Startseite Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

  • Ulrich Langer ORCID logo , Olaf Steinbach ORCID logo EMAIL logo und Huidong Yang
Veröffentlicht/Copyright: 10. Oktober 2021

Abstract

We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 -norm regularization is replaced by the H - 1 -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ h to the state u and the desired state u ¯ in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

Funding source: Austrian Science Fund

Award Identifier / Grant number: NFN S117-03

Funding statement: The third author was partly supported by the Austrian Science Fund (FWF), Grant No. NFN S117-03.

Acknowledgements

We would like to thank Fredi Tröltzsch (TU Berlin) for fruitful discussions during his visits at Linz and via ZOOM during the corona time. Special thanks goes to Volker John (WIAS Berlin) who drew our attention to the results known for differential filters.

References

[1] S. Badia, A. F. Martín and J. Principe, Implementation and scalability analysis of balancing domain decomposition methods, Arch. Comput. Methods Eng. 20 (2013), no. 3, 239–262. 10.1007/s11831-013-9086-4Suche in Google Scholar

[2] S. Badia, A. F. Martín and J. Principe, Multilevel balancing domain decomposition at extreme scales, SIAM J. Sci. Comput. 38 (2016), no. 1, C22–C52. 10.1137/15M1013511Suche in Google Scholar

[3] L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Sci. Comput., Springer, Berlin, 2006. Suche in Google Scholar

[4] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd ed., Cambridge University, Cambridge, 2007. 10.1017/CBO9780511618635Suche in Google Scholar

[5] A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for sparse matrix equations, Sparsity and its Applications, Cambridge University, Cambridge (1985), 257–284. Suche in Google Scholar

[6] W. L. Briggs, V. E. Henson and S. F. McCormick, A Multigrid Tutorial, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, 2000. 10.1137/1.9780898719505Suche in Google Scholar

[7] E. Casas, A review on sparse solutions in optimal control of partial differential equations, SeMA J. 74 (2017), no. 3, 319–344. 10.1007/s40324-017-0121-5Suche in Google Scholar

[8] C. R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput. 25 (2003), no. 1, 246–258. 10.1137/S1064827502412887Suche in Google Scholar

[9] H. Goering, A. Felgenhauer, G. Lube, H.-G. Roos and L. Tobiska, Singularly Perturbed Differential Equations, Math. Res. 13, Akademie-Verlag, Berlin, 1983. Suche in Google Scholar

[10] G. Haase and U. Langer, Multigrid methods: From geometrical to algebraic versions, Modern Methods in Scientific Computing and Applications, Springer, Dordrecht (2002), 103–153. 10.1007/978-94-010-0510-4_4Suche in Google Scholar

[11] W. Hackbusch, Multi-Grid Methods and Applications, Springer, Heidelberg, 1985. 10.1007/978-3-662-02427-0Suche in Google Scholar

[12] V. John, Finite Element Methods for Incompressible Flow Problems, Springer Ser. Comput. Math. 51, Springer, Cham, 2016. 10.1007/978-3-319-45750-5Suche in Google Scholar

[13] G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput. 20 (1998), no. 1, 359–392. 10.1137/S1064827595287997Suche in Google Scholar

[14] S. Kaya and C. C. Manica, Convergence analysis of the finite element method for a fundamental model in turbulence, Math. Models Methods Appl. Sci. 22 (2012), no. 11, Article ID 1250033. 10.1142/S0218202512500339Suche in Google Scholar

[15] F. Kickinger, Algebraic multi-grid for discrete elliptic second-order problems, Multigrid Methods V, Lect. Notes Comput. Sci. Eng. 3, Springer, Berlin (1998), 157–172. 10.1007/978-3-642-58734-4_9Suche in Google Scholar

[16] J. Kraus and M. Wolfmayr, On the robustness and optimality of algebraic multilevel methods for reaction-diffusion type problems, Comput. Vis. Sci. 16 (2013), no. 1, 15–32. 10.1007/s00791-014-0221-zSuche in Google Scholar

[17] U. Langer, M. Neumüller and A. Schafelner, Space-time finite element methods for parabolic evolution problems with variable coefficients, Advanced Finite Element Methods with Applications, Springer, Cham (2019), 247–275. 10.1007/978-3-030-14244-5_13Suche in Google Scholar

[18] U. Langer, O. Steinbach, F. Tröltzsch and H. Yang, Space-time finite element discretization of parabolic optimal control problems with energy regularization, SIAM J. Numer. Anal. 59 (2021), no. 2, 675–695. 10.1137/20M1332980Suche in Google Scholar

[19] U. Langer and H. Yang, Robust and efficient monolithic fluid-structure-interaction solvers, Internat. J. Numer. Methods Engrg. 108 (2016), no. 4, 303–325. 10.1002/nme.5214Suche in Google Scholar

[20] U. Langer and H. Yang, BDDC preconditioners for a space-time finite element discretization of parabolic problems, Domain Decomposition Methods in Science and Engineering XXV, Lect. Notes Comput. Sci. Eng. 138, Springer, Cham (2020), 367–374. 10.1007/978-3-030-56750-7_42Suche in Google Scholar

[21] W. J. Layton and L. G. Rebholz, Approximate Deconvolution Models of Turbulence. Analysis, Phenomenology and Numerical Analysis, Lecture Notes in Math. 2042, Springer, Heidelberg, 2012. 10.1007/978-3-642-24409-4Suche in Google Scholar

[22] J. Li and O. B. Widlund, FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg. 66 (2006), no. 2, 250–271. 10.1002/nme.1553Suche in Google Scholar

[23] S. MacLachlan and N. Madden, Robust solution of singularly perturbed problems using multigrid methods, SIAM J. Sci. Comput. 35 (2013), no. 5, A2225–A2254. 10.1137/120889770Suche in Google Scholar

[24] J. Mandel and C. R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl. 10 (2003), no. 7, 639–659. 10.1002/nla.341Suche in Google Scholar

[25] J. Mandel, C. R. Dohrmann and R. Tezaur, An algebraic theory for primal and dual substructuring methods by constraints, Appl. Numer. Math. 54 (2005), no. 2, 167–193. 10.1016/j.apnum.2004.09.022Suche in Google Scholar

[26] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University, Cambridge, 2000. Suche in Google Scholar

[27] F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal. 18 (1984), no. 1–2, 29–37. 10.1080/00036818408839508Suche in Google Scholar

[28] M. Neumüller and O. Steinbach, Regularization error estimates for distributed control problems in energy spaces, Math. Methods Appl. Sci. 44 (2021), no. 5, 4176–4191. 10.1002/mma.7021Suche in Google Scholar

[29] T. A. Nhan, S. MacLachlan and N. Madden, Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems, Numer. Algorithms 79 (2018), no. 1, 281–310. 10.1007/s11075-017-0437-3Suche in Google Scholar

[30] T. A. Nhan and N. Madden, An analysis of diagonal and incomplete Cholesky preconditioners for singularly perturbed problems on layer-adapted meshes, J. Appl. Math. Comput. 65 (2021), no. 1–2, 245–272. 10.1007/s12190-020-01390-zSuche in Google Scholar

[31] M. A. Olshanskii and A. Reusken, On the convergence of a multigrid method for linear reaction-diffusion problems, Computing 65 (2000), no. 3, 193–202. 10.1007/s006070070006Suche in Google Scholar

[32] C. Popa, Algebraic multigrid smoothing property of Kaczmarz’s relaxation for general rectangular linear systems, Electron. Trans. Numer. Anal. 29 (2007/08), 150–162. Suche in Google Scholar

[33] J. W. Ruge and K. Stüben, Algebraic multigrid, Multigrid Methods, Frontiers Appl. Math. 3, SIAM, Philadelphia (1987), 73–130. 10.1137/1.9781611971057.ch4Suche in Google Scholar

[34] A. H. Schatz and L. B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp. 40 (1983), no. 161, 47–89. 10.1090/S0025-5718-1983-0679434-4Suche in Google Scholar

[35] G. Stadler, Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices, Comput. Optim. Appl. 44 (2009), no. 2, 159–181. 10.1007/s10589-007-9150-9Suche in Google Scholar

[36] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements, Springer, New York, 2008. 10.1007/978-0-387-68805-3Suche in Google Scholar

[37] O. Steinbach, Space-time finite element methods for parabolic problems, Comput. Methods Appl. Math. 15 (2015), no. 4, 551–566. 10.1515/cmam-2015-0026Suche in Google Scholar

[38] O. Steinbach and H. Yang, Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D, Numer. Linear Algebra Appl. 25 (2018), no. 3, Article ID e2143. 10.1002/nla.2143Suche in Google Scholar

[39] A. Toselli and O. Widlund, Domain Decomposition Methods—Algorithms and Theory, Springer Ser. Comput. Math. 34, Springer, Berlin, 2005. 10.1007/b137868Suche in Google Scholar

[40] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Grad. Stud. Math. 112, American Mathematical Society, Providence, 2010. 10.1090/gsm/112/07Suche in Google Scholar

[41] R. Verfürth, A Posteriori Error Esimtation Techniques for Finite Element Methods, Oxford University, Oxford, 2013. 10.1093/acprof:oso/9780199679423.001.0001Suche in Google Scholar

[42] J. Xu and L. Zikatanov, Algebraic multigrid methods, Acta Numer. 26 (2017), 591–721. 10.1017/S0962492917000083Suche in Google Scholar

[43] H. Yang and W. Zulehner, Numerical simulation of fluid-structure interaction problems on hybrid meshes with algebraic multigrid methods, J. Comput. Appl. Math. 235 (2011), no. 18, 5367–5379. 10.1007/978-3-642-12535-5_12Suche in Google Scholar

[44] S. Zampini and X. Tu, Multilevel balancing domain decomposition by constraints deluxe algorithms with adaptive coarse spaces for flow in porous media, SIAM J. Sci. Comput. 39 (2017), no. 4, A1389–A1415. 10.1137/16M1080653Suche in Google Scholar

Received: 2021-09-10
Accepted: 2021-09-16
Published Online: 2021-10-10
Published in Print: 2022-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 29.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/cmam-2021-0169/html
Button zum nach oben scrollen