Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE

[1] M. Arioli, D. Kourounis, and D. Loghin. Discrete fractional Sobolev norms for domain decomposition preconditioning. IMA J. Numer. Anal., 33(1):318{342, 2013.10.1093/imanum/drr024Search in Google Scholar

[2] M. Arioli and D. Loghin. Discrete interpolation norms with applications. SIAM J. Num. Anal., 47(4):2924{2951, 2009.10.1137/080729360Search in Google Scholar

[3] Roscoe A. Bartlett, Matthias Heinkenschloss, Denis Ridzal, and Bart G. van Bloemen Waanders. Domain decomposition methods for advection dominated linear-quadratic elliptic optimal control problems. Comput. Methods Appl. Mech. Engrg., 195(44-47):6428{6447, 2006.10.1016/j.cma.2006.01.009Search in Google Scholar

[4] George Biros and Omar Ghattas. Parallel Lagrange-Newton-Krylov- Schur methods for PDE-constrained optimization. I. The Krylov-Schur solver. SIAM J. Sci. Comput., 27(2):687{713 (electronic), 2005.10.1137/S106482750241565XSearch in Google Scholar

[5] George Biros and Omar Ghattas. Parallel Lagrange-Newton-Krylov- Schur methods for PDE-constrained optimization. II. The Lagrange- Newton solver and its application to optimal control of steady viscous ows. SIAM J. Sci. Comput., 27(2):714{739 (electronic), 2005.10.1137/S1064827502415661Search in Google Scholar

[6] Alfio Borzi and Volker Schulz. Multigrid methods for PDE optimization. SIAM Rev., 51(2):361{395, 2009.10.1137/060671590Search in Google Scholar

[7] H. C. Elman. Iterative methods for large sparse non-symmetric systems of linear equations. PhD thesis, Yale University, New Haven, 1982.Search in Google Scholar

[8] Matthias Heinkenschloss and Hoang Nguyen. Balancing Neumann- Neumann methods for elliptic optimal control problems. In Domain de- composition methods in science and engineering, volume 40 of Lect. Notes Comput. Sci. Eng., pages 589{596. Springer, Berlin, 2005.10.1007/3-540-26825-1_62Search in Google Scholar

[9] Matthias Heinkenschloss and Hoang Nguyen. Neumann-Neumann domain decomposition preconditioners for linear-quadratic elliptic optimal control problems. SIAM J. Sci. Comput., 28(3):1001{1028, 2006.10.1137/040612774Search in Google Scholar

[10] Michael A. Heroux, Padma Raghavan, and Horst D. Simon, editors. Par- allel processing for scientific computing, volume 20 of Software, Envi- ronments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.10.1137/1.9780898718133Search in Google Scholar

[11] R. Herzog and O. Rheinbach. FETI-DP for optimal control problems. In J. Erhel, M.J. Gander, L. Halpern, G. Pichot, T. Sassi, and O.B. Widlund, editors, Domain Decomposition Methods in Science and Engi- neering XXI, Lecture Notes in Computational Science and Engineering. Springer International Publishing, 2014.Search in Google Scholar

[12] Roland Herzog and Susann Mach. Preconditioned solution of state gradient constrained elliptic optimal control problems. SIAM J. Numer. Anal., 54(2):688-718, 2016.10.1137/130948045Search in Google Scholar

[13] Roland Herzog and Ekkehard Sachs. Preconditioned conjugate gradient method for optimal control problems with control and state constraints. SIAM J. Matrix Anal. Appl., 31(5):2291{2317, 2010.10.1137/090779127Search in Google Scholar

[14] M. Kovcvara, D. Loghin, and J. Turner. Constraint interface preconditioning for topology optimization problems. SIAM J. Sci. Comput., 38(1):A128{A145, 2016.10.1137/140980387Search in Google Scholar

[15] J. L. Lions and E. Magenes. Problèmes aux limites non homogènes et applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris, 1968.Search in Google Scholar

[16] T.P. Mathew, M. Sarkis, and C.E. Schaerer. Analysis of block matrix preconditioners for elliptic optimal control problems. Numerical Linear Algebra with Applications, 14(4):257{279, 2007. cited By 13.10.1002/nla.526Search in Google Scholar

[17] M. F. Murphy, G. H. Golub, and A. J.Wathen. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comp., 21:1969-1972, 2000.10.1137/S1064827599355153Search in Google Scholar

[18] John W. Pearson. On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems. Electron. Trans. Numer. Anal., 44:53-72, 2015.Search in Google Scholar

[19] John W. Pearson, Martin Stoll, and Andrew J. Wathen. Preconditioners for state-constrained optimal control problems with Moreau-Yosida penalty function. Numer. Linear Algebra Appl., 21(1):81{97, 2014.10.1002/nla.1863Search in Google Scholar

[20] John W. Pearson and Andrew J. Wathen. A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl., 19(5):816-829, 2012.10.1002/nla.814Search in Google Scholar

[21] John W. Pearson and Andrew J. Wathen. Fast iterative solvers for convection-diffusion control problems. Electron. Trans. Numer. Anal., 40:294-310, 2013.Search in Google Scholar

[22] Ernesto E. Prudencio, Richard Byrd, and Xiao-Chuan Cai. Parallel full space SQP Lagrange-Newton-Krylov-Schwarz algorithms for PDEconstrained optimization problems. SIAM J. Sci. Comput., 27(4):1305- 1328 (electronic), 2006.10.1137/040602997Search in Google Scholar

[23] Ernesto E. Prudencio and Xiao-Chuan Cai. Parallel multilevel restricted Schwarz preconditioners with pollution removing for PDE-constrained optimization. SIAM J. Sci. Comput., 29(3):964{985, 2007.10.1137/050635663Search in Google Scholar

[24] Yue Qiu, Martin B. van Gijzen, Jan-Willem van Wingerden, Michel Verhaegen, and Cornelis Vuik. Efficient preconditioners for PDE-constrained optimization problem with a multilevel sequentially semiseparable matrix structure. Electron. Trans. Numer. Anal., 44:367-400, 2015.Search in Google Scholar

[25] Tyrone Rees, H. Sue Dollar, and Andrew J. Wathen. Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput., 32(1):271-298, 2010.10.1137/080727154Search in Google Scholar

[26] Tyrone Rees and Martin Stoll. Block-triangular preconditioners for PDEconstrained optimization. Numer. Linear Algebra Appl., 17(6):977-996, 2010.10.1002/nla.693Search in Google Scholar

[27] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing Co., Boston, 1996.Search in Google Scholar

[28] Martin Stoll and Andy Wathen. Preconditioning for partial differential equation constrained optimization with control constraints. Numer. Lin- ear Algebra Appl., 19(1):53{71, 2012.10.1002/nla.823Search in Google Scholar

[29] Shlomo. Ta'asan, Institute for Computer Applications in Science, Engineering., and Langley Research Center. "One Shot" methods for optimal control of distributed parameter systems I [microform] : finite dimensional control / Shlomo Ta'asan. Institute for Computer Applications in Science and Engineering : NASA Langley Research Center ; National Technical Information Service, distributor Hampton, Va. : [Spring field, Va], 1991.Search in Google Scholar

[30] Walter Zulehner. Nonstandard norms and robust estimates for saddle point problems. SIAM J. Matrix Anal. Appl., 32(2):536{560, 2011.10.1137/100814767Search in Google Scholar