A topological derivative-based algorithm to solve optimal control problems with \(L^0(\Omega)\) control cost. (English) Zbl 07876116
The paper derives a novel descent method for PDE-constrained optimal control problems that involve the \(L^0\)-cost of the control, i.e., the measure of the support of the control. A fundamental tool is the topological derivative of the value function with respect to variations of the support. For instance, it is shown that the pointwise a.e. non-negativity of this topological derivative is a necessary optimality condition, and the construction of the descent direction is based on this derivative. As main result it is proved that the algorithm generates a minimizing sequence for the value function. Interesting numerical examples like a binary control problem are also included.
Reviewer: Florian Mannel (Lübeck)
MSC:
| 49M05 | Numerical methods based on necessary conditions |
| 49K40 | Sensitivity, stability, well-posedness |
| 65K10 | Numerical optimization and variational techniques |
Keywords:
topological derivative; control support optimization; sparse optimal control; \(L^0\) optimizationReferences:
| [1] | S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var. 11 (2005), 401-425, doi:10.1051/cocv:2005012. · Zbl 1123.35040 · doi:10.1051/cocv:2005012 |
| [2] | S. Amstutz, A semismooth Newton method for topology optimization, Nonlinear Anal. 73 (2010), 1585-1595, doi:10.1016/j.na.2010.04.065. · Zbl 1201.49029 · doi:10.1016/j.na.2010.04.065 |
| [3] | S. Amstutz, Analysis of a level set method for topology optimization, Optim. Methods Softw. 26 (2011), 555-573, doi:10.1080/10556788.2010.521557. · Zbl 1227.49045 · doi:10.1080/10556788.2010.521557 |
| [4] | S. Amstutz, An introduction to the topological derivative, Engineering Computations 39 (2021), 3-33, doi:10.1108/ec-07-2021-0433. · doi:10.1108/EC-07-2021-0433 |
| [5] | S. Amstutz and H. Andrä, A new algorithm for topology optimization using a level-set method, J. Comput. Phys. 216 (2006), 573-588, doi:10.1016/j.jcp.2005.12.015. · Zbl 1097.65070 · doi:10.1016/j.jcp.2005.12.015 |
| [6] | S. Amstutz and A. Laurain, A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints, Comput. Optim. Appl. 56 (2013), 369-403, doi:10.1007/s10589-013-9555-6. · Zbl 1312.49031 · doi:10.1007/s10589-013-9555-6 |
| [7] | A. Ben Abda, M. Hassine, M. Jaoua, and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control Optim. 48 (2009/10), 2871-2900, doi: 10.1137/070704332. · Zbl 1202.49054 · doi:10.1137/070704332 |
| [8] | Daniel Wachsmuth A topological derivative-based algorithm for 𝐿 0 problems |
| [9] | V. I. Bogachev, Measure Theory. Vol. I, Springer-Verlag, Berlin, 2007, doi:10.1007/978-3-540-34514-5. · Zbl 1120.28001 · doi:10.1007/978-3-540-34514-5 |
| [10] | H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in 𝐿 1 , J. Math. Soc. Japan 25 (1973), 565-590, doi:10.2969/jmsj/02540565. · Zbl 0278.35041 · doi:10.2969/jmsj/02540565 |
| [11] | A. Carpio and M. L. Rapún, Hybrid topological derivative and gradient-based methods for electri-cal impedance tomography, Inverse Problems 28 (2012), 095010, 22, doi:10.1088/0266-5611/28/9/ 095010. · Zbl 1253.35215 · doi:10.1088/0266-5611/28/9/095010 |
| [12] | E. Casas and D. Wachsmuth, A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations, SIAM J. Control Optim. 61 (2023), 1095-1112, doi:10.1137/22m1486418. · Zbl 1518.35333 · doi:10.1137/22M1486418 |
| [13] | J. Céa, S. Garreau, P. Guillaume, and M. Masmoudi, The shape and topological optimizations connection, volume 188, 2000, 713-726, doi:10.1016/s0045-7825(99)00357-6. IV WCCM, Part II (Buenos Aires, 1998). · Zbl 0972.74057 · doi:10.1016/S0045-7825(99)00357-6 |
| [14] | J. Céa, A. Gioan, and J. Michel, Quelques résultats sur l’identification de domaines, Calcolo 10 (1973), 207-232, doi:10.1007/bf02575843. · Zbl 0303.93023 · doi:10.1007/BF02575843 |
| [15] | J. Céa, A. Gioan, and J. Michel, Adaptation de la méthode du gradient à un problème d’identification de domaine, in Computing Methods in Applied Sciences and Engineering (Proc. Internat. Sympos., Versailles, 1973), Part 2, Lecture Notes in Comput. Sci., Vol. 11, Springer, Berlin, 1974, 391-402. · Zbl 0304.93011 |
| [16] | J. Diestel and J. J. Uhl, Jr., Vector Measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. · Zbl 0369.46039 |
| [17] | I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353, doi:10.1016/ 0022-247x(74)90025-0. · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0 |
| [18] | I. Ekeland and R. Témam, Convex Analysis and Variational Problems, volume 28 of Classics in Applied Mathematics, SIAM, Philadelphia, PA, 1999, doi:10.1137/1.9781611971088. · Zbl 0939.49002 · doi:10.1137/1.9781611971088 |
| [19] | L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010, doi:10.1090/gsm/019. · Zbl 1194.35001 · doi:10.1090/gsm/019 |
| [20] | P. Fulmański, A. Laurain, J. F. Scheid, and J. Sokołowski, A level set method in shape and topology optimization for variational inequalities, Int. J. Appl. Math. Comput. Sci. 17 (2007), 413-430, doi:10.2478/v10006-007-0034-z. · Zbl 1237.49058 · doi:10.2478/v10006-007-0034-z |
| [21] | S. Garreau, P. Guillaume, and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim. 39 (2001), 1756-1778, doi:10.1137/s0363012900369538. · Zbl 0990.49028 · doi:10.1137/S0363012900369538 |
| [22] | P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem, SIAM J. Control Optim. 41 (2002), 1042-1072, doi:10.1137/s0363012901384193. · Zbl 1053.49031 · doi:10.1137/S0363012901384193 |
| [23] | M. Hahn, S. Leyffer, and S. Sager, Binary optimal control by trust-region steepest descent, Math. Program. 197 (2023), 147-190, doi:10.1007/s10107-021-01733-z. · Zbl 07662896 · doi:10.1007/s10107-021-01733-z |
| [24] | M. Hassine, S. Jan, and M. Masmoudi, From differential calculus to 0-1 topological optimization, SIAM J. Control Optim. 45 (2007), 1965-1987, doi:10.1137/050631720. · Zbl 1139.49039 · doi:10.1137/050631720 |
| [25] | M. Hintermüller and A. Laurain, A shape and topology optimization technique for solving a class of linear complementarity problems in function space, Comput. Optim. Appl. 46 (2010), 535-569, doi:10.1007/s10589-008-9201-x. · Zbl 1230.90187 · doi:10.1007/s10589-008-9201-x |
| [26] | Daniel Wachsmuth A topological derivative-based algorithm for 𝐿 0 problems |
| [27] | M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005), 45-61, doi:10.1007/s10589-005-4559-5. · Zbl 1074.65069 · doi:10.1007/s10589-005-4559-5 |
| [28] | M. Hrizi and M. Hassine, One-iteration reconstruction algorithm for geometric inverse source problem, J. Elliptic Parabol. Equ. 4 (2018), 177-205, doi:10.1007/s41808-018-0015-4. · Zbl 1497.65266 · doi:10.1007/s41808-018-0015-4 |
| [29] | K. Ito and K. Kunisch, Optimal control with 𝐿 𝑝 (Ω), 𝑝 ∈ [0, 1), control cost, SIAM J. Control Optim. 52 (2014), 1251-1275, doi:10.1137/120896529. · Zbl 1306.49010 · doi:10.1137/120896529 |
| [30] | D. Kalise, K. Kunisch, and K. Sturm, Optimal actuator design based on shape calculus, Math. Models Methods Appl. Sci. 28 (2018), 2667-2717, doi:10.1142/s0218202518500586. Corrected version (2024) at https://arxiv.org/abs/1711.01183v2. · Zbl 1411.49030 · doi:10.1142/S0218202518500586 |
| [31] | P. Manns, M. Hahn, C. Kirches, S. Leyffer, and S. Sager, On convergence of binary trust-region steepest descent, J. Nonsmooth Anal. Optim. 4 (2023), doi:10.46298/jnsao-2023-10164. · Zbl 1542.49013 · doi:10.46298/jnsao-2023-10164 |
| [32] | A. Münch, Optimal design of the support of the control for the 2-D wave equation: a numerical method, Int. J. Numer. Anal. Model. 5 (2008), 331-351. · Zbl 1242.49091 |
| [33] | A. Münch, Optimal location of the support of the control for the 1-D wave equation: numerical investigations, Comput. Optim. Appl. 42 (2009), 443-470, doi:10.1007/s10589-007-9133-x. · Zbl 1208.49052 · doi:10.1007/s10589-007-9133-x |
| [34] | A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013, doi:10.1007/978-3-642-35245-4. · Zbl 1276.35002 · doi:10.1007/978-3-642-35245-4 |
| [35] | A. A. Novotny, J. Sokołowski, and A. Żochowski, Applications of the Topological Derivative Method, volume 188 of Studies in Systems, Decision and Control, Springer, Cham, 2019, doi:10.1007/ 978-3-030-05432-8. With a foreword by Michel Delfour. · Zbl 1460.74001 · doi:10.1007/978-3-030-05432-8 |
| [36] | A. A. Novotny, J. Sokołowski, and A. Żochowski, Topological derivatives of shape functionals. Part II: First-order method and applications, J. Optim. Theory Appl. 180 (2019), 683-710, doi: 10.1007/s10957-018-1419-x. · Zbl 1414.49045 · doi:10.1007/s10957-018-1419-x |
| [37] | R. T. Rockafellar, Integral functionals, normal integrands and measurable selections, in Nonlinear Operators and the Calculus Of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lecture Notes in Math., Vol. 543, Springer, Berlin, 1976, 157-207. · Zbl 0374.49001 |
| [38] | J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim. 37 (1999), 1251-1272, doi:10.1137/s0363012997323230. · Zbl 0940.49026 · doi:10.1137/S0363012997323230 |
| [39] | J. Sokołowski and A. Żochowski, Topological derivative for optimal control problems, Control Cybernet. 28 (1999), 611-625. Recent advances in control of PDEs. · Zbl 0960.49027 |
| [40] | G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coeffi-cients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189-258, http://www.numdam.org/item? id=AIF_1965__15_1_189_0. · Zbl 0151.15401 |
| [41] | D. Wachsmuth, Iterative hard-thresholding applied to optimal control problems with 𝐿 0 (Ω) control cost, SIAM J. Control Optim. 57 (2019), 854-879, doi:10.1137/18m1194602. · Zbl 1410.49011 · doi:10.1137/18M1194602 |
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