zbMATH — the first resource for mathematics

POD-Galerkin model order reduction for parametrized time dependent linear quadratic optimal control problems in saddle point formulation. (English) Zbl 1444.49013
In a parametrized optimal control problem OCP(\(\mu\)), the parameter \(\mu\) could represent some physical, natural or geometrical features. The present article investigates OCP(\(\mu\))s to quadratic optimization models constrained to linear time dependent PDE(\(\mu\)), in the form they are formulated in the papers of [F. Negri et al., Comput. Math. Appl. 69, No. 4, 319–336 (2015; Zbl 1421.49026); SIAM J. Sci. Comput. 35, No. 5, A2316–A2340 (2013; Zbl 1280.49046)]. The second section starts with the presentation of parabolic time dependent OCP(\(\mu\))s in saddle point formulation. Under some special assumptions and using an earlier “Brezzi theorem” the well-posedness of the saddle point structure in space-time formulation is proved. A discretized version of time dependent OCP(\(\mu\))s by means of an all-at-once space-time discretization procedure is obtained. The third section is devoted to the application of R0M (reduce order methods) approximations to time dependent OCP(\(\mu\)). The P0D-Galerkin algorithm is presented and it is extended to parametrized time dependent OCP(\(\mu\))s. In order to validate the performances of theoretical results, in the forth and fifth sections, two examples are presented and largely discussed: a time dependent boundary optimal control problem for Graetz flow and a time dependent distributed optimal control problem for Stokes equations, both examples having geometrical and phisical parametrizations. Numerical results for a time dependent OCP(\(\mu\)) for a Cavity Viscous Flow are shown in the sixth section. Conclusions, perspectives, next improvements and References with an impressive number of 60 titles, end the article.

49N10 Linear-quadratic optimal control problems
49M25 Discrete approximations in optimal control
35J20 Variational methods for second-order elliptic equations
65N06 Finite difference methods for boundary value problems involving PDEs
90C20 Quadratic programming
Full Text: DOI
[1] Ali, S.; Ballarin, F.; Rozza, G., Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations, Comput. Math. Appl. (2020)
[2] Babuška, I., Error-bounds for finite element method, Numer. Math., 16, 4, 322-333 (1971) · Zbl 0214.42001
[3] Bader, E.; Kärcher, M.; Grepl, MA; Veroy, K., Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints, SIAM J. Sci. Comput., 38, 6, A3921-A3946 (2016) · Zbl 1426.49029
[4] Bader, E.; Kärcher, M.; Grepl, MA; Veroy-Grepl, K., A certified reduced basis approach for parametrized linear-quadratic optimal control problems with control constraints, IFAC-PapersOnLine, 48, 1, 719-720 (2015)
[5] Ballarin, F.; Faggiano, E.; Manzoni, A.; Quarteroni, A.; Rozza, G.; Ippolito, S.; Antona, C.; Scrofani, R., Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts, Biomech. Model. Mechanobiol., 16, 4, 1373-1399 (2017)
[6] Ballarin, F.; Manzoni, A.; Quarteroni, A.; Rozza, G., Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Eng., 102, 5, 1136-1161 (2015) · Zbl 1352.76039
[7] Barrault, M.; Maday, Y.; Nguyen, NC; Patera, AT, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339, 9, 667-672 (2004) · Zbl 1061.65118
[8] Benzi, M.; Golub, GH; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034
[9] Bochev, PB; Gunzburger, MD, Least-Squares Finite Element Methods (2009), New York: Springer, New York
[10] Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), Berlin: Springer, Berlin · Zbl 1277.65092
[11] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, ESAIM Math. Model. Numer. Anal., 8, R2, 129-151 (1974) · Zbl 0338.90047
[12] Burkardt, J.; Gunzburger, M.; Lee, H., POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comput. Methods Appl. Mech. Eng., 196, 1-3, 337-355 (2006) · Zbl 1120.76323
[13] Chapelle, D.; Gariah, A.; Moireau, P.; Sainte-Marie, J., A Galerkin strategy with proper orthogonal decomposition for parameter-dependent problems: analysis, assessments and applications to parameter estimation, ESAIM Math. Model. Numer. Anal., 47, 6, 1821-1843 (2013) · Zbl 1295.65096
[14] De los Reyes, J.C., Tröltzsch, F.: Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints. SIAM J. Control Optim. 46(2), 604-629 (2007) · Zbl 1356.49034
[15] Dedè, L., Optimal flow control for Navier-Stokes equations: drag minimization, Int. J. Numer. Methods Fluids, 55, 4, 347-366 (2007) · Zbl 1388.76074
[16] Dedè, L., Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems, SIAM J. Sci. Comput., 32, 2, 997-1019 (2010) · Zbl 1221.35030
[17] Delfour, MC; Zolésio, J., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization (2011), Philadelphia: SIAM, Philadelphia · Zbl 1251.49001
[18] Eriksson, K.; Johnson, C., Error estimates and automatic time step control for nonlinear parabolic problems. I, SIAM J. Numer. Anal., 24, 1, 12-23 (1987) · Zbl 0618.65104
[19] Gerner, AL; Veroy, K., Certified reduced basis methods for parametrized saddle point problems, SIAM J. Sci. Comput., 34, 5, A2812-A2836 (2012) · Zbl 1255.76024
[20] Glas, S.; Mayerhofer, A.; Urban, K., Two Ways to Treat Time in Reduced Basis Methods, 1-16 (2017), Cham: Springer, Cham · Zbl 1448.65158
[21] Guberovic, R.; Schwab, C.; Stevenson, R., Space-time variational saddle point formulations of stokes and Navier-Stokes equations, ESAIM Math. Model. Numer. Anal., 48, 3, 875-894 (2014) · Zbl 1295.35354
[22] Haslinger, J.; Mäkinen, RAE, Introduction to Shape Optimization: Theory, Approximation, and Computation (2003), Philadelphia: SIAM, Philadelphia · Zbl 1020.74001
[23] Hesthaven, JS; Rozza, G.; Stamm, B., Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics (2015), Milan: Springer, Milan
[24] Hinze, M., Köster, M., Turek, S.: A hierarchical space-time solver for distributed control of the Stokes equation. Technical Report, SPP1253-16-01 (2008)
[25] Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S., Optimization with PDE Constraints (2008), Antwerp: Springer, Antwerp · Zbl 1167.49001
[26] Iapichino, L.; Trenz, S.; Volkwein, S.; Karasözen, B.; Manguoğlu, M.; Tezer-Sezgin, M.; Göktepe, S.; Uğur, Ö., Reduced-order multiobjective optimal control of semilinear parabolic problems, Numerical Mathematics and Advanced Applications ENUMATH 2015, 389-397 (2016), Cham: Springer, Cham
[27] Iapichino, L.; Ulbrich, S.; Volkwein, S., Multiobjective PDE-constrained optimization using the reduced-basis method, Adv. Comput. Math., 43, 5, 945-972 (2017) · Zbl 1386.35068
[28] Kärcher, M.; Grepl, MA, A certified reduced basis method for parametrized elliptic optimal control problems, ESAIM Control Optim. Calc. Var., 20, 2, 416-441 (2014) · Zbl 1287.49032
[29] Kärcher, M.; Tokoutsi, Z.; Grepl, MA; Veroy, K., Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls, J. Sci. Comput., 75, 1, 276-307 (2018) · Zbl 1388.49023
[30] Kunisch, K.; Volkwein, S., Proper orthogonal decomposition for optimality systems, ESAIM Math. Model. Numer. Anal., 42, 1, 1-23 (2008) · Zbl 1141.65050
[31] Lassila, T.; Manzoni, A.; Quarteroni, A.; Rozza, G., A reduced computational and geometrical framework for inverse problems in hemodynamics, Int. J. Numer. Methods Biomed. Eng., 29, 7, 741-776 (2013)
[32] Leugering, G.; Benner, P.; Engell, S.; Griewank, A.; Harbrecht, H.; Hinze, M.; Rannacher, R.; Ulbrich, S., Trends in PDE Constrained Optimization (2014), New York: Springer, New York · Zbl 1306.49001
[33] Lions, JL, Optimal Control of System Governed by Partial Differential Equations (1971), Berlin: Springer, Berlin
[34] Logg, A.; Mardal, K.; Wells, G., Automated Solution of Differential Equations by the Finite Element Method (2012), Berlin: Springer, Berlin
[35] Mohammadi, B.; Pironneau, O., Applied Shape Optimization for Fluids (2010), New York: Oxford University Press, New York
[36] Negri, F.; Manzoni, A.; Rozza, G., Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations, Comput. Math. Appl., 69, 4, 319-336 (2015) · Zbl 1421.49026
[37] Negri, F.; Rozza, G.; Manzoni, A.; Quarteroni, A., Reduced basis method for parametrized elliptic optimal control problems, SIAM J. Sci. Comput., 35, 5, A2316-A2340 (2013) · Zbl 1280.49046
[38] Pošta, M.; Roubíček, T., Optimal control of Navier-Stokes equations by Oseen approximation, Comput. Math. Appl., 53, 3, 569-581 (2007) · Zbl 1133.49024
[39] Prud’Homme, C.; Rovas, DV; Veroy, K.; Machiels, L.; Maday, Y.; Patera, A.; Turinici, G., Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, J. Fluids Eng., 124, 1, 70-80 (2002)
[40] Quarteroni, A.; Rozza, G.; Dedè, L.; Quaini, A.; Ceragioli, F.; Dontchev, A.; Futura, H.; Marti, K.; Pandolfi, L., Numerical approximation of a control problem for advection-diffusion processes, IFIP Conference on System Modeling and Optimization, System Modeling and Optimization, CSMO, 261-273 (2005), Boston: Springer, Boston
[41] Quarteroni, A., Rozza, G., Quaini, A.: Reduced basis methods for optimal control of advection-diffusion problems. In: Advances in Numerical Mathematics, pp. 193-216. RAS and University of Houston, Moscow (2007)
[42] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (2008), Berlin: Springer, Berlin · Zbl 1151.65339
[43] RBniCS—reduced order modelling in FEniCS. http://mathlab.sissa.it/rbnics (2015)
[44] Rozza, G.; Huynh, D.; Manzoni, A., Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants, Numer. Math., 125, 1, 115-152 (2013) · Zbl 1318.76006
[45] Rozza, G.; Huynh, D.; Patera, A., 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, 3, 229-275 (2008) · Zbl 1304.65251
[46] Rozza, G., Manzoni, A., Negri, F.: Reduction strategies for PDE-constrained optimization problems in haemodynamics. In: ECCOMAS: Congress Proceedings, Vienna, pp. 1749-1768 (2012)
[47] Rozza, G.; Veroy, K., On the stability of the reduced basis method for Stokes equations in parametrized domains, Comput. Methods Appl. Mech. Eng., 196, 7, 1244-1260 (2007) · Zbl 1173.76352
[48] Schöberl, J.; Zulehner, W., Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimisation problems, SIAM J. Matrix Anal. Appl., 29, 3, 752-773 (2007) · Zbl 1154.65029
[49] Schwab, C.; Stevenson, R., Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comput., 78, 267, 1293-1318 (2009) · Zbl 1198.65249
[50] Seymen, ZK; Yücel, H.; Karasözen, B., Distributed optimal control of time-dependent diffusion-convection-reaction equations using space-time discretization, J. Comput. Appl. Math., 261, 146-157 (2014) · Zbl 1278.49036
[51] Stoll, M., Wathen, A.: All-at-once solution of time-dependent PDE-constrained optimization problems. Unspecified, Tech. Rep (2010) · Zbl 1195.65083
[52] Stoll, M.; Wathen, A., All-at-once solution of time-dependent Stokes control, J. Comput. Phys., 232, 1, 498-515 (2013)
[53] Strazzullo, M.; Ballarin, F.; Mosetti, R.; Rozza, G., Model reduction for parametrized optimal control problems in environmental marine sciences and engineering, SIAM J. Sci. Comput., 40, 4, B1055-B1079 (2018) · Zbl 1395.49015
[54] Strazzullo, M., Zainib, Z., Ballarin, F., Rozza, G.: Reduced order methods for parametrized nonlinear and time dependent optimal flow control problems: towards applications in biomedical and environmental sciences. ENUMATH 2019 Proceedings (2020)
[55] Tröltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics, vol. 112. Verlag, Wiesbad (2010) · Zbl 1195.49001
[56] Urban, K.; Patera, AT, A new error bound for reduced basis approximation of parabolic partial differential equations, C. R. Math., 350, 3-4, 203-207 (2012) · Zbl 1242.35157
[57] Yano, M., A space-time Petrov-Galerkin certified reduced basis method: application to the Boussinesq equations, SIAM J. Sci. Comput., 36, 1, A232-A266 (2014) · Zbl 1288.35275
[58] Yano, M.; Patera, AT; Urban, K., A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation, Math. Models Methods Appl. Sci., 24, 9, 1903-1935 (2014) · Zbl 1295.65098
[59] Yilmaz, F., Karasözen, B.: An all-at-once approach for the optimal control of the unsteady Burgers equation. J. Comput. Appl. Math. 259, 771-779 (2014). Recent Advances in Applied and Computational Mathematics: ICACM-IAM-METU · Zbl 1318.49052
[60] Zainib, Z.; Ballarin, F.; Rozza, G.; Triverio, P.; Jiménez-Juan, L.; Fremes, S., Reduced order methods for parametric optimal flow control in coronary bypass grafts, towards patient-specific data assimilation, Int. J. Numer. Methods Biomed. Eng. (2020)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.