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Preconditioned iterative method for boundary value method discretizations of a parabolic optimal control problem

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Abstract

A distributed optimal control problem with the constraint of a parabolic partial differential equation is considered. Boundary value methods are used to solve the coupled initial/final value problems arising from the first order optimality conditions for this problem. We use a block triangular preconditioning strategy for solving the resulting two-by-two linear system. By making use of a matching strategy and a Kronecker product-based splitting technique we establish a Kronecker product-based approximation to the Schur complement. Since the Schur complement approximation is in a form of one Kronecker product structure, the preconditioner can be implemented efficiently. Numerical experiments are presented to illustrate the accuracy and computational efficiency of the proposed approach.

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References

  1. Abbeloos, D., Diehl, M., Hinze, M., Vandewalle, S.: Nested multigrid methods for time-periodic parabolic optimal control problems. Comput. Vis. Sci. 14, 27–38 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amodio, P., Brugnano, L.: A note on the efficient implementation of implicit methods for ODEs. J. Comput. Appl. Math. 87, 1–9 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Axelsson, O., Verwer, J.G.: Boundary value techniques for initial value problems in ordinary differential equations. Math. Comput. 45, 153–171 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J. Numer. Anal. 33, 343–369 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problem. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benzi, M., Ferragut, L., Pennacchio, M., Simoncini, V.: Solution of linear systems from an optimal control problem arising in wind simulation. Numer. Linear. Algebra. Appl. 17, 895–915 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benzi, M., Haber, E., Taralli, L.: A preconditioning technique for a class of PDE-constrained optimization problems. Adv. Comput. Math. 35, 149–173 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bertaccini, D.: A circulant preconditioner for the systems of LMF-based ODE codes. SIAM J. Sci. Comput. 22, 767–786 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bertaccini, D.: Reliable preconditioned iterative linear solvers for some numerical integrators. Numer. Linear. Algebra. Appl. 8, 111–125 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bertaccini, D., Ng, M.K.: The convergence rate of block preconditioned systems arising from LMF-based ODE codes. BIT 41, 433–450 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bertaccini, D.: The spectrum of circulant-like preconditioners for some general linear multistep formulas for linear boundary value problems. SIAM J. Numer. Anal. 40, 1798–1822 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bertaccini, D., Ng, M.K.: Band-Toeplitz preconditioned GMRES iterations for time-dependent PDEs. BIT 43, 901–914 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bertaccini, D., Golub, G.H., Serra-Capizzano, S.: Spectral analysis of a preconditioned iterative method for the convection-diffusion equation. SIAM J. Matrix. Anal. Appl. 29, 260–278 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B.: Large-Scale PDE-Constrained Optimization: An Introduction. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  16. Borzì, A.: Multigrid methods for parabolic distributed optimal control problems. J. Comput. Appl. Math. 157, 365–382 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Borzì, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361–395 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, Philadelphia (2012)

    MATH  Google Scholar 

  19. Brugnano, L., Trigiante, D.: A parallel preconditioning technique for BVM methods. Appl. Numer. Math. 13, 277–290 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Brugnano, L., Trigiante, D.: High-order multistep methods for boundary value problems. Appl. Numer. Math. 18, 79–94 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Brugnano, L., Trigiante, D.: Convergence and stability of boundary value methods for ordinary differential equations. J. Comput. Appl. Math. 66, 97–109 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Brugnano, L., Trigiante, D.: Boundary value methods: the third way between linear multistep and Runge–Kutta methods. Comput. Math. Appl. 36, 269–284 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordan and Breach, Amsterdam (1998)

    MATH  Google Scholar 

  24. Chan, R.H., Ng, M.K., Jin, X.: Strang-type preconditioners for systems of LMF-based ODE codes. IMA J. Numer. Anal. 21, 451–462 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chen, H., Zhang, C.-J.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218, 2619–2630 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Chen, H., Zhang, C.-J.: Block boundary value methods for solving Volterra integral and integro-differential equations. J. Comput. Appl. Math. 236, 2822–2837 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chen, H., Zhang, C.-J.: Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Appl. Numer. Math. 62, 141–154 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chen, H.: A splitting preconditioner for the iterative solution of implicit Runge–Kutta and boundary value methods. BIT 54, 607–621 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Chen, H.: Generalized Kronecker product splitting iteration for the solution of implicit Runge–Kutta and boundary value methods. Numer. Linear. Algebra. Appl. 22, 357–370 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen, H.: Kronecker product splitting preconditioners for implicit Runge–Kutta discretizations of viscous wave equations. Appl. Math. Model. 40, 4429–4440 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Chen, H.: A splitting preconditioner for implicit Runge–Kutta discretizations of a partial differential-algebraic equation. Numer. Algorithms. 73, 1037–1054 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, H., Lv, W., Zhang, T.-T.: A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations. J. Comput. Phys. 360, 1–14 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Chen, H., Zhang, T.-T., Lv, W.: Block preconditioning strategies for time-space fractional diffusion equations. Appl. Math. Comput. 337, 41–53 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Chen, H., Wang, X.-L., Li, X.-L.: A note on efficient preconditioner of implicit Runge–Kutta methods with application to fractional diffusion equations. Appl. Math. Comput. 351, 116–123 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Chen, H., Lv, W.: Kronecker product-based structure preserving preconditioner for three-dimensional space-fractional diffusion equations. Int. J. Comput. Math. (2019). https://doi.org/10.1080/00207160.2019.1581177

    Article  Google Scholar 

  36. Güttel, S., Pearson, J.W.: A rational deferred correction approach to parabolic optimal control problems. IMA J. Numer. Anal. 38, 1861–1892 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Heinkenschloss, M.: A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems. J. Comput. Appl. Math. 173, 169–198 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2009)

    MATH  Google Scholar 

  39. Hockney, R.W.: A fast direct solution of Poisson’s equation using fourier analysis. J. ACM 12, 95–113 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  40. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  41. Iavernaro, F., Mazzia, F.: Convergence and stability of multistep methods solving nonlinear initial value problems. SIAM J. Sci. Comput. 18, 270–285 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  42. Iavernaro, F., Mazzia, F.: Block boundary value methods for the solution of ordinary differential equations. SIAM J. Sci. Comput. 21, 323–399 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ipsen, I.C.F.: A note on preconditioning nonsymmetric matrices. SIAM J. Sci. Comput. 23, 1050–1051 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  45. Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  46. Van Lent, J., Vandewalle, S.: Multigrid methods for implicit Runge–Kutta and boundary value method discretizations of PDEs. SIAM J. Sci. Comput. 27, 67–92 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Liang, Z.-Z., Axelsson, O., Neytcheva, M.: A robust structured preconditioner for time-hormonic parabolic optimal control problems. Numer. Algor. 79, 575–596 (2018)

    Article  MATH  Google Scholar 

  48. Mathew, T.P., Sarkis, M., Schaerer, C.E.: Analysis of block parareal preconditioners for parabolic optimal control problems. SIAM J. Sci. Comput. 32, 1180–1200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear. Algebra. Appl. 19, 816–829 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Pearson, J.W., Stoll, M., Wathen, A.J.: Regularization-robust preconditioners for time-dependent PDE-constrained optimization problems. SIAM J. Matrix. Anal. Appl. 33, 1126–1152 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Pearson, J.W., Stoll, M.: Fast iterative solution of reaction-diffusion control problems arising from chemical processes. SIAM J. Sci. Comput. 35, B987–B1009 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  52. Pearson, J.W.: Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems. Appl. Numer. Math. 108, 87–101 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  53. Rees, T., Stoll, M.: Block-triangular preconditioners for PDE-constrained optimization. Numer. Linear. Algebra. Appl. 17, 977–996 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  54. Ruge, J.W., Stüben, K.: Algebraic multigrid. In: McCormick, S.F.(ed.) Multigrid Methods, Frontiers in Applied Mathematics and Statistics, vol. 3, pp. 73–130. SIAM, Philadelphia (1987)

  55. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  56. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  57. Spotz, W.F., Carey, G.F.: A high-order compact formulation for the 3D Poisson equation. Numer. Methods Partial Differ. Equ. 12, 235–243 (1996)

    Article  MATH  Google Scholar 

  58. Stoll, M., Wathen, A.J.: All-at-once solution of time-dependent PDE-constrained optimization problems. Technical report, University of Oxford, Oxford (2010)

  59. Stoll, M., Wathen, A.J.: All-at-once solution of time-dependent Stokes control. J. Comput. Phys. 232, 498–515 (2013)

    Article  MathSciNet  Google Scholar 

  60. Stoll, M.: All-at-once solution of a time-dependent time-periodic PDE-constrained optimization problems. IMA J. Numer. Anal. 34, 1554–1577 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. Stoll, M., Breiten, T.: A low-rank in time approach to PDE-constrained optimization. SIAM J. Sci. Comput. 37, B1–B29 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  63. Wathen, A.J., Rees, T.: Chebyshev semi-iteration in preconditioning for problems including the mass matrix. Electron. Trans. Numer. Anal. 34, 125–135 (2008)

    MathSciNet  MATH  Google Scholar 

  64. Zeng, M., Zhang, H.: A new preconditioning strategy for solving a class of time-dependent PDE-constrained optimization problems. J. Comput. Math. 32, 215–232 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  65. Zhang, C.-J., Chen, H.: Block boundary value methods for delay differential equations. Appl. Numer. Math. 60, 915–923 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  66. Zhang, C.-J., Chen, H.: Asymptotic stability of block boundary value methods for delay differential-algebraic equations. Math. Comput. Simul. 81, 100–108 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  67. Zhang, C.-J., Chen, H., Wang, L.-M.: Strang-type preconditioners applied ordinary and neutral differential-algebraic equations. Numer. Linear. Algebra. Appl. 18, 843–855 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referee for valuable suggestions, which improved the original manuscript of the paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11301575, 11971085), the Natural Science Foundation Project of CQ CSTC (No. cstc2018jcyjAX0113),the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-M201800501), the Program of Chongqing Innovation Research Group Project in University (No. CXQT19018), and the Talent Project of Chongqing Normal University (Grant No. 02030307-0054).

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  1. College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

    Hao Chen & Qiuyue Huang

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  1. Hao Chen
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Chen, H., Huang, Q. Preconditioned iterative method for boundary value method discretizations of a parabolic optimal control problem. Calcolo 57, 5 (2020). https://doi.org/10.1007/s10092-019-0353-0

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