Skip to main content
SpringerLink
Log in

A Brief Introduction to PDE-Constrained Optimization

  • Chapter
Frontiers in PDE-Constrained Optimization

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 163))

Abstract

In this chapter we give a brief overview of optimization problems with partial differential equation (PDE) constraints, i.e., PDE-constrained optimization (PDECO). We start with three potentially different formulations of a general PDECO problem and focus on the so-called reduced form. We present a derivation of the optimality conditions. Later we discuss the linear and the semilinear quadratic PDECO problems. We conclude with the discretization and the convergence rates for these problems. For illustration, we make a MATLAB code available at

https://bitbucket.org/harbirantil/pde_constrained_opt that solves the semilinear PDECO problem with control constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The chain rule only requires the outer function to be Hadamard directionally differentiable and the inner function to be Hadamard (Gâteaux) directionally differentiable [79, Proposition 3.6].

  2. 2.

    Note both the choices of spaces for S are motivated by the theory of Nemytskii or superposition operators. Care must to taken to ensure their differentiability [81, Section 4.3].

References

  1. R.A. Adams and J.J.F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 2003.

    MATH  Google Scholar 

  2. J.-J. Alibert and J.-P. Raymond. Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim., 18(3–4):235–250, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Antil, M. Heinkenschloss, and R. H.W. Hoppe. Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system. Optimization Methods and Software, 26(4–5):643–669, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Antil, M. Heinkenschloss, R.H.W. Hoppe, C. Linsenmann, and A. Wixforth. Reduced order modeling based shape optimization of surface acoustic wave driven microfluidic biochips. Math. Comput. Simul., 82(10):1986–2003, June 2012.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Antil, M. Heinkenschloss, R.H.W. Hoppe, and D. C. Sorensen. Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables. Comput. Vis. Sci., 13(6):249–264, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Antil, M. Heinkenschloss, and D. C. Sorensen. Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems. volume 8 of Springer MS&A series: Reduced Order Methods for modeling and computational G. Rozza, Eds. Springer-Verlag Italia, Milano, 2013.

    Google Scholar 

  7. H. Antil, M. Hintermüller, R. Nochetto, T. Surowiec, and D. Wegner. Finite horizon model predictive control of electrowetting on dielectric with pinning. Interfaces Free Bound., 19(1): 1–30, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Antil, R.H. Nochetto, and P. Venegas. Controlling the Kelvin force: Basic strategies and applications to magnetic drug targeting. arXiv preprint arXiv:1704.06872, 2017.

    Google Scholar 

  9. H. Antil, R.H. Nochetto, and P. Venegas. Optimizing the Kelvin force in a moving target subdomain. Math. Models Methods Appl. Sci., 28(1):95–130, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Antil, J. Pfefferer, and M. Warma. A note on semilinear fractional elliptic equation: analysis and discretization. Math. Model. Numer. Anal. (ESAIM: M2AN), 51:2049–2067, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Antil and M. Warma. Optimal control of fractional semilinear PDEs. arXiv preprint arXiv:1712.04336, 2017.

    Google Scholar 

  12. T. Apel, M. Mateos, J. Pfefferer, and A. Rösch. On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J. Control Optim., 53(6):3620–3641, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Arada, E. Casas, and F. Tröltzsch. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl., 23(2):201–229, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Attouch, G. Buttazzo, and G. Michaille. Variational analysis in Sobolev and BV spaces, volume 17 of MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, second edition, 2014. Applications to PDEs and optimization.

    Google Scholar 

  15. M. Benzi, G.H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numer., 14:1–137, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Berggren and M. Heinkenschloss. Parallel solution of optimal-control problems by time-domain decomposition. In Computational science for the 21st century. Symposium, pages 102–112, 1997.

    Google Scholar 

  17. P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal., 43(3):1457–1472, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  18. J.F. Bonnans and A. Shapiro. Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer-Verlag, New York, 2000.

    Google Scholar 

  19. S.C. Brenner and L.R. Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York, third edition, 2008.

    Book  MATH  Google Scholar 

  20. E. Casas. Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim., 24(6):1309–1318, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Casas. Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim., 31(4):993–1006, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Casas. Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems. Adv. Comput. Math., 26(1–3):137–153, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Casas, C. Clason, and K. Kunisch. Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim., 50(4):1735–1752, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Casas, C. Clason, and K. Kunisch. Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim., 51(1):28–63, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  25. E. Casas and K. Kunisch. Optimal control of semilinear elliptic equations in measure spaces. SIAM J. Control Optim., 52(1):339–364, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  26. E. Casas, M. Mateos, and J.-P. Raymond. Penalization of Dirichlet optimal control problems. ESAIM Control Optim. Calc. Var., 15(4):782–809, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Casas, M. Mateos, and F. Tröltzsch. Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl., 31(2):193–219, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Casas, M. Mateos, and B. Vexler. New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM Control Optim. Calc. Var., 20(3):803–822, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  29. E. Casas and F. Tröltzsch. A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl., 53(1):173–206, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Cohen, M. Hoffmann, and M. Reiß. Adaptive wavelet Galerkin methods for linear inverse problems. SIAM J. Numer. Anal., 42(4):1479–1501, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  31. W. Dahmen, A. Kunoth, and K. Urban. A wavelet Galerkin method for the Stokes equations. Computing, 56(3):259–301, 1996. International GAMM-Workshop on Multi-level Methods (Meisdorf, 1994).

    Article  MathSciNet  MATH  Google Scholar 

  32. X. Deng, X.-C. Cai, and J. Zou. Two-level space-time domain decomposition methods for three-dimensional unsteady inverse source problems. J. Sci. Comput., 67(3):860–882, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  33. X. Deng and M. Heinkenschloss. A parallel-in-time gradient-type method for discrete time optimal control problems. 2016.

    Google Scholar 

  34. A. Ern and J.L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.

    Book  MATH  Google Scholar 

  35. R.S. Falk. Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl., 44:28–47, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  36. P.E. Farrell, D.A. Ham, S.W. Funke, and M.E. Rognes. Automated derivation of the adjoint of high-level transient finite element programs. SIAM J. Sci. Comput., 35(4):C369–C393, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  37. G.B. Folland. Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999. Modern techniques and their applications, A Wiley-Interscience Publication.

    Google Scholar 

  38. D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

    Chapter  MATH  Google Scholar 

  39. V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1986. Theory and algorithms.

    Google Scholar 

  40. P. Grisvard. Singularities in boundary value problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris, 1992.

    MATH  Google Scholar 

  41. M. Gubisch and S. Volkwein. Proper orthogonal decomposition for linear-quadratic optimal control. In Model reduction and approximation, volume 15 of Comput. Sci. Eng., pages 3–63. SIAM, Philadelphia, PA, 2017.

    MATH  Google Scholar 

  42. M.D. Gunzburger. Perspectives in flow control and optimization, volume 5 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.

    Google Scholar 

  43. J. Haslinger and R. A. E. Mäkinen. Introduction to shape optimization, volume 7 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. Theory, approximation, and computation.

    Google Scholar 

  44. M. Heinkenschloss. Numerical solution of implicitly constrained optimization problems. 2008.

    Google Scholar 

  45. J.S. Hesthaven, G. Rozza, and B. Stamm. Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016. BCAM SpringerBriefs.

    Book  MATH  Google Scholar 

  46. J.S. Hesthaven and T. Warburton. Nodal discontinuous Galerkin methods, volume 54 of Texts in Applied Mathematics. Springer, New York, 2008. Algorithms, analysis, and applications.

    Google Scholar 

  47. M. Hintermüller and M. Hinze. Moreau-Yosida regularization in state-constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal., 47(3):1666–1683, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Hintermüller, K. Ito, and K. Kunisch. The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim., 13(3):865–888 (2003), 2002.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Hintermüller and I. Kopacka. Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim., 20(2):868–902, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Hintermüller and I. Kopacka. A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Comput. Optim. Appl., 50(1):111–145, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  51. M. Hintermüller and M. Ulbrich. A mesh-independence result for semismooth Newton methods. Math. Program., 101(1, Ser. B):151–184, 2004.

    Google Scholar 

  52. M. Hinze. A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl., 30(1):45–61, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  53. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints, volume 23 of Mathematical Modelling: Theory and Applications. Springer, New York, 2009.

    Book  MATH  Google Scholar 

  54. M. Hinze and S. Volkwein. Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In Dimension reduction of large-scale systems, volume 45 of Lect. Notes Comput. Sci. Eng., pages 261–306. Springer, Berlin, 2005.

    MATH  Google Scholar 

  55. M. Hinze and S. Volkwein. Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl., 39(3):319–345, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  56. K. Ito and K. Kunisch. Lagrange multiplier approach to variational problems and applications, volume 15 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008.

    Google Scholar 

  57. C.T. Kelley. Iterative methods for optimization, volume 18 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.

    Google Scholar 

  58. D. Kinderlehrer and G. Stampacchia. An introduction to variational inequalities and their applications, volume 31 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1980 original.

    Google Scholar 

  59. K. Kohls, A. Rösch, and K.G. Siebert. A posteriori error analysis of optimal control problems with control constraints. SIAM J. Control Optim., 52(3):1832–1861, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  60. D. Kouri, D. Ridzal, and G.von Winckel. Webpage. https://trilinos.org/packages/rol/.

  61. G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich, editors. Constrained optimization and optimal control for partial differential equations, volume 160 of International Series of Numerical Mathematics. Birkhäuser/Springer Basel AG, Basel, 2012.

    Google Scholar 

  62. J.-L. Lions. Optimal control of systems governed by partial differential equations. Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170. Springer-Verlag, New York-Berlin, 1971.

    Google Scholar 

  63. S. May, R. Rannacher, and B. Vexler. Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim., 51(3):2585–2611, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  64. I. Neitzel, J. Pfefferer, and A. Rösch. Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation. SIAM J. Control Optim., 53(2):874–904, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  65. I. Neitzel and F. Tröltzsch. Numerical analysis of state-constrained optimal control problems for PDEs. In Constrained optimization and optimal control for partial differential equations, volume 160 of Internat. Ser. Numer. Math., pages 467–482. Birkhäuser/Springer Basel AG, Basel, 2012.

    Google Scholar 

  66. J. Nocedal and S.J. Wright. Numerical optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York, second edition, 2006.

    MATH  Google Scholar 

  67. R.H. Nochetto, K.G. Siebert, and A. Veeser. Theory of adaptive finite element methods: an introduction. In Multiscale, nonlinear and adaptive approximation, pages 409–542. Springer, Berlin, 2009.

    Chapter  MATH  Google Scholar 

  68. E. Rocca G. Schimperna J. Sprekels P. Colli, A. Favini, editor. Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, volume 22 of Springer INdAM Series. Springer, Cham, 2017. In Honour of Prof. Gianni Gilardi.

    Google Scholar 

  69. A. Quarteroni, A. Manzoni, and F. Negri. Reduced basis methods for partial differential equations, volume 92 of Unitext. Springer, Cham, 2016. An introduction, La Matematica per il 3+2.

    Google Scholar 

  70. B. Rivière. Discontinuous Galerkin methods for solving elliptic and parabolic equations, volume 35 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Theory and implementation.

    Google Scholar 

  71. S.M. Robinson. Stability theory for systems of inequalities. II. Differentiable nonlinear systems. SIAM J. Numer. Anal., 13(4):497–513, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  72. A. Rösch. Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw., 21(1):121–134, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  73. A. Rösch and R. Simon. Linear and discontinuous approximations for optimal control problems. Numer. Funct. Anal. Optim., 26(3):427–448, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  74. A. Rösch and R. Simon. Superconvergence properties for optimal control problems discretized by piecewise linear and discontinuous functions. Numer. Funct. Anal. Optim., 28(3–4): 425–443, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  75. A. Rösch and D. Wachsmuth. A-posteriori error estimates for optimal control problems with state and control constraints. Numer. Math., 120(4):733–762, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  76. A. Schiela and S. Ulbrich. Operator preconditioning for a class of inequality constrained optimal control problems. SIAM J. Optim., 24(1):435–466, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  77. R. Schneider and G. Wachsmuth. A posteriori error estimation for control-constrained, linear-quadratic optimal control problems. SIAM J. Numer. Anal., 54(2):1169–1192, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  78. C. Schwab and C.J. Gittelson. Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer., 20:291–467, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  79. A. Shapiro. On concepts of directional differentiability. J. Optim. Theory Appl., 66(3):477–487, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  80. J. Sokoł owski and J.-P. Zolésio. Introduction to shape optimization, volume 16 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1992. Shape sensitivity analysis.

    Google Scholar 

  81. F. Tröltzsch. Optimal control of partial differential equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.

    Google Scholar 

  82. M. Ulbrich. Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, volume 11 of MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2011.

    Google Scholar 

  83. S. Ulbrich. Preconditioners based on “parareal” time-domain decomposition for time-dependent PDE-constrained optimization. In Multiple Shooting and Time Domain Decomposition Methods, pages 203–232. Springer, 2015.

    Google Scholar 

  84. A.J. Wathen. Preconditioning. Acta Numer., 24:329–376, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  85. J. Zowe and S. Kurcyusz. Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim., 5(1):49–62, 1979.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Harbir Antil

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Dmitriy Leykekhman

Authors
  1. Harbir Antil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitriy Leykekhman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harbir Antil .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Harbir Antil

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Drew P. Kouri

  3. Corporate Strategic Research, ExxonMobil Research and Engineering Company, Annandale, NJ, USA

    Martin-D. Lacasse

  4. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Denis Ridzal

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Science+Business Media, LLC, part of Springer Nature

About this chapter

Cite this chapter

Antil, H., Leykekhman, D. (2018). A Brief Introduction to PDE-Constrained Optimization. In: Antil, H., Kouri, D.P., Lacasse, MD., Ridzal, D. (eds) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8636-1_1

Download citation

Publish with us

Policies and ethics

Search

Navigation