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Maximal discrete sparsity in parabolic optimal control with measures
1. | Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz, Universitätsstraße 1, 56070 Koblenz, Germany |
2. | Institut für Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany |
We consider variational discretization [
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
J. J. Ahlberg and E. N. Nilson,
Convergence properties of the spline fit, J. Soc. Indust. Appl. Math., 11 (1963), 95-104.
doi: 10.1137/0111007. |
[3] |
O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. I, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. |
[4] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() ![]() |
[5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[6] |
E. Casas, C. Clason and K. Kunisch,
Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 50 (2012), 1735-1752.
doi: 10.1137/110843216. |
[7] |
E. Casas, C. Clason and K. Kunisch,
Parabolic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 51 (2013), 28-63.
doi: 10.1137/120872395. |
[8] |
E. Casas and K. Kunisch,
Parabolic control problems in space-time measure spaces, ESAIM. Control, Optimisation and Calculus of Variations, 22 (2016), 355-370.
doi: 10.1051/cocv/2015008. |
[9] |
E. Casas, B. Vexler and E. Zuazua,
Sparse initial data identification for parabolic PDE and its finite element approximations, Mathematical Control and Related Fields, 5 (2015), 377-399.
doi: 10.3934/mcrf.2015.5.377. |
[10] |
C. Clason, Nonsmooth Analysis and Optimization, eprint, arXiv: 1708.04180. |
[11] |
C. Clason and K. Kunisch,
A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control, Optimisation and Calculus of Variations, 17 (2011), 243-266.
doi: 10.1051/cocv/2010003. |
[12] |
N. von Daniels, M. Hinze and M. Vierling,
Crank-Nicolson time stepping and variational discretization of control-constrained parabolic optimal control problems, SIAM Journal on Control and Optimization, 53 (2015), 1182-1198.
doi: 10.1137/14099680X. |
[13] |
I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.
doi: 10.1137/1.9781611971088. |
[14] |
L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998. |
[15] |
C. Goll, R. Rannacher and W. Wollner,
The damped Crank-Nicolson time-marching scheme for the adaptive solution of the Black-Scholes equation, Journal of Computational Finance, 18 (2015), 1-37.
|
[16] |
W. Gong,
Error estimates for finite element approximations of parabolic equations with measure data, Mathematics of Computation, 82 (2013), 69-98.
doi: 10.1090/S0025-5718-2012-02630-5. |
[17] |
W. Gong, M. Hinze and Z. Zhou,
A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control, SIAM Journal on Control and Optimization, 52 (2014), 97-119.
doi: 10.1137/110840133. |
[18] |
M. Hinze,
A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications. An International Journal, 30 (2005), 45-61.
doi: 10.1007/s10589-005-4559-5. |
[19] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009. |
[20] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, Providence, RI, 2008.
doi: 10.1090/gsm/096. |
[21] |
K. Kunisch, K. Pieper and B. Vexler,
Measure valued directional sparsity for parabolic optimal control problems, SIAM Journal on Control and Optimization, 52 (2014), 3078-3108.
doi: 10.1137/140959055. |
[22] |
J. R. Munkres, Topology, Prentice Hall Inc., Upper Saddle River, NJ, 2000. |
[23] |
J. A. Nitsche, $L_{\infty }$-convergence of Finite Element Approximation in Journées "Éléments Finis", Univ. Rennes, Rennes, 1975. |
[24] |
K. Pieper and B. Vexler,
A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM Journal on Control and Optimization, 51 (2013), 2788-2808.
doi: 10.1137/120889137. |
[25] |
R. A. Polyak,
Complexity of the regularized Newton's method, Pure and Applied Functional Analysis, 3 (2018), 327-347.
|
[26] |
W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987. |
[27] |
W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-71333-3. |
[28] |
G. Stadler,
Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Computational Optimization and Applications. An International Journal, 44 (2009), 159-181.
doi: 10.1007/s10589-007-9150-9. |
[29] |
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 2006. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
J. J. Ahlberg and E. N. Nilson,
Convergence properties of the spline fit, J. Soc. Indust. Appl. Math., 11 (1963), 95-104.
doi: 10.1137/0111007. |
[3] |
O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. I, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. |
[4] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() ![]() |
[5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[6] |
E. Casas, C. Clason and K. Kunisch,
Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 50 (2012), 1735-1752.
doi: 10.1137/110843216. |
[7] |
E. Casas, C. Clason and K. Kunisch,
Parabolic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 51 (2013), 28-63.
doi: 10.1137/120872395. |
[8] |
E. Casas and K. Kunisch,
Parabolic control problems in space-time measure spaces, ESAIM. Control, Optimisation and Calculus of Variations, 22 (2016), 355-370.
doi: 10.1051/cocv/2015008. |
[9] |
E. Casas, B. Vexler and E. Zuazua,
Sparse initial data identification for parabolic PDE and its finite element approximations, Mathematical Control and Related Fields, 5 (2015), 377-399.
doi: 10.3934/mcrf.2015.5.377. |
[10] |
C. Clason, Nonsmooth Analysis and Optimization, eprint, arXiv: 1708.04180. |
[11] |
C. Clason and K. Kunisch,
A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control, Optimisation and Calculus of Variations, 17 (2011), 243-266.
doi: 10.1051/cocv/2010003. |
[12] |
N. von Daniels, M. Hinze and M. Vierling,
Crank-Nicolson time stepping and variational discretization of control-constrained parabolic optimal control problems, SIAM Journal on Control and Optimization, 53 (2015), 1182-1198.
doi: 10.1137/14099680X. |
[13] |
I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.
doi: 10.1137/1.9781611971088. |
[14] |
L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998. |
[15] |
C. Goll, R. Rannacher and W. Wollner,
The damped Crank-Nicolson time-marching scheme for the adaptive solution of the Black-Scholes equation, Journal of Computational Finance, 18 (2015), 1-37.
|
[16] |
W. Gong,
Error estimates for finite element approximations of parabolic equations with measure data, Mathematics of Computation, 82 (2013), 69-98.
doi: 10.1090/S0025-5718-2012-02630-5. |
[17] |
W. Gong, M. Hinze and Z. Zhou,
A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control, SIAM Journal on Control and Optimization, 52 (2014), 97-119.
doi: 10.1137/110840133. |
[18] |
M. Hinze,
A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications. An International Journal, 30 (2005), 45-61.
doi: 10.1007/s10589-005-4559-5. |
[19] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009. |
[20] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, Providence, RI, 2008.
doi: 10.1090/gsm/096. |
[21] |
K. Kunisch, K. Pieper and B. Vexler,
Measure valued directional sparsity for parabolic optimal control problems, SIAM Journal on Control and Optimization, 52 (2014), 3078-3108.
doi: 10.1137/140959055. |
[22] |
J. R. Munkres, Topology, Prentice Hall Inc., Upper Saddle River, NJ, 2000. |
[23] |
J. A. Nitsche, $L_{\infty }$-convergence of Finite Element Approximation in Journées "Éléments Finis", Univ. Rennes, Rennes, 1975. |
[24] |
K. Pieper and B. Vexler,
A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM Journal on Control and Optimization, 51 (2013), 2788-2808.
doi: 10.1137/120889137. |
[25] |
R. A. Polyak,
Complexity of the regularized Newton's method, Pure and Applied Functional Analysis, 3 (2018), 327-347.
|
[26] |
W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987. |
[27] |
W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-71333-3. |
[28] |
G. Stadler,
Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Computational Optimization and Applications. An International Journal, 44 (2009), 159-181.
doi: 10.1007/s10589-007-9150-9. |
[29] |
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 2006. |





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