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Keywords:
preconditioning; Schur complement; transformation; optimal control; implicit time integration
preconditioning; Schur complement; transformation; optimal control; implicit time integration
Summary:
Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods. Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks. Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications.
References:
[1] Axelsson, O.: On the efficiency of a class of $A$-stable methods. BIT, Nord. Tidskr. Inf.-behandl. 14 (1974), 279-287. DOI 10.1007/BF01933227 | MR 0391518 | Zbl 0289.65028
[2] Axelsson, O., Blaheta, R., Kohut, R.: Preconditioning methods for high-order strongly stable time integration methods with an application for a DAE problem. Numer. Linear Algebra Appl. 22 (2015), 930-949. DOI 10.1002/nla.2015 | MR 3426322 | Zbl 06604516
[3] Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Poisson and convection-diffusion control. Numer. Algorithms 73 (2016), 631-663. DOI 10.1007/s11075-016-0111-1 | MR 3564863 | Zbl 1353.65059
[4] Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Stokes control. Numer. Algorithms 74 (2017), 19-37. DOI 10.1007/s11075-016-0136-5 | MR 3590387 | Zbl 1365.65167
[5] Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7 (2000), 197-218. DOI 10.1002/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S | MR 1762967 | Zbl 1051.65025
[6] Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66 (2014), 811-841. DOI 10.1007/s11075-013-9764-1 | MR 3240302 | Zbl 1307.65034
[7] Axelsson, O., Vassilevski, P. S.: A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning. SIAM J. Matrix Anal. Appl. 12 (1991), 625-644. DOI 10.1137/0612048 | MR 1121697 | Zbl 0748.65028
[8] Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93 (2015), 41-60. DOI 10.1007/s10665-013-9670-5 | MR 3386091 | Zbl 1360.65089
[9] Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: 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. DOI 10.1093/imanum/drs001 | MR 3020961 | Zbl 1271.65100
[10] Bai, Z.-Z., Chen, F., Wang, Z.-Q.: Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices. Numer. Algorithms 62 (2013), 655-675. DOI 10.1007/s11075-013-9696-9 | MR 3034831 | Zbl 1267.65034
[11] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics 15, Springer, New York (1991). DOI 10.1007/978-1-4612-3172-1 | MR 1115205 | Zbl 0788.73002
[12] Butcher, J. C.: Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, Chichester (2008). DOI 10.1002/9780470753767 | MR 2401398 | Zbl 1167.65041
[13] Cahouet, J., Chabard, J.-P.: Some fast 3D finite element solvers for the generalized Stokes problem. Int. J. Numer. Methods Fluids 8 (1988), 869-895. DOI 10.1002/fld.1650080802 | MR 0953141 | Zbl 0665.76038
[14] Greenbaum, A., Pták, V., Strakoš, Z.: Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17 (1996), 465-469. DOI 10.1137/S0895479894275030 | MR 1397238 | Zbl 0857.65029
[15] Lions, J. L.: Some Aspects of the Optimal Control of Distributed Parameter Systems. CBMS-NSF Regional Conference Series in Applied Mathematics 6, Society for Industrial and Applied Mathematics, Philadelphia (1972). DOI 10.1137/1.9781611970616 | MR 0479526 | Zbl 0275.49001
[16] Paige, C. C., Saunders, M. A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12 (1975), 617-629. DOI 10.1137/0712047 | MR 0383715 | Zbl 0319.65025
[17] Saad, Y.: A flexible inner-outer preconditioned GMRES-algorithm. SIAM J. Sci. Comp. 14 (1993), 461-469. DOI 10.1137/0914028 | MR 1204241 | Zbl 0780.65022
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