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doi: 10.3934/mcrf.2022013
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Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation

Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz, Universitätsstraße 1, 56070 Koblenz, Germany

*Corresponding author: Evelyn Herberg

Received  July 2021 Revised  February 2022 Early access March 2022

We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments.

Citation: Evelyn Herberg, Michael Hinze. Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022013
References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, The Clarendon Press, Oxford University Press, 2000. 
[2]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, Mathematical Programming Society (MPS), Philadelphia, PA, 2006.

[3]

C. Bahriawati and C. Carstensen, Three MATLAB implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control, Computational Methods in Applied Mathematics, 5 (2005), 333-361.  doi: 10.2478/cmam-2005-0016.

[4]

S. Bartels, Total variation minimization with finite elements: Convergence and iterative solution, SIAM Journal on Numerical Analysis, 50 (2012), 1162-1180.  doi: 10.1137/11083277X.

[5]

S. Bartels and M. Milicevic, Iterative finite element solution of a constrained total variation regularized model problem, Discrete & Continuous Dynamical Systems S, 10 (2017), 1207-1232.  doi: 10.3934/dcdss.2017066.

[6]

K. Bredies and D. Vicente, A perfect reconstruction property for PDE-constrained totalvariation minimization with application in Quantitative Susceptibility Mapping, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019), Paper No. 83, 19 pp. doi: 10.1051/cocv/2018009.

[7]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[8]

E. CasasP. I. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the p-Laplace equation. Ⅰ. Convergence result, SIAM Journal on Control and Optimization, 54 (2016), 1406-1422.  doi: 10.1137/15M1028108.

[9]

E. CasasF. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM Journal on Control and Optimization, 55 (2017), 1752-1788.  doi: 10.1137/16M1056511.

[10]

E. Casas and K. Kunisch, Analysis of optimal control problems of semilinear elliptic equations by bv-functions, Set-Valued and Variational Analysis, 27 (2019), 355-379.  doi: 10.1007/s11228-018-0482-7.

[11]

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.

[12]

E. Casas and K. Kunisch, Using sparse control methods to identify sources in linear diffusion-convection equations, Inverse Problems, 35 (2019), 114002, 17pp. doi: 10.1088/1361-6420/ab331c.

[13]

E. CasasK. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Applied Mathematics and Optimization, 40 (1999), 229-257.  doi: 10.1007/s002459900124.

[14]

Y. Chen and W. Liu, Error estimates and superconvergence of mixed finite element for quadratic optimal control, International Journal of Numerical Analysis and Modeling, 3 (2006), 311-321. 

[15]

C. ClasonF. Kruse and K. Kunisch, Total variation regularization of multi-material topology optimization, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 275-303.  doi: 10.1051/m2an/2017061.

[16]

J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Mathematics of Computation, 44 (1985), 39-52.  doi: 10.1090/S0025-5718-1985-0771029-9.

[17]

R. G. Durán, Error analysis in $ L^p \le p \le \infty $, for mixed finite element methods for linear and quasi-linear elliptic problems, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 22 (1988), 371-387.  doi: 10.1051/m2an/1988220303711.

[18]

L. Gastaldi and R. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, Modélisation Mathématique et Analyse Numérique, 23 (1989), 103-128.  doi: 10.1051/m2an/1989230101031.

[19]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, Berlin Springer Verlag, 1979.

[20]

W. Gong and N. Yan, Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs, SIAM Journal on Control and Optimization, 49 (2011), 984-1014.  doi: 10.1137/100795632.

[21]

D. HafemeyerF. MannelI. Neitzel and B. Vexler, Finite element error estimates for one-dimensional elliptic optimal control by BV-functions, Mathematical Control and Related Fields, 10 (2010), 333-363.  doi: 10.3934/mcrf.2019041.

[22]

E. Herberg and M. Hinze, 6 Variational discretization approach applied to an optimal control problem with bounded measure controls, Optimization and Control for Partial Differential Equations, 2022, arXiv: 2003.14380. doi: 10.1515/9783110695984-006.

[23]

E. HerbergM. Hinze and H. Schumacher, Maximal discrete sparsity in parabolic optimal control with measures, Mathematical Control and Related Field, 10 (2020), 735-759.  doi: 10.3934/mcrf.2020018.

[24]

M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[25]

M. HinzeB. Kaltenbacher and T. N. T. Quyen, Identifying conductivity in electrical impedance tomography with total variation regularization, Numerische Mathematik, 138 (2018), 723-765.  doi: 10.1007/s00211-017-0920-8.

[26]

M. Hinze and T. N. T. Quyen, Finite element approximation of source term identification with TV-regularization, Inverse Problems, 35 (2019), 124004, 27pp. doi: 10.1088/1361-6420/ab3478.

[27]

J. Peypouquet, Convex Optimization in Normed Spaces: Theory, Methods and Examples, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0.

[28]

P.-A. Raviart and J.-M. Thomas, A mixed finite element method for 2-nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Springer, 1977,292–315.

[29]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Computational Optimization and Applications, 44 (2009), 159-181.  doi: 10.1007/s10589-007-9150-9.

[30]

P. Trautmann and D. Walter, A fast Primal-Dual-Active-Jump method for minimization in ${\text{BV}}((0, T);\mathbb{R}^d)$, arXiv preprint, arXiv: 2106.00633, 2021.

[31]

W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, The Clarendon Press, Oxford University Press, 2000. 
[2]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, Mathematical Programming Society (MPS), Philadelphia, PA, 2006.

[3]

C. Bahriawati and C. Carstensen, Three MATLAB implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control, Computational Methods in Applied Mathematics, 5 (2005), 333-361.  doi: 10.2478/cmam-2005-0016.

[4]

S. Bartels, Total variation minimization with finite elements: Convergence and iterative solution, SIAM Journal on Numerical Analysis, 50 (2012), 1162-1180.  doi: 10.1137/11083277X.

[5]

S. Bartels and M. Milicevic, Iterative finite element solution of a constrained total variation regularized model problem, Discrete & Continuous Dynamical Systems S, 10 (2017), 1207-1232.  doi: 10.3934/dcdss.2017066.

[6]

K. Bredies and D. Vicente, A perfect reconstruction property for PDE-constrained totalvariation minimization with application in Quantitative Susceptibility Mapping, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019), Paper No. 83, 19 pp. doi: 10.1051/cocv/2018009.

[7]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[8]

E. CasasP. I. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the p-Laplace equation. Ⅰ. Convergence result, SIAM Journal on Control and Optimization, 54 (2016), 1406-1422.  doi: 10.1137/15M1028108.

[9]

E. CasasF. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM Journal on Control and Optimization, 55 (2017), 1752-1788.  doi: 10.1137/16M1056511.

[10]

E. Casas and K. Kunisch, Analysis of optimal control problems of semilinear elliptic equations by bv-functions, Set-Valued and Variational Analysis, 27 (2019), 355-379.  doi: 10.1007/s11228-018-0482-7.

[11]

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.

[12]

E. Casas and K. Kunisch, Using sparse control methods to identify sources in linear diffusion-convection equations, Inverse Problems, 35 (2019), 114002, 17pp. doi: 10.1088/1361-6420/ab331c.

[13]

E. CasasK. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Applied Mathematics and Optimization, 40 (1999), 229-257.  doi: 10.1007/s002459900124.

[14]

Y. Chen and W. Liu, Error estimates and superconvergence of mixed finite element for quadratic optimal control, International Journal of Numerical Analysis and Modeling, 3 (2006), 311-321. 

[15]

C. ClasonF. Kruse and K. Kunisch, Total variation regularization of multi-material topology optimization, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 275-303.  doi: 10.1051/m2an/2017061.

[16]

J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Mathematics of Computation, 44 (1985), 39-52.  doi: 10.1090/S0025-5718-1985-0771029-9.

[17]

R. G. Durán, Error analysis in $ L^p \le p \le \infty $, for mixed finite element methods for linear and quasi-linear elliptic problems, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 22 (1988), 371-387.  doi: 10.1051/m2an/1988220303711.

[18]

L. Gastaldi and R. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, Modélisation Mathématique et Analyse Numérique, 23 (1989), 103-128.  doi: 10.1051/m2an/1989230101031.

[19]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, Berlin Springer Verlag, 1979.

[20]

W. Gong and N. Yan, Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs, SIAM Journal on Control and Optimization, 49 (2011), 984-1014.  doi: 10.1137/100795632.

[21]

D. HafemeyerF. MannelI. Neitzel and B. Vexler, Finite element error estimates for one-dimensional elliptic optimal control by BV-functions, Mathematical Control and Related Fields, 10 (2010), 333-363.  doi: 10.3934/mcrf.2019041.

[22]

E. Herberg and M. Hinze, 6 Variational discretization approach applied to an optimal control problem with bounded measure controls, Optimization and Control for Partial Differential Equations, 2022, arXiv: 2003.14380. doi: 10.1515/9783110695984-006.

[23]

E. HerbergM. Hinze and H. Schumacher, Maximal discrete sparsity in parabolic optimal control with measures, Mathematical Control and Related Field, 10 (2020), 735-759.  doi: 10.3934/mcrf.2020018.

[24]

M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[25]

M. HinzeB. Kaltenbacher and T. N. T. Quyen, Identifying conductivity in electrical impedance tomography with total variation regularization, Numerische Mathematik, 138 (2018), 723-765.  doi: 10.1007/s00211-017-0920-8.

[26]

M. Hinze and T. N. T. Quyen, Finite element approximation of source term identification with TV-regularization, Inverse Problems, 35 (2019), 124004, 27pp. doi: 10.1088/1361-6420/ab3478.

[27]

J. Peypouquet, Convex Optimization in Normed Spaces: Theory, Methods and Examples, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0.

[28]

P.-A. Raviart and J.-M. Thomas, A mixed finite element method for 2-nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Springer, 1977,292–315.

[29]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Computational Optimization and Applications, 44 (2009), 159-181.  doi: 10.1007/s10589-007-9150-9.

[30]

P. Trautmann and D. Walter, A fast Primal-Dual-Active-Jump method for minimization in ${\text{BV}}((0, T);\mathbb{R}^d)$, arXiv preprint, arXiv: 2106.00633, 2021.

[31]

W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

Figure 1.  The variationally discrete solution to the data from Example 1 for $ h = \tfrac{1}{2048} $. The inclusions in (31) are clearly visible
Figure 2.  Example 1: Convergence plots of the errors of the solutions to the variationally discrete problem compared to the known exact solution
Figure 3.  The variationally discrete solution to the data from Example 2 for $ h = \tfrac{1}{1024} $. The inclusions in (31) are clearly visible
Figure 4.  Example 2: Convergence plots of the errors of the solutions to the variationally discrete problem compared to the approximation of the exact solution. The reference solution is computed on a grid with $ h = \tfrac{1}{1024} $
Table 1.  Example 1: Convergence order (potency of gridsize $ h $) of the respective errors when the grid is refined from gridsize $ h_1 $ to gridsize $ h_2 $. We remark that the gridsizes are rounded
$ h_1 $ $ h_2 $ $ || \bar u - \bar u_h||_{1} $ $ || \bar u - \bar u_h||_{2} $ $ || \bar y - \bar y_h||_{2} $ $ || \bar p - \bar p_h||_{{\infty}} $ $ || \bar \Phi - \Phi_h||_{{\infty}} $
0.2500 0.1250 0.1943 -0.0171 0.2403 -0.2224 -0.4324
0.1250 0.0625 1.3436 0.7759 1.4444 1.1389 2.6278
0.0625 0.0313 1.1471 0.9368 1.7284 0.8966 1.5183
0.0313 0.0156 0.9982 0.4874 1.0286 1.0597 0.7761
0.0156 0.0078 1.4732 2.7648 -0.1603 0.4420 -2.0774
0.0078 0.0039 0.0178 -2.3127 1.6393 1.4590 3.8838
0.0039 0.0020 0.9832 0.4948 0.9920 0.9936 1.3328
0.0020 0.0010 0.9975 0.4975 0.9957 1.0184 0.3887
0.0010 0.0005 0.9353 0.4984 0.9025 0.9738 -0.6235
mean 0.8989 0.4584 0.9790 0.8622 0.8216
slope of best fit 0.9307 0.4854 1.0089 0.9241 0.9608
$ h_1 $ $ h_2 $ $ || \bar u - \bar u_h||_{1} $ $ || \bar u - \bar u_h||_{2} $ $ || \bar y - \bar y_h||_{2} $ $ || \bar p - \bar p_h||_{{\infty}} $ $ || \bar \Phi - \Phi_h||_{{\infty}} $
0.2500 0.1250 0.1943 -0.0171 0.2403 -0.2224 -0.4324
0.1250 0.0625 1.3436 0.7759 1.4444 1.1389 2.6278
0.0625 0.0313 1.1471 0.9368 1.7284 0.8966 1.5183
0.0313 0.0156 0.9982 0.4874 1.0286 1.0597 0.7761
0.0156 0.0078 1.4732 2.7648 -0.1603 0.4420 -2.0774
0.0078 0.0039 0.0178 -2.3127 1.6393 1.4590 3.8838
0.0039 0.0020 0.9832 0.4948 0.9920 0.9936 1.3328
0.0020 0.0010 0.9975 0.4975 0.9957 1.0184 0.3887
0.0010 0.0005 0.9353 0.4984 0.9025 0.9738 -0.6235
mean 0.8989 0.4584 0.9790 0.8622 0.8216
slope of best fit 0.9307 0.4854 1.0089 0.9241 0.9608
Table 2.  Example 2: Convergence order (potency of gridsize $ h $) of the respective errors when the grid is refined from gridsize $ h_1 $ to gridsize $ h_2 $. We remark that the gridsizes are rounded
$ h_1 $ $ h_2 $ $ || \bar u - \bar u_h||_{1} $ $ || \bar u - \bar u_h||_{2} $ $ || \bar y - \bar y_h||_{2} $ $ || \bar p -\bar p_h||_{{\infty}} $ $ || \bar \Phi - \Phi_h||_{{\infty}} $
0.2500 0.1250 0.3110 0.2454 1.0464 0.7448 1.5684
0.1250 0.0625 0.9990 0.5319 1.0788 0.8119 1.6061
0.0625 0.0313 0.9763 0.4961 1.0147 1.0266 -0.2077
0.0313 0.0156 0.9348 0.4682 0.9376 0.9737 1.3400
0.0156 0.0078 1.1204 0.5630 1.0757 1.1106 1.0238
0.0078 0.0039 0.7379 0.3679 0.6267 0.8702 0.3120
mean 0.8466 0.4454 0.9633 0.9230 0.9404
slope of best fit 0.9004 0.4679 0.9823 0.9450 0.9137
$ h_1 $ $ h_2 $ $ || \bar u - \bar u_h||_{1} $ $ || \bar u - \bar u_h||_{2} $ $ || \bar y - \bar y_h||_{2} $ $ || \bar p -\bar p_h||_{{\infty}} $ $ || \bar \Phi - \Phi_h||_{{\infty}} $
0.2500 0.1250 0.3110 0.2454 1.0464 0.7448 1.5684
0.1250 0.0625 0.9990 0.5319 1.0788 0.8119 1.6061
0.0625 0.0313 0.9763 0.4961 1.0147 1.0266 -0.2077
0.0313 0.0156 0.9348 0.4682 0.9376 0.9737 1.3400
0.0156 0.0078 1.1204 0.5630 1.0757 1.1106 1.0238
0.0078 0.0039 0.7379 0.3679 0.6267 0.8702 0.3120
mean 0.8466 0.4454 0.9633 0.9230 0.9404
slope of best fit 0.9004 0.4679 0.9823 0.9450 0.9137
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