# American Institute of Mathematical Sciences

June  2020, 10(2): 333-363. doi: 10.3934/mcrf.2019041

## Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

 1 Department of Mathematics, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany 2 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, 8010 Graz, Austria 3 Institute for Numerical Simulation, Universität Bonn, Endenicher Allee 19b, 53115 Bonn, Germany

* Corresponding author: Dominik Hafemeyer

Received  February 2019 Revised  June 2019 Published  June 2020 Early access  August 2019

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the $L^2$- and $L^\infty$-error for the state and the adjoint state are of order ${\mathcal O}(h^2)$ and that the $L^1$-error of the control behaves like ${\mathcal O}(h^2)$, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain $L^2$-error estimates of order $\mathcal{O}(h)$ for the state and $W^{1, \infty}$-error estimates of order $\mathcal{O}(h)$ for the adjoint state. Under the same structural assumption as before we derive an $L^1$-error estimate of order $\mathcal{O}(h)$ for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.

Citation: Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control and Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041
##### References:
 [1] W. Alt, R. Baier, F. Lempio and M. Gerdts, Approximations of linear control problems with bang-bang solutions, Optimization, 62 (2013), 9-32.  doi: 10.1080/02331934.2011.568619. [2] W. Alt, U. Felgenhauer and M. Seydenschwanz, Euler discretization for a class of nonlinear optimal control problems with control appearing linearly, Comput. Optim. Appl., 69 (2018), 825-856.  doi: 10.1007/s10589-017-9969-7. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [4] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, Second edition. MOS-SIAM Series on Optimization, 17. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2014. doi: 10.1137/1.9781611973488. [5] S. Bartels, Total variation minimization with finite elements: Convergence and iterative solution, SIAM J. Numer. Anal., 50 (2012), 1162-1180.  doi: 10.1137/11083277X. [6] S. Bartels and M. Milicevic, Iterative finite element solution of a constrained total variation regularized model problem, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1207-1232.  doi: 10.3934/dcdss.2017066. [7] L. Bonifacius, K. Pieper and B Vexler, Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls, SIAM J. Control Optim., 57 (2019), 1730-1756.  doi: 10.1137/18M1213816. [8] K. Bredies and D. Vicente, A perfect reconstruction property for pde-constrained total-variation minimization with application in quantitative susceptibility mapping, ESAIM Control Optim. Calc. Var., Accepted for Publication. [9] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [10] J. Casado-Díaz, C. Castro, M. Luna-Laynez and E. Zuazua, Numerical approximation of a one-dimensional elliptic optimal design problem, Multiscale Model. Simul., 9 (2011), 1181-1216.  doi: 10.1137/10081928X. [11] E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 50 (2012), 1735-1752.  doi: 10.1137/110843216. [12] E. Casas, C. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63.  doi: 10.1137/120872395. [13] E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108. [14] E. Casas, F. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM J. Control Optim., 55 (2017), 1752-1788.  doi: 10.1137/16M1056511. [15] E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces, SIAM J. Control Optim., 52 (2014), 339-364.  doi: 10.1137/13092188X. [16] E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces, ESAIM Control Optim. Calc. Var., 22 (2016), 355-370.  doi: 10.1051/cocv/2015008. [17] E. Casas and K. Kunisch, Analysis of optimal control problems of semilinear elliptic equations by BV-functions, Set-Valued Var. Anal., 27 (2019), 355-379.  doi: 10.1007/s11228-018-0482-7. [18] E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40 (1999), 229-257.  doi: 10.1007/s002459900124. [19] E. Casas, M. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0. [20] E. Casas, D. Wachsmuth and G. Wachsmuth, Second-order analysis and numerical approximation for bang-bang bilinear control problems, SIAM J. Control Optim., 56 (2018), 4203-4227.  doi: 10.1137/17M1139953. [21] I. Chryssoverghi, Approximate gradient/penalty methods with general discretization schemes for optimal control problems, in Large-Scale Scientific Computing, Lecture Notes in Comput. Sci., Springer, Berlin, 3743 (2006), 199–207. doi: 10.1007/11666806_21. [22] C. Clason, F. Kruse and K. Kunisch, Total variation regularization of multi-material topology optimization, ESAIM, Math. Model. Numer. Anal., 52 (2018), 275-303.  doi: 10.1051/m2an/2017061. [23] C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.  doi: 10.1051/cocv/2010003. [24] K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls, Comput. Optim. Appl., 51 (2012), 931-939.  doi: 10.1007/s10589-010-9365-z. [25] A. L. Dontchev, An a priori estimate for discrete approximations in nonlinear optimal control, SIAM J. Control Optim., 34 (1996), 1315-1328.  doi: 10.1137/S036301299426948X. [26] A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order runge-kutta approximations in control constrained optimal control, SIAM J. Numer. Anal., 38 (2000), 202-226.  doi: 10.1137/S0036142999351765. [27] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer, New York, 2004. doi: 10.1007/978-1-4757-4355-5. [28] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. [29] M. Hinze and T. Quyen, Iterated total variation regularization with finite element methods for reconstruction the source term in elliptic systems, preprint, arXiv: 1901.10278. [30] K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.  doi: 10.1137/140959055. [31] K. Kunisch, P. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244.  doi: 10.1137/141001366. [32] J. Peypouquet, Convex Optimization in Normed Spaces: Theory, Methods and Examples, With a foreword by Hedy Attouch, SpringerBriefs in Optimization, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0. [33] K. Pieper, B. Quoc Tang, P. Trautmann and D. Walter, Inverse point source location with the helmholtz equation on a bounded domain, preprint, arXiv: 1805.03310. [34] K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM J. Control Optim., 51 (2013), 2788-2808.  doi: 10.1137/120889137. [35] P. Trautmann, B. Vexler and A. Zlotnik, Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients, Math. Control Relat. Fields, 8 (2018), 411-449.  doi: 10.3934/mcrf.2018017. [36] M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, MOS-SIAM Series on Optimization, 11. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2011. doi: 10.1137/1.9781611970692. [37] V. Veliov, On the time-discretization of control systems, SIAM J. Control Optim., 35 (1997), 1470-1486.  doi: 10.1137/S0363012995288987. [38] V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: The linear case, Control Cybern., 34 (2005), 967-982. [39] D. Walter, On Sparse Sensor Placement for Parameter Identifcation Problems with Partial Differential Equations, PhD thesis, Technische Universität München, 2018. [40] M. F. Wheeler, An optimal $L_{\infty }$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal., 10 (1973), 914-917.  doi: 10.1137/0710077. [41] W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

##### References:
 [1] W. Alt, R. Baier, F. Lempio and M. Gerdts, Approximations of linear control problems with bang-bang solutions, Optimization, 62 (2013), 9-32.  doi: 10.1080/02331934.2011.568619. [2] W. Alt, U. Felgenhauer and M. Seydenschwanz, Euler discretization for a class of nonlinear optimal control problems with control appearing linearly, Comput. Optim. Appl., 69 (2018), 825-856.  doi: 10.1007/s10589-017-9969-7. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [4] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, Second edition. MOS-SIAM Series on Optimization, 17. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2014. doi: 10.1137/1.9781611973488. [5] S. Bartels, Total variation minimization with finite elements: Convergence and iterative solution, SIAM J. Numer. Anal., 50 (2012), 1162-1180.  doi: 10.1137/11083277X. [6] S. Bartels and M. Milicevic, Iterative finite element solution of a constrained total variation regularized model problem, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1207-1232.  doi: 10.3934/dcdss.2017066. [7] L. Bonifacius, K. Pieper and B Vexler, Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls, SIAM J. Control Optim., 57 (2019), 1730-1756.  doi: 10.1137/18M1213816. [8] K. Bredies and D. Vicente, A perfect reconstruction property for pde-constrained total-variation minimization with application in quantitative susceptibility mapping, ESAIM Control Optim. Calc. Var., Accepted for Publication. [9] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [10] J. Casado-Díaz, C. Castro, M. Luna-Laynez and E. Zuazua, Numerical approximation of a one-dimensional elliptic optimal design problem, Multiscale Model. Simul., 9 (2011), 1181-1216.  doi: 10.1137/10081928X. [11] E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 50 (2012), 1735-1752.  doi: 10.1137/110843216. [12] E. Casas, C. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63.  doi: 10.1137/120872395. [13] E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108. [14] E. Casas, F. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM J. Control Optim., 55 (2017), 1752-1788.  doi: 10.1137/16M1056511. [15] E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces, SIAM J. Control Optim., 52 (2014), 339-364.  doi: 10.1137/13092188X. [16] E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces, ESAIM Control Optim. Calc. Var., 22 (2016), 355-370.  doi: 10.1051/cocv/2015008. [17] E. Casas and K. Kunisch, Analysis of optimal control problems of semilinear elliptic equations by BV-functions, Set-Valued Var. Anal., 27 (2019), 355-379.  doi: 10.1007/s11228-018-0482-7. [18] E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40 (1999), 229-257.  doi: 10.1007/s002459900124. [19] E. Casas, M. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0. [20] E. Casas, D. Wachsmuth and G. Wachsmuth, Second-order analysis and numerical approximation for bang-bang bilinear control problems, SIAM J. Control Optim., 56 (2018), 4203-4227.  doi: 10.1137/17M1139953. [21] I. Chryssoverghi, Approximate gradient/penalty methods with general discretization schemes for optimal control problems, in Large-Scale Scientific Computing, Lecture Notes in Comput. Sci., Springer, Berlin, 3743 (2006), 199–207. doi: 10.1007/11666806_21. [22] C. Clason, F. Kruse and K. Kunisch, Total variation regularization of multi-material topology optimization, ESAIM, Math. Model. Numer. Anal., 52 (2018), 275-303.  doi: 10.1051/m2an/2017061. [23] C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.  doi: 10.1051/cocv/2010003. [24] K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls, Comput. Optim. Appl., 51 (2012), 931-939.  doi: 10.1007/s10589-010-9365-z. [25] A. L. Dontchev, An a priori estimate for discrete approximations in nonlinear optimal control, SIAM J. Control Optim., 34 (1996), 1315-1328.  doi: 10.1137/S036301299426948X. [26] A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order runge-kutta approximations in control constrained optimal control, SIAM J. Numer. Anal., 38 (2000), 202-226.  doi: 10.1137/S0036142999351765. [27] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer, New York, 2004. doi: 10.1007/978-1-4757-4355-5. [28] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. [29] M. Hinze and T. Quyen, Iterated total variation regularization with finite element methods for reconstruction the source term in elliptic systems, preprint, arXiv: 1901.10278. [30] K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.  doi: 10.1137/140959055. [31] K. Kunisch, P. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244.  doi: 10.1137/141001366. [32] J. Peypouquet, Convex Optimization in Normed Spaces: Theory, Methods and Examples, With a foreword by Hedy Attouch, SpringerBriefs in Optimization, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0. [33] K. Pieper, B. Quoc Tang, P. Trautmann and D. Walter, Inverse point source location with the helmholtz equation on a bounded domain, preprint, arXiv: 1805.03310. [34] K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM J. Control Optim., 51 (2013), 2788-2808.  doi: 10.1137/120889137. [35] P. Trautmann, B. Vexler and A. Zlotnik, Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients, Math. Control Relat. Fields, 8 (2018), 411-449.  doi: 10.3934/mcrf.2018017. [36] M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, MOS-SIAM Series on Optimization, 11. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2011. doi: 10.1137/1.9781611970692. [37] V. Veliov, On the time-discretization of control systems, SIAM J. Control Optim., 35 (1997), 1470-1486.  doi: 10.1137/S0363012995288987. [38] V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: The linear case, Control Cybern., 34 (2005), 967-982. [39] D. Walter, On Sparse Sensor Placement for Parameter Identifcation Problems with Partial Differential Equations, PhD thesis, Technische Universität München, 2018. [40] M. F. Wheeler, An optimal $L_{\infty }$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal., 10 (1973), 914-917.  doi: 10.1137/0710077. [41] W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
Example 1: A semi-discrete solution to the data from Section 5.3. The discretization parameter $h$ is roughly $3.8\cdot 10^{-6}$. The inclusions provided in Corollary 2 are clearly visible
Example 1: Convergence plots of the errors of the solutions to the semi-discrete problem (Pvd) compared to the exact solution. The exact solution is known
Example 1: Convergence plots of the errors of the solutions to the fully discrete problem (Pcd) compared to the exact solution. The exact solution is known
Example 2: A variationally discrete solution to the data from Section 5.4. The discretization parameter $h$ is roughly $3.8\cdot 10^{-6}$. The inclusions provided in Corollary 2 are clearly visible
Example 2: Convergence plots of the errors of the solutions to the semi-discrete problem (Pvd) compared to an approximation of the exact solution. The reference solution is computed as solution to (Pvd) with $h_{\text{ref}}\approx 3.8\cdot 10^{-6}$
Example 2: Convergence plots of the errors of the solutions to the fully discrete problem (Pcd) compared to an approximation of the exact solution. The reference solution is computed as solution to (Pcd) with $h_{\text{ref}}\approx 2.4\cdot 10^{-7}$
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