 2.2 Let A R 2m 2m be the coefficient matrix of the linear system 5 and P bctd R 2m 2m be defined in 7. Denote by ν l the eigenvalue of K M R m m, where ν l >0,l =,2,...,m, and by y l C m the eigenvector of K M 2 for l =,2,...,m. Then i the eigenvalues of the preconditioned matrix P bctd A are with algebraic multiplicity m, and the remaining eigenvalues equal 2β ν2 l +, l =,2,...,m; ii the eigenvectors of the preconditioned matrix P bctd A areoftheform x, x C m \{0}, and ν K My l, l =,2,...,m. 0 Proof i A straightforward calculation shows that bctd A = 0 K K K MK P y l M K K 2β M = I 2β K M 0 I + 2β K. MK M It follows from the above equality that the eigenvalues of the preconditioned matrix P bctd A are with algebraic multiplicity m, andm eigenvalues with the form + 2β ν2,whereν is any eigenvalue of the matrix K M. This demonstrates the conclusions of i. ii Let {λ,x, y T } be an eigenpair for the matrix P bctd A, then from the expression of P bctd A in, we have I 2β K M x 0 I + 2β K MK M y = = λe, or the above equation can be rewritten as x 2β K My=λx, I + 2β K MK My = λy. x y It is easy to see that if λ =,theny = 0, thus the corresponding eigenvectors take the form x e =, with x C m \{0}. 2 0 If λ,wehave and x = 2β λ K My 3 I +2βM KM K y = λy. 4
5 Zeng et al. Journal of Inequalities and Applications :355 Page 5 of 4 Thus, λ = 2β ν2 +,withν an eigenvalue of the matrix K M. Direct computations lead to x = ν K My. 5 Obviously, y C m is an eigenvector of the matrix K M 2. This completes the proof. Furthermore, using the theory of finite element methods, we can use the results stated in the above two theorems to obtain more concrete bounds that involved the discretization mesh size h. Since in our numerical tests we use the P conforming finite elements to discretize problem, we recall the following lemma from [5, 20] which will be used frequently in our theory analysis. Lemma 2.3 Let T h = {T i : i =,2,...,E h } be a family of quasi-uniform and shape regularity triangulations of the domain R 2. Denote by h Ti the diameter of the element T i and h = max Ti T h h Ti. For the P conforming finite element, the mass matrix M and the stiffness matrix K satisfy c h 2 xt Mx x T x c 2h 2, 6 d h 2 xt Kx x T x d 2, 7 x 0 R m. Constants c, c 2, d and d 2 are positive and independent of h and β. With the above prepared techniques, we can obtain the following two corollaries. Corollary 2.4 Let λ be an eigenvalue of P bcd A, with P bcd and A being defined in 6 and 5. Then the following bound holds: + c2 h4 2βd2 2 λ + c2 2 2βd 2. 8 Proof From i in Theorem 2., the eigenvalues of P bcd A are λ l k = A direct calculation shows that λ l k 2β ν le 2k+πi 2, k =0,,l =,2,...,m. 2 =+ 2β ν2 l 2 2k +π cos =+ 2β 2 2β ν2 l. 9 In the last step of the above equation we have used the fact that cos 2k+π 2 =0. Thus, from the above equality, we know that if we want to bound λ l k,itissufficientto bound ν l. Recalling that ν l is an eigenvalue of K M,weget K Mx = ν l x, i.e., Mx = ν l Kx,
6 Zeng et al. Journal of Inequalities and Applications :355 Page 6 of 4 thus x T Mx = ν l x T Kx, ν l = xt Mx x T Kx. 20 From 7 in Lemma 2.3,we obtain d 2 xt x x T Kx d h 2. 2 Thus, from 6, 20and2weget c h 2 ν l c d 2 d Then the corollary follows by 9and22. Corollary 2.5 Let λ be an eigenvalue of P bctd A, with P bctd and A defined in 7 and 5. Then the eigenvalues of the preconditioned matrix P bctd A are with algebraic multiplicity m, the remaining m eigenvalues can be bounded as follows: + c2 h4 2βd2 2 λ + c2 2 2βd Proof From i in Theorem 2.2, using the same proof procedure as in Corollary 2.4, we can obtain the corresponding conclusion. Remark 2.6 From 8and23 in Corollaries2.4 and 2.5, we can see that the eigenvalues of the preconditioned matrices P bcd A and P bctd A satisfy λ + c2 2 2βd 2 as h Furthermore, if β is not too small, then from 24 we know that the eigenvalues of the preconditioned matrices P bcd A and P bctda are clustered. When the parameter β is not too small, 24alsomeansthatP bcd and P btcd may perform as optimal preconditioners in the sense that their performances are independent of the mesh size h. Remark 2.7 In practical computations, we may choose good approximations K and M for K and M, respectively. Then the corresponding preconditioners P bcd and P bctd are replaced by P bcd = 0 K K 0 and P bctd = M K, K 0 respectively.
7 Zeng et al. Journal of Inequalities and Applications :355 Page 7 of 4 3 Preconditioners for the Tikhonov parameter β sufficiently small In this section, we discuss block diagonal and block-triangular diagonal preconditioning techniques for the linear system 5 and analyze the eigenvalues and eigenvectors of the preconditioned matrices. The block diagonal preconditioner is given by P bd = M 0 0 2β M, 25 and the block-triangular diagonal preconditioner is of the form P btd = M K 0 2β M. 26 Moreover, we find that both the preconditioned matrices P bd A and P btda have eigenvalues around when the Tikhonov parameter β is sufficiently small i.e., β. These properties are stated in the following two theorems. The spectral analysis for preconditioned matrices is largely based on the references [5, 9]. First, we consider the block diagonal preconditioner. Theorem 3. Let A R 2m 2m be the coefficient matrix of the linear system 5 and P bd R 2m 2m be defined in 25. Set μ l the eigenvalue M K R m m, where μ l >0,l =,2,...,m, and x l C m the corresponding eigenvector for l =,2,...,m. Then i the eigenvalues of the preconditioned matrix P bd A are λ l k = 2βμ l e 2k+πi 2, k =0,,l =,2,...,m, where i denotes the imaginary unit; ii the eigenvectors of the preconditioned matrix P bd A areoftheform M Kx l 2βμl e 2k+πi 2, l =,2,...,m, k =0,. x l Proof i A simple calculation shows that P bd A = M 0 M K I 0 2βM K 2β M = 2βM K M K. 27 I To proceed, let C = I P bd A = 0 M K 0 B =, 28 2βM K 0 2βB 0 where B = M K. Therefore, C 2 2βB 2 0 = βB 2
8 Zeng et al. Journal of Inequalities and Applications :355 Page 8 of 4 From 29 we can know that if μ l is the eigenvalue of M K, then the eigenvalues of C are 2βμ l e 2k+πi 2, k = 0,. Thus, the eigenvalues of the preconditioned matrix P bd A are 2βμ l e 2k+πi 2, k = 0,. This leads to i. ii Let μ l be the eigenvalue of M K R m m and x l C m be the corresponding eigenvector for l =,2,...,m,then M KM Kx l = μ 2 l xl. Multiplying both sides of the above equation by 2β yields 2βM KM Kx l = 2βμ 2 l xl = 2βμ l e 2k+πi 2 2x l, i.e., 2βM K 2βμl e 2k+πi 2 M Kx l = 2βμ l e 2k+πi 2 x l. 30 Let y = M Kx l, 2βμl e 2k+πi 2 then M Kx l = 2βμ l e 2k+πi 2 y. 3 From the expressions of 30and3weobtain Bx l = 2βμ l e 2k+πi 2 y, 2βBy = 2βμ l e 2k+πi 2 x l. Thus, y 0 B y C = x l 2βB 0 x l = 2βμ l e 2k+πi y 2. x l Therefore, if x l C m is an eigenvector of the matrix B,then y x is the eigenvector of the l matrix C R 2m 2m.Thisfinishestheproof. Then we turn our attention to the block-triangular diagonal preconditioner. Theorem 3.2 Let A R 2m 2m be the coefficient matrix of the linear system 5 and P btd R 2m 2m be defined in 26. Set μ l theeigenvalueofm K R m m, where μ l >0, l =,2,...,m, and x l C m the eigenvector of the matrix M K 2 for l =,2,...,m. Then i the eigenvalues of the preconditioned matrix P btd A are with algebraic multiplicity m, and the remaining eigenvalues are equal to 2βμ 2 l +, l =,2,...,m;
9 Zeng et al. Journal of Inequalities and Applications :355 Page 9 of 4 ii the eigenvectors of the preconditioned matrix P btd A areoftheform 0, y C m \{0}, and y Proof i By a direct calculation we get that x l μ M Kx l, l =,2,...,m. P btd A = M 2βM KM M K I +2βM KM K 0 0 2βM K 2β M =. 32 2βM K I It follows from the above equality that the eigenvalues of the preconditioned matrix P btd A are with algebraic multiplicity m,andmeigenvalues with the form +2βμ2,where μ is any eigenvalue of the matrix M K. This demonstrates the conclusions of i. ii Let {λ,x, y T } be an eigenpair for the matrix P btd A, then from the expression of P btd A in 32, we have I +2βM KM K 0 2βM K I x y = λ = λe, or the above equation can be rewritten as I +2βM KM Kx = λx, 2βM Kx + y = λy. x y It is easy to see that if λ =,thenx = 0, thus the corresponding eigenvectors take the form 0 e =, with y C m \{0}. 33 y If λ,wehave and y = 2β λ M Kx 34 I +2βM KM K x = λx. 35 Thus, λ =2βμ 2 +,withμ an eigenvalue of the matrix M K. A direct calculation shows that y = μ M Kx. 36 Obviously, x C m is an eigenvector of the matrix M K 2. This completes the proof. Furthermore, we provide more concrete bounds which involved the discretization mesh size h. The results are stated in the following corollaries.
10 Zeng et al. Journal of Inequalities and Applications :355 Page 0 of 4 Corollary 3.3 Let λ be an eigenvalue of P bd A, with P bd and A defined in 25 and 5. Then the following bound holds: + 2βd2 c 2 2 λ + 2βd2 2 c h4 Proof From i in Theorem 3., the eigenvalues of P bd A are λ l k = 2βμ l e 2k+πi 2, k =0,,l =,2,...,m. A direct calculation leads to λ l 2 k =+2βμ 2 l 2 2β cos 2k +π 2 =+2βμ 2 l. 38 In the last step of the above equation we have used the fact that cos 2k+π 2 =0. Thus, from the above equation, we know that if we want to bound λ l k,itisenoughto bound μ l. Recalling that μ l is an eigenvalue of M K,weget M Kx = μ l x, i.e., Kx = μ l Mx, thus x T Kx = μ l x T Mx, μ l = xt Kx x T Mx. 39 From 6 in Lemma 2.3,we obtain c 2 h xt x 2 x T Mx c h Thus, from 7, 39and40we get d c 2 μ l d 2 c h 2. 4 Then the corollary follows by 38and4. Corollary 3.4 Let λ be an eigenvalue of P btd A, with P btd and A defined in 26 and 5. Then the eigenvalues of the preconditioned matrix P btd A are with algebraic multiplicity m, the remaining m eigenvalues can be bounded as follows: + 2βd2 c 2 2 λ + 2βd2 2 c h4 Proof From i in Theorem 3.2, using the same proof procedure as in Corollary 3.3, we can obtain the corresponding conclusion.
11 Zeng et al. Journal of Inequalities and Applications :355 Page of 4 Remark 3.5 From Theorems 3. and 3.2, we can see that the eigenvalues of the preconditioned matrices P bd A and P btda are around when the Tikhonov parameter β is sufficiently small. Remark 3.6 In actual computations, we may choose good approximations K and M for K and M, respectively. Then the corresponding preconditioners P bd and P btd are replaced by P bd = M 0 0 2β and P btd = M K 0 M, 2β respectively. Some examples of proper M and K were presented in [5, 7]. 4 Numerical experiments The test problem we consider is the following linear elliptic optimal control problem: min u,f 2 u û β f 2 2 subject to u = f in, u =0 on, where =0, 0,, û L 2 isgivenby ifx, y [0, 2 û = ]2, 0, otherwise. Our numerical experiments are performed by MATLAB. We consider four uniformly refined meshes, which are constructed by subsequently splitting each triangle into four triangles by connecting the midpoints of the edges of the triangle. First, by setting the regular parameter β =0 2 not very small, we show the eigenvalues of the preconditioned matrices P bcd A and P bctda in Figures and 2, they are both clustered. These results demonstrate the theoretical analysis in Theorems 2. and 2.2. Also, by setting the regular parameter β =0 8 very small, we show the eigenvalues of the preconditioned matrices Figure The eigenvalue distribution of the preconditioned matrix P bcd A with β =0 2 and h =2 3.
12 Zeng et al. Journal of Inequalities and Applications :355 Page 2 of 4 Figure 2 The eigenvalue distribution of the preconditioned matrix P bctd A with β =0 2 and h =2 3. Figure 3 The eigenvalue distribution of the preconditioned matrix P bd A with β =0 8 and h =2 3. Figure 4 The eigenvalue distribution of the preconditioned matrix P btd A with β =0 8 and h =2 3. P bd A and P btda in Figures 3 and 4, both of them are clustered. These results validate the theoretical analysis in Theorems 3.and 3.2. Furthermore, for comparison, we show the number of iterations for different preconditioners with different regular parameters in Table. In all implementations, we use zero as initial guess and stop the iteration when r k = b Ax k 2 / r 0 = b The iteration methods we use are preconditioned GMRES methods. From the results stated in Table, first we find that when the regular parameter β is not very small, the number of iterations of P bcd and P bctd preconditioners is smaller than P bd and P btd,thusweshould choose P bcd or P bctd as preconditioners. Moreover, in this case, we observe that the number of iterations of the preconditioned GMRES methods is hardly sensitive to the changes in the mesh size h,thisvalidatestheanalysisinremark2.6. On the other hand, when the
13 Zeng et al. Journal of Inequalities and Applications :355 Page 3 of 4 Table The number of iterations for different preconditioners β h P bd P btd P bcd P bctd , , , , , , regular parameter β is very small, the number of iterations of P bd and P btd preconditioners is smaller than P bcd and P bctd,thusp bd or P btd preconditioners are our choices. Competing interests The authors declare that they have no competing interests. Authors contributions The main idea of this paper was proposed by YZ. All authors contributed equally in writing this article and read and approved the final manuscript. Acknowledgements We thank the anonymous referees for their valuable comments and suggestions which led to an improved presentation of this paper. This work was supported by the opening fund of Jiangsu Key Lab for NSLSCS Grant No , the Training Program for Outstanding Young Teachers in Guangdong Province Grant No , and the Educational Commission of Guangdong Province Grant No. 204KQNCX20. Received: 30 March 205 Accepted: 22 October 205 References. Lions, JL: Optimal Control of Systems. Springer, New York 968
14 Zeng et al. Journal of Inequalities and Applications :355 Page 4 of 4 2. Bergounioux, M, Ito, K, Kunisch, K: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, Bergounioux, M, Kunisch, K: Primal-dual strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22, Casas, E: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 3, Rees, T, Dollar, HS, Wathen, AJ: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32, Bai, Z: Block preconditioners for elliptic PDE-constrained problems. Computing 9, Rees, T, Stoll, M: Block-triangular preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 7, Herzog, R, Sachs, E: Preconditioned conjugate gradient method for optimal control problems with control and state constraints. SIAM J. Matrix Anal. Appl. 3, Pearson, JW, Wathen, A: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 9, Schöberl, J, Zulehner, W: Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems. SIAM J. Matrix Anal. Appl. 29, Zhang, G, Zheng, Z: Block-symmetric and block-lower-triangular preconditioner PDE-constrained optimization problems. J. Comput. Math. 3, Bai, Z, Benzi, M, Chen, F, Wang, Z: 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, Borzì, A, Schulz, V: Multigrid methods for PDE optimization. SIAM Rev. 5, Lass, O, Vallejos, M, Borzi, A, Douglas, CC: Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems. Computing 84, Schöberl, J, Simon, R, Zulehner, W: A robust multigrid method for elliptic optimal control problems. SIAM J. Numer. Anal. 49, Simoncini, V: Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems. Comput. Optim. Appl. 53, Vallejos, M, Borzì, A: Multigrid optimization methods for linear and bilinear elliptic optimal control problems. Computing 82, Zulehner, W: Nonstandard norms and robust estimates for saddle point problems. SIAM J. Matrix Anal. Appl. 32, Wang, S: A class of preconditioned iterative methods for the optimal control of the Stokes equations. Master thesis, Nanjing normal university May Elman, HC, Silvester, DJ, Wathen, AJ: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation Series. Oxford University Press, Oxford 2005
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