Abstract
In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.
Funding statement: We acknowledge the support by European Union Funding for Research and Innovation – Horizon 2020 Program – in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics”. We also acknowledge the PRIN 2017 “Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM-GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”.
Acknowledgements
The computations in this work have been performed with RBniCS [58] library, developed at SISSA mathLab, which is an implementation in FEniCS [31] of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.
References
[1] T. Akman, B. Karasözen and Z. Kanar-Seymen, Streamline upwind/Petrov Galerkin solution of optimal control problems governed by time dependent diffusion-convection-reaction equations, TWMS J. Appl. Eng. Math. 7 (2017), no. 2, 221–235. Search in Google Scholar
[2] S. Ali, Stabilized reduced basis methods for the approximation of parametrized viscous flows, Ph.D. Thesis, SISSA, 2018. Search in Google Scholar
[3] P. R. Amestoy, A. Buttari, J.-Y. L’Excellent and T. Mary, Performance and scalability of the block low-rank multifrontal factorization on multicore architectures, ACM Trans. Math. Software 45 (2019), no. 1, 1–26. 10.1145/3242094Search in Google Scholar
[4] P. R. Amestoy, I. S. Duff, J.-Y. L’Excellent and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl. 23 (2001), no. 1, 15–41. 10.1137/S0895479899358194Search in Google Scholar
[5] K. Atkinson and W. Han, Theoretical Numerical Analysis, Texts Appl. Math. 39, Springer, New York, 2005. 10.1007/978-0-387-28769-0Search in Google Scholar
[6] F. Ballarin, G. Rozza and M. Strazzullo, Space-time POD-Galerkin approach for parametric flow control, Numerical Control. Part A, Handb. Numer. Anal. 23, North-Holland, Amsterdam (2022), 307–338. 10.1016/bs.hna.2021.12.009Search in Google Scholar
[7] P. Benner, M. Ohlberger, A. Patera, G. Rozza and K. Urban, Model Reduction of Parametrized Systems, Model. Simul. Appl. 17, Springer, Cham, 2017. 10.1007/978-3-319-58786-8Search in Google Scholar
[8] A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), no. 1–3, 199–259. 10.1016/0045-7825(82)90071-8Search in Google Scholar
[9] G. Carere, Reduced order methods for optimal control problems constrained by PDEs with random inputs and applications, Master’s thesis, University of Amsterdam and SISSA, 2019. Search in Google Scholar
[10] R. Chakir, Y. Maday and P. Parnaudeau, A non-intrusive reduced basis approach for parametrized heat transfer problems, J. Comput. Phys. 376 (2019), 617–633. 10.1016/j.jcp.2018.10.001Search in Google Scholar
[11] S. S. Collis and M. Heinkenschloss, Analysis of the Streamline Upwind/Petrov Galerkin method applied to the solution of optimal control problems, Report CAAM TR02-01, Rice University, 2002. Search in Google Scholar
[12] L. Dedè, Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems, SIAM J. Sci. Comput. 32 (2010), no. 2, 997–1019. 10.1137/090760453Search in Google Scholar
[13] K. Eriksson and C. Johnson, Error estimates and automatic time step control for nonlinear parabolic problems. I, SIAM J. Numer. Anal. 24 (1987), no. 1, 12–23. 10.1137/0724002Search in Google Scholar
[14] S. Ganesan, An operator-splitting Galerkin/SUPG finite element method for population balance equations: Stability and convergence, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 6, 1447–1465. 10.1051/m2an/2012012Search in Google Scholar
[15] F. Gelsomino and G. Rozza, Comparison and combination of reduced-order modelling techniques in 3D parametrized heat transfer problems, Math. Comput. Model. Dyn. Syst. 17 (2011), no. 4, 371–394. 10.1080/13873954.2011.547672Search in Google Scholar
[16] A.-L. Gerner and K. Veroy, Certified reduced basis methods for parametrized saddle point problems, SIAM J. Sci. Comput. 34 (2012), no. 5, A2812–A2836. 10.1137/110854084Search in Google Scholar
[17] S. Giere, T. Iliescu, V. John and D. Wells, SUPG reduced order models for convection-dominated convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Engrg. 289 (2015), 454–474. 10.1016/j.cma.2015.01.020Search in Google Scholar
[18] R. Guberovic, C. Schwab and R. Stevenson, Space-time variational saddle point formulations of Stokes and Navier–Stokes equations, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 3, 875–894. 10.1051/m2an/2013124Search in Google Scholar
[19] M. Heinkenschloss and D. Leykekhman, Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems, SIAM J. Numer. Anal. 47 (2010), no. 6, 4607–4638. 10.1137/090759902Search in Google Scholar
[20] J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs Math., Springer, Cham, 2016. 10.1007/978-3-319-22470-1Search in Google Scholar
[21] M. Hinze, M. Köster and S. Turek, A hierarchical space-time solver for distributed control of the Stokes equation, Technical Report SPP 1253-16-01, Universität Koblenz, 2008. Search in Google Scholar
[22] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Math. Model. Theory Appl. 23, Springer, Dordrecht, 2008. Search in Google Scholar
[23] T. J. R. Hughes, Finite Element Methods for Convection Dominated Flows, ASME, New York, 1979. Search in Google Scholar
[24] T. J. R. Hughes, Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier–Stokes equations, Internat. J. Numer. Methods Fluids 7 (1987), no. 11, 1261–1275. 10.1002/fld.1650071108Search in Google Scholar
[25] V. John and J. Novo, Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations, SIAM J. Numer. Anal. 49 (2011), no. 3, 1149–1176. 10.1137/100789002Search in Google Scholar
[26] Z. Kanar Seymen, H. Yücel and B. Karasözen, Distributed optimal control of time-dependent diffusion-convection-reaction equations using space-time discretization, J. Comput. Appl. Math. 261 (2014), 146–157. 10.1016/j.cam.2013.11.006Search in Google Scholar
[27] M. Kärcher, Z. Tokoutsi, M. A. Grepl and K. Veroy, Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls, J. Sci. Comput. 75 (2018), no. 1, 276–307. 10.1007/s10915-017-0539-zSearch in Google Scholar
[28] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems, M2AN Math. Model. Numer. Anal. 42 (2008), no. 1, 1–23. 10.1051/m2an:2007054Search in Google Scholar
[29] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss. 170, Springer, New York, 1971. 10.1007/978-3-642-65024-6Search in Google Scholar
[30] J.-L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems, Society for Industrial and Applied Mathematics, Philadelphia, 1972. 10.1137/1.9781611970616Search in Google Scholar
[31] A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method, Lect. Notes Comput. Sci. Eng. 84, Springer, Heidelberg, 2012. 10.1007/978-3-642-23099-8Search in Google Scholar
[32] Y. Maday and E. Tadmor, Analysis of the spectral vanishing viscosity method for periodic conservation laws, SIAM J. Numer. Anal. 26 (1989), no. 4, 854–870. 10.1137/0726047Search in Google Scholar
[33] A. Manzoni, A. Quarteroni and S. Salsa, Optimal Control of Partial Differential Equations—Analysis, Approximation, and Applications, Appl. Math. Sci. 207, Springer, Cham, 2021. 10.1007/978-3-030-77226-0Search in Google Scholar
[34] F. Negri, Reduced basis method for parametrized optimal control problems governed by PDEs, Master’s thesis, Politecnico di Milano and EPFL, 2011. Search in Google Scholar
[35] F. Negri, A. Manzoni and G. Rozza, Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations, Comput. Math. Appl. 69 (2015), no. 4, 319–336. 10.1016/j.camwa.2014.12.010Search in Google Scholar
[36] F. Negri, G. Rozza, A. Manzoni and A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems, SIAM J. Sci. Comput. 35 (2013), no. 5, A2316–A2340. 10.1137/120894737Search in Google Scholar
[37] P. Pacciarini and G. Rozza, Stabilized reduced basis method for parametrized advection-diffusion PDEs, Comput. Methods Appl. Mech. Engrg. 274 (2014), 1–18. 10.1016/j.cma.2014.02.005Search in Google Scholar
[38] A. Quarteroni, Numerical Models for Differential Problems, Model. Simul. Appl. 2, Springer, Milan, 2009. 10.1007/978-88-470-1071-0Search in Google Scholar
[39] A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, Unitext 92, Springer, Cham, 2016. 10.1007/978-3-319-15431-2Search in Google Scholar
[40] A. Quarteroni and G. Rozza, Reduced Order Methods for Modeling and Computational Reduction, Model. Simul. Appl. 9, Springer, Cham, 2014. 10.1007/978-3-319-02090-7Search in Google Scholar
[41] A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications, J. Math. Ind. 1 (2011), 1–49. 10.1186/2190-5983-1-3Search in Google Scholar
[42] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 1994. 10.1007/978-3-540-85268-1Search in Google Scholar
[43] G. Rozza, D. B. P. Huynh and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics, Arch. Comput. Methods Eng. 15 (2008), no. 3, 229–275. 10.1007/s11831-008-9019-9Search in Google Scholar
[44] G. Rozza, N.-C. Nguyen, A. T. Patera and S. Deparis, Reduced basis methods and a posteriori error estimators for heat transfer problems, Heat Transfer Summer Conference, ASME, New York (2009), 753–762. 10.1115/HT2009-88211Search in Google Scholar
[45] M. Stoll and A. J. Wathen, All-at-once solution of time-dependent PDE-constrained optimization problems, Technical Report, 2010. Search in Google Scholar
[46] M. Stoll and A. Wathen, All-at-once solution of time-dependent Stokes control, J. Comput. Phys. 232 (2013), 498–515. 10.1016/j.jcp.2012.08.039Search in Google Scholar
[47] M. Strazzullo, Model Order Reduction for Nonlinear and Time-Dependent Parametric Optimal Flow Control Problems, PhD. Thesis, SISSA, 2021. Search in Google Scholar
[48] M. Strazzullo, F. Ballarin, R. Mosetti and G. Rozza, Model reduction for parametrized optimal control problems in environmental marine sciences and engineering, SIAM J. Sci. Comput. 40 (2018), no. 4, B1055–B1079. 10.1137/17M1150591Search in Google Scholar
[49] M. Strazzullo, F. Ballarin and G. Rozza, POD-Galerkin model order reduction for parametrized time dependent linear quadratic optimal control problems in saddle point formulation, J. Sci. Comput. 83 (2020), no. 3, Paper No. 55. 10.1007/s10915-020-01232-xSearch in Google Scholar
[50] M. Strazzullo, F. Ballarin and G. Rozza, A certified reduced basis method for linear parametrized parabolic optimal control problems in space-time formulation, preprint (2021), https://arxiv.org/abs/2103.00460. Search in Google Scholar
[51] M. Strazzullo, F. Ballarin and G. Rozza, POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: An application to shallow water equations, J. Numer. Math. 30 (2022), no. 1, 63–84. 10.1515/jnma-2020-0098Search in Google Scholar
[52] D. Torlo, Stabilized reduced basis method for transport PDEs with random inputs, Master’s thesis, University of Trieste and SISSA, 2016. Search in Google Scholar
[53] D. Torlo, F. Ballarin and G. Rozza, Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs, SIAM/ASA J. Uncertain. Quantif. 6 (2018), no. 4, 1475–1502. 10.1137/17M1163517Search in Google Scholar
[54] F. Tröltzsch, Optimal Control Of Partial Differential Equations, Theory Methods Appl. 112, American Mathematical Society, Providence, 2010. 10.1090/gsm/112/07Search in Google Scholar
[55] K. Urban and A. T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations, C. R. Math. Acad. Sci. Paris 350 (2012), no. 3–4, 203–207. 10.1016/j.crma.2012.01.026Search in Google Scholar
[56] L. Venturi, Weighted Reduced Order Methods for parametrized PDEs in uncertainty quantification problems, Master’s thesis, University of Trieste and SISSA, 2016. Search in Google Scholar
[57] Multiphenics – Easy prototyping of multiphysics problems in FEniCS, https://mathlab.sissa.it/multiphenics. Search in Google Scholar
[58] RBniCS – Reduced order modelling in FEniCS, https://www.rbnicsproject.org/. Search in Google Scholar
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