Abstract
In this paper we present a new steepest-descent type algorithm for convex optimization problems. Our algorithm pieces the unknown into sub-blocs of unknowns and considers a partial optimization over each sub-bloc. In quadratic optimization, our method involves Newton technique to compute the step-lengths for the sub-blocs resulting descent directions. Our optimization method is fully parallel and easily implementable, we first presents it in a general linear algebra setting, then we highlight its applicability to a parabolic optimal control problem, where we consider the blocs of unknowns with respect to the time dependency of the control variable. The parallel tasks, in the last problem, turn “on” the control during a specific time-window and turn it “off” elsewhere. We show that our algorithm significantly improves the computational time compared with recognized methods. Convergence analysis of the new optimal control algorithm is provided for an arbitrary choice of partition. Numerical experiments are presented to illustrate the efficiency and the rapid convergence of the method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alla, A., Falcone, M.: A time-adaptive pod method for optimal control problems. In: Proceedings of the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, pp 245–250 (2013)
Alla, A., Falcone, M.: An adaptive pod approximation method for the control of advection-diffusion equations. In: Control and Optimization with PDE Constraints, pp 1–17. Springer (2013)
Arian, E., Fahl, M., Sachs, E.W.: Trust-region proper orthogonal decomposition for flow control. Technical report, DTIC Document (2000)
Baksalary, J.K., Puntanen, S.: Generalized matrix versions of the cauchy-schwarz and kantorovich inequalities. Aequationes Mathematicae 41(1), 103–110 (1991)
Bank, R.E., Welfert, B.D., Yserentant, H.: A class of iterative methods for solving saddle point problems. Numer. Math. 56(7), 645–666 (1990)
Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer.l Anal. 8(1), 141–148 (1988)
Belgacem, F.B., Kaber, S.M.: On the Dirichlet boundary controllability of the one-dimensional heat equation: Semi-analytical calculations and ill-posedness degree. Inverse Probl. 27(5), 055012,19 (2011)
Bjorstad, P., Gropp, W.: Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (2004)
Borzì, A., Ito, K., Kunisch, K.: An optimal control approach to optical flow computation. Internat. J. Numer. Methods Fluids 40(1–2), 231–240 (2002). ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 2001)
Byrd, R.H., Lopez-Calva, G., Nocedal, J.: A line search exact penalty method using steering rules. Math. Program. 133(1–2, Ser. A), 39–73 (2012)
Carthel, C., Glowinski, R., Lions, J.-L.: On exact and approximate boundary controllabilities for the heat equation: a numerical approach. J. Optim. Theory Appl. 82(3), 429–484 (1994)
Cauchy, A.-L.: Methode générale pour la résolution des systèmes d’équations simultanéeś. Compte Rendu des Scieances de L’Académie des Sciences XXV S’erie A(25), 536–538 (1847)
Choi, H., Hinze, M., Kunisch, K.: Instantaneous control of backward-facing step flows. Appl. Numer. Math. 31(2), 133–158 (1999)
Choi, H., Temam, R., Moin, P., Kim, J.: Feedback control for unsteady flow and its application to the stochastic Burgers equation. J. Fluid Mech. 253, 509–543 (1993)
Ciarlet, P.G.: Introduction à l’analyse numérique matricielle et à l’optimisation. Collection Mathématiques Appliquées pour la Maî trise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris (1982)
Coron, J.-M., Trélat, E.: Global steady-state controllability of one-dimensional semilinear heat equations. SIAM J. Control Optim. 43(2), 549–569 (2004)
Ervedoza, S., Zuazua, E.: The wave equation: control and numerics. In: Control of Partial Differential Equations, Lecture Notes in Mathematics, pp 245–339. Springer, Berlin, Heidelberg (2012)
Evans, D.J.: Preconditioning methods: theory and applications. Gordon and Breach Science Publishers, Inc. (1983)
Fattorini, H.O.: Infinite-dimensional optimization and control theory, volume 62 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1999)
Graham, W.R., Peraire, J., Tang, K.Y.: Optimal control of vortex shedding using low-order models. I, Open-loop model development. Internat. J. Numer. Methods Engrg. 44(7), 945–972 (1999)
Graham, W.R., Peraire, J., Tang, K.Y.: Optimal control of vortex shedding using low-order models. II, Model-based control. Internat. J. Numer. Methods Engrg. 44(7), 973–990 (1999)
Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for newton’s method. SIAM J. Numer.l Anal. 23(4), 707–716 (1986)
Grote, M.J., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18(3), 838–853 (1997)
Gubisch, M., Stefan, V.: POD a-posteriori error analysis for optimal control problems with mixed control-state constraints. Comput. Optim. Appl. 58(3), 619–644 (2014)
Heinkenschloss, M.: A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems. J. Comput. Appl. Math. 173(1), 169–198 (2005)
Hinze, M., Kunisch, K.: Suboptimal control strategies for backward facing step ows. In: Proceedings of 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Editor A. Sydow, vol. 3, pp 53–58. Citeseer (1997)
Hinze, M.: Optimal and Instantaneous Control of the Instationary Navier-Stokes Equations (2000)
Hinze, M., Kunisch, K.: Three control methods for time-dependent fluid flow. Flow Turbul. Combust. 65(3–4), 273–298 (2000)
Ito, K., Ravindran, S.S.: A reduced basis method for control problems governed by PDEs. In: Control and Estimation of Distributed Parameter Systems (Vorau, 1996), Volume 126 of Internat. Ser. Numer. Math., pp 153–168. Basel, Birkhäuser (1998)
Ito, K., Ravindran, S.S.: Reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid Dyn. 15(2), 97–113 (2001)
Kantorovich, L.V.: Functional analysis and applied mathematics. NBS Rep. 1509. U. S. Department of Commerce National Bureau of Standards, Los Angeles, Calif. Translated by C. D. Benster (1952)
Kunisch, K., Volkwein, S., Xie, L.: HJB-POD-based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3(4), 701–722 (2004)
Lagnese, J.E., Leugering, G.: Time-domain decomposition of optimal control problems for the wave equation. Syst. Control Lett. 48(3–4), 229–242 (2003). Optimization and control of distributed systems
Lasiecka, I., Triggiani, R.: Control theory for partial differential equations: continuous and approximation theories. I-II, Volume 47-75 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2000). Abstract hyperbolic-like systems over a finite time horizon
Lee, C., Kim, J., Choi, H.: Suboptimal control of turbulent channel flow for drag reduction. J. Fluid Mech. 358, 245–258 (1998)
Lions, J., Maday, Y., Turinici, G.: A “parareal” in time discretization of pde’s. Comptes Rendus de l’Academie des Sciences Series I Mathematics 332(7), 661–668 (2001)
Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band, vol. 170. Springer, New York (1971)
Lions, J.-L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1-2, volume 8-9 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch (1988)
Maday, Y., Salomon, J., Turinici, G.: Monotonic parareal control for quantum systems. SIAM J. Numer. Anal. 45(6), 2468–2482 (2007)
Maday, Y., Riahi, M.-K., Salomon, J.: Parareal in time intermediate targets methods for optimal control problems. In: Bredies, K., Clason, C., Kunisch, K., von Winckel, G. (eds.) Control and Optimization with PDE Constraints, Volume 164 of International Series of Numerical Mathematics, pp 79–92. Springer, Basel (2013)
Maday, Y., Turinici, G.: A parareal in time procedure for the control of partial differential equations. C. R. Math. Acad. Sci. Paris 335(4), 387–392 (2002)
Malas, T., Gürel, L.: Incomplete lu preconditioning with the multilevel fast multipole algorithm for electromagnetic scattering. SIAM J. Sci. Comput. 29(4), 1476–1494 (2007)
Marduel, X., Kunisch, K.: Suboptimal control of transient nonisothermal viscoelastic fluid flows. Phys. Fluids (1994-present) 13(9), 2478–2491 (2001)
Micu, S., Zuazua, E.: Regularity issues for the null-controllability of the linear 1-d heat equation. Syst. Control Lett. 60(6), 406–413 (2011)
Pironneau, O., Hecht, F., Morice, J.: freefem++, www.freefem.org/ (2013)
Quarteroni, A., Valli, A.: Domain Decomposition Methods For Partial Differential Equations. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York (1999). Oxford Science Publications
Ravindran, S.S.: Proper Orthogonal Decomposition in Optimal Control of Fluids, volume 99. National Aeronautics and Space Administration, Langley Research Center (1999)
Rees, T., Stoll, M., Wathen, A.: All-at-once preconditioning in PDE-constrained optimization. Kybernetika (Prague) 46(2), 341–360 (2010)
Rüde, U.: Mathematical and computational techniques for multilevel adaptive methods. SIAM (1993)
Saad, Y.: Iterative methods for sparse linear systems. Siam (2003)
Scilab Enterprises: Scilab: Le logiciel open source gratuit de calcul numrique. Scilab Enterprises, Orsay, France (2012)
Silvester, D., Wathen, A.: Fast iterative solution of stabilised Stokes systems. II, Using general block preconditioners. SIAM J. Numer. Anal. 31(5), 1352–1367 (1994)
Stoll, M., Wathen, A.: All-at-once solution of time-dependent Stokes control. J. Comput. Phys. 232, 498–515 (2013)
Toselli, A., Widlund, O.: Domain decomposition methods: Algorithms and theory, vol. 3. Springer (2005)
Tröltzsch, F.: Optimal control of partial differential equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels (2010)
Ulbrich, S.: Generalized sqp-methods with ?parareal? time-domain decomposition for time-dependent pde-constrained optimization. In: Real-Time PDE-constrained Optimization, vol. 3, pp 145–168. SIAM, Philadelphia (2007)
Wathen, A., Silvester, D.: Fast iterative solution of stabilised Stokes systems. I, Using simple diagonal preconditioners. SIAM J. Numer. Anal. 30(3), 630–649 (1993)
Yuan, G., Wei, Z.: The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems. Math. Model. Anal. 17(2), 203–216 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Riahi, M.K. A new approach to improve ill-conditioned parabolic optimal control problem via time domain decomposition. Numer Algor 72, 635–666 (2016). https://doi.org/10.1007/s11075-015-0060-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-015-0060-0