Xiaoliang Song^{1}, Bo Chen^{2}, Bo Yu^{3}
[1] G. Stadler, Elliptic optimal control problems with L^{1}control cost and applications for the placement of control devices. Comp. Optim. Appls., 44(2009), 159181. [2] G. Wachsmuth and D. Wachsmuth, Convergence and regularisation results for optimal control problems with sparsity functional. ESAIM Control Optim. Calc. Var., 17(2011), 858886. [3] E. Casas, R. Herzog, G. Wachsmuth, Approximation of sparse controls in semilinear equations by piecewise linear functions. Numer. Math., 122(2012), 645669. [4] E. Casas, R. Herzog, G, Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with L^{1} cost functional. SIAM J. Optim., 22(2012), 795820. [5] M. Ulbrich, Nonsmooth Newtonlike methods for variational inequalities and constrained optimization problems in function spaces. Habilitation thesis, Fakultät für Mathematik, Technische Universität München, 2002. [6] M. Ulbrich, Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim., 13(2003), 805842. [7] M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints. Springer Science and Business Media, 23(2008). [8] T. Blumensath, M.E. Davies, Iterative Thresholding for Sparse Approximations. J. Fourier Anal. Appl., 14(2008), 629654. [9] K. Jiang, D.F. Sun, K.C. Toh, An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP. SIAM J. Optim., 22(2012), 10421064. [10] A. Beck, M. Teboulle, A fast iterative shrinkagethresholding algorithm for linear inverse problems. SIAM J. Imaging Sci., 2(2009), 183202. [11] K.C. Toh, S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems. Pac. J. Optim., 6(2010), 615640. [12] M. Fazel, T.K. Pong, D.F. Sun, P. Tseng, Hankel matrix rank minimization with applications to system identification and realization. SIAM J. Matrix Anal. Appl., 34(2013), 946977. [13] Q. Fan, Y.L. Jiao and X.L. Lu, A primal dual active set algorithm with continuation for compressed sensing, IEEE Trans. Signal Process., 62:(2014), 62766285. [14] L. Chen, D.F. Sun, K.C. Toh, An efficient inexact symmetric GaussSeidel based majorized ADMM for highdimensional convex composite conic programming. Math. Program., (2015), 134. [15] Y.L. Jiao, B. Jin and X.L. Lu, A primal dual active set with continuation algorithm for the l0regularized optimization problem, Appl. Comput. Harmon. Anal., 39(2015), 400426. [16] X.D. Li, D.F. Sun, K.C. Toh, A Schur complement based semiproximal ADMM for convex quadratic conic programming and extensions. Math. Program., 155(2016), 333373. [17] X.L. Song, B. Yu, Y.Y. Wang, X.Z. Zhang, An inexact heterogeneous ADMM algorithm for elliptic optimal control problems with L^{1}control cost.arXiv preprint arXiv:1610.00306, 2016. [18] A. Schindele, A. Borzì, Proximal methods for elliptic optimal control problems with sparsity cost functional. Appl. Math., 7(2016), 967992. [19] X.L. Song, B. Chen, B. Yu, An efficient dualitybased approach for PDEconstrained sparse optimization. Comput Optim Appl., 69(2018), 461500. [20] X. L. Song, B. Chen, B. Yu:Mesh Independence of an Accelerated Block Coordinate Descent Method for Sparse Optimal Control Problems, arXiv:1610.00308, 2017. [21] Y. Cui, Large scale composite optimization problems with coupled objective functions:theory, algorithms and applications. PhD thesis, National University of Singapore, 2016. [22] X.D. Li, D.F. Sun, K.C. Toh, A block symmetric GaussSeidel decomposition theorem for convex composite quadratic programming and its applications. arXiv:1703.06629, 2017 [23] A.J. Wathen, Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal., 7(1987), 449457. [24] P.G. Ciarlet, The finite element method for elliptic problems. Society for Industrial and Applied Mathematics, 2002. [25] H.C. Elman, D.J. Silvester and A.J. Wathen, Finite elements and fast iterative solvers:with applications in incompressible fluid dynamics. Oxford University Press (UK), 2014. [26] C. Carstensen, Quasiinterpolation and a posteriori error analysis in finite element methods. ESAIM:Math. Model. Numer. Anal., 33(1999), 11871202. [27] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, vol. 31, Siam, 1980. 
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