Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T05:23:30.428Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fast Solver for an H1 Regularized PDE-Constrained Optimization Problem

Part of: Partial differential equations, boundary value problems Numerical linear algebra Numerical methods in calculus of variations and optimal control Optimality conditions

Published online by Cambridge University Press:  15 January 2016

Andrew T. Barker ,
Tyrone Rees  and
Martin Stoll
Andrew T. Barker
Affiliation:
Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Mail Stop L-561, Livermore, CA 94551, USA
Tyrone Rees*
Affiliation:
Numerical Analysis Group, Scientific Computing Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, United Kingdom
Martin Stoll
Affiliation:
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
*
*Corresponding author. Email addresses:barker29@llnl.gov (A. T. Barker), tyrone.rees@stfc.ac.uk (T. Rees), stollm@mpi-magdeburg.mpg.de (M. Stoll)
Get access

Abstract

In this paper we consider PDE-constrained optimization problems which incorporate an H1 regularization control term. We focus on a time-dependent PDE, and consider both distributed and boundary control. The problems we consider include bound constraints on the state, and we use a Moreau-Yosida penalty function to handle this. We propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.

Keywords

PreconditioningKrylov methodsPDE-constrained optimizationoptimal control of PDEs

MSC classification

Secondary: 49M25: Discrete approximations 49K20: Problems involving partial differential equations 65F10: Iterative methods for linear systems 65N22: Solution of discretized equations 65F50: Sparse matrices 65N55: Multigrid methods; domain decomposition
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 143 - 167
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Zhong-Zhi, Bai: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91(4), 379–395 (2011)Google Scholar
[2]Bangerth, W., Hartmann, R., Kanschat, G.: deal. II—a general-purpose object-oriented finite element library. ACM Trans. Math. Software 33(4), Art. 24, 27 (2007)Google Scholar
[3]Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., and der Vorst, H. V.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. 2nd Edition, SIAM, Philadelphia, PA, 1994.Google Scholar
[4]Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer 14, 1–137 (2005)CrossRefGoogle Scholar
[5]Benzi, M., Haber, E., Taralli, L.: A preconditioning technique for a class of PDE-constrained optimization problems. Advances in Computational Mathematics 35, 149–173 (2011)Google Scholar
[6]Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4), 1176–1194 (1999)CrossRefGoogle Scholar
[7]Bergounioux, M., Kunisch, K.:Primal-dual strategy for state-constrained optimal control problems, Comput. Optim. Appl. 22(2), 193–224 (2002) DOI 10.1023/A:1015489608037Google Scholar
[8]Bochev, P., Lehoucq, R.: On the finite element solution of the pure neumann problem. SIAM review 47(1), 50–66 (2005)CrossRefGoogle Scholar
[9]Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp 50(181), 1–17 (1988)Google Scholar
[10]Cai, Xiao-Chuan and Liu, Si and Zou, Jun: An overlapping domain decomposition method for parameter identification problems. Domain Decomposition Methods in Science and Engineering XVII, 60, 451–458 (2008)Google Scholar
[11]Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986). DOI 10.1137/0324078. URL http://dx.doi.org/10.1137/0324078Google Scholar
[12]Casas, E., Herzog, R. and Wachsmuth, G.: Approximation of sparse controls in semilinear equations by piecewise linear functions. Numerische Mathematik 122, 645–669 (2012)Google Scholar
[13]Chan, R.H. and Chan, T.F. and Wan, WL and others: Multigrid for differential-convolution problems arising from image processing. Proc. Workshop on Scientific Computing, pp. 58–72 (1997)Google Scholar
[14]Chan, T.F. and Tai, X.C.: Identification of discontinuous coefficients in elliptic problems using total variation regularization. SIAM Journal on Scientific Computing 25(3) 881–904 (2003)Google Scholar
[15]Choi, Y., Farhat, C., Murray, W. and Saunders, M.: A practical factorization of a Schur complement for PDE-constrained Distributed Optimal Control. arXiv preprint arXiv:1312.5653 (2013)Google Scholar
[16]Christofides, P.: Nonlinear and robust control of PDE systems: Methods and applications to transport-reaction processes. Birkhauser (2001)Google Scholar
[17]Cimrák, I. and Melicher, V.: Mixed Tikhonov regularization in Banach spaces based on domain decomposition submitted to Applied Mathematics and Computation (2012)Google Scholar
[18]Collis, S.S., Ghayour, K., Heinkenschloss, M., Ulbrich, M. and Ulbrich, S.: Numerical solution of optimal control problems governed by the compressible Navier-Stokes equations International series of numerical mathematics, 43–56 (2002)Google Scholar
[19]van den Doel, K. and Ascher, U. and Haber, E.: The lost honour of l2-based regularization Submitted (2012)Google Scholar
[20]Du, X., Sarkis, M., Schaerer, C.E., Szyld, D.B.: Inexact and truncated parareal-in-time Krylov subspace methods for parabolic optimal control problems. Tech. Rep. 12-02-06, Department of Mathematics, Temple University (2012)Google Scholar
[21]Duff, I.S., Erisman, A.M., Reid, J.K.: Direct methods for sparse matrices. Monographs on Numerical Analysis. The Clarendon Press Oxford University Press, New York (1989)Google Scholar
[22]Duff, Iain S.: MA57—a code for the solution of sparse symmetric definite and indefinite systems. ACM Transactions on Mathematical Software (TOMS) 30(2) 118–144 (2004)Google Scholar
[23]Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)Google Scholar
[24]Falgout, R.: An Introduction to Algebraic Multigrid. Computing in Science and Engineering, 8 (2006), pp. 24–33. Special Issue on Multigrid Computing.Google Scholar
[25]Fletcher, R.: Conjugate gradient methods for indefinite systems, Lecture Notes in Mathematics, vol. 506. Springer-Verlag, Berlin (1976), 73–89.Google Scholar
[26]Freund, R.W., Nachtigal, N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Num. Math. 60(1) 315–339 (1991)Google Scholar
[27]Gee, M., Siefert, C., Hu, J., Tuminaro, R., Sala, M.: ML 5.0 smoothed aggregation user's guide. Tech. Rep. SAND2006-2649, Sandia National Laboratories (2006)Google Scholar
[28]Gunzburger, Max D.: Perspectives in flow control and optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003)Google Scholar
[29]Greenbaum, A.: Iterative methods for solving linear systems, Frontiers in Applied Mathematics, vol. 17. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1997)Google Scholar
[30]Griewank, A., Walther, A.: Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Transactions on Mathematical Software (TOMS) 26(1), 19–45 (2000)CrossRefGoogle Scholar
[31]Haber, E. and Hanson, L.: Model Problems in PDE-Constrained Optimization Emory University TR-2007-009 (2007)Google Scholar
[32]Hackbusch, W.:Multigrid methods and applications. vol. 4 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1985.Google Scholar
[33]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). DOI 10.1016/j.cam.2004.03.005Google Scholar
[34]Heinkenschloss, M.: Formulation and analysis of a sequential quadratic programming method for the optimal Dirichlet boundary control of Navier-Stokes flow Optimal Control, Theory, Algorithms, and Applications (1998)Google Scholar
[35]Heinkenschloss, Matthias, Ridzal, Denis: A matrix-free trust-region SQP method for equality constrained optimization. Technical Report 11-17, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, 2011.Google Scholar
[36]Herzog, R., Sachs, E.W.: Preconditioned conjugate gradient method for optimal control problems with control and state constraints. SIAM J. Matrix Anal. Appl. 31(5), 2291–2317 (2010)Google Scholar
[37]Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand 49, 409–436 (1953) (1952)Google Scholar
[38]Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3), 865–888 (2002)Google Scholar
[39]Hintermüller, M., Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159–187 (2006)Google Scholar
[40]Hinze, M., Köster, M., Turek, S.: A Hierarchical Space-Time Solver for Distributed Control of the Stokes Equation. Tech. rep., SPP1253-16-01 (2008)Google Scholar
[41]Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications. Springer-Verlag, New York (2009)Google Scholar
[42]Hogg, Jonathan D., and Scott, Jennifer A.. HSL_MA97: A bit-compatible multifrontal code for sparse symmetric systems. Science and Technology Facilities Council, 2011.Google Scholar
[43]Ito, K., Kunisch, K.: Semi-smooth Newton methods for state-constrained optimal control problems. Systems Control Lett. 50(3), 221–228 (2003). DOI 10.1016/S0167-6911(03)00156-7Google Scholar
[44]Ito, K., Kunisch, K.: Lagrange multiplier approach to variational problems and applications, Advances in Design and Control, vol. 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008)Google Scholar
[45]Kanzow, C.: Inexact semismooth Newton methods for large-scale complementarity problems. Optimization Methods and Software 19(3-4), 309–325 (2004). DOI 10.1080/10556780310001636369. URL http://www.tandfonline.com/doi/abs/10.1080/10556780310001636369CrossRefGoogle Scholar
[46]Keung, Y.L. and Zou, J.: Numerical identifications of parameters in parabolic systems Inverse Problems 14(1), 83–100 (1999)Google Scholar
[47]Kollmann, M., Kolmbauer, M.: A Preconditioned MinRes Solver for Time-Periodic Parabolic Optimal Control Problems. Sumitted, Numa-Report 2011-06 (August 2011)Google Scholar
[48]Li, F. and Shen, C. and Li, C.: Multiphase Soft Segmentation with Total Variation and H1 Regularization Journal of Mathematical Imaging and Vision 37(2), 98–111 (2010)Google Scholar
[49]Ng, M.K. and Chan, R.H. and Chan, T.F. and Yip, A.M.: Cosine transform preconditioners for high resolution image reconstruction Linear Algebra and its Applications 316(1) 89–104 (2000)Google Scholar
[50]Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput 21(6), 1969–1972 (2000)Google Scholar
[51]Paige, C.C., Saunders, M.A.: Solutions of sparse indefinite systems of linear equations. SIAM J. Numer. Anal 12(4), 617–629 (1975)Google Scholar
[52]Pearson, J.W., Stoll, M., Wathen, A.: Preconditioners for state constrained optimal control problems with Moreau-Yosida penalty function. Numerical Linear Algebra with Applications 21(1), 81–97, (2014)CrossRefGoogle Scholar
[53]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)Google Scholar
[54]Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numerical Linear Algebra with Applications 19, 816–829 (2012). DOI 10.1002/nla.814. URL http://dx.doi.org/10.1002/nla.814Google Scholar
[55]Peirce, A., Dahleh, M., Rabitz, H.: Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications. Physical Review A 37(12), 4950 (1988)Google Scholar
[56]Pironneau, O.: Optimal shape design for elliptic systems. System Modeling and Optimization pp. 42–66 (1982)Google Scholar
[57]Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM Journal on Scientific Computing 32(1), 271–298 (2010). DOI http://dx.doi.org/10.1137/080727154Google Scholar
[58]Rees, T., Stoll, M., Wathen, A.: All-at-once preconditioners for PDE-constrained optimization. Kybernetika 46, 341–360 (2010)Google Scholar
[59]Reyes, De los, Juan-Carlos, and Carola-Bibiane, Schönlieb: Image denoising: learning noise distribution via PDE-constrained optimization, http://arxiv.org/abs/1207.3425 (2012)Google Scholar
[60]Ruge, J. W. and Stüben, K.: Algebraic multigrid. in Multigrid methods, vol. 3 of Frontiers Appl. Math., SIAM, Philadelphia, PA, 1987, pp. 73–130.Google Scholar
[61]Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2003)Google Scholar
[62]Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM Journal on Scientific Computing, 14 (1993), pp. 461–461.Google Scholar
[63]Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3), 856–869 (1986).Google Scholar
[64]Simoncini, V., Szyld, D.: Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl 14(1), 1–61 (2007).Google Scholar
[65]Stoll, M.: All-at-once solution of a time-dependent time-periodic PDE-constrained optimization problems. IMA J Numer Anal (2013)Google Scholar
[66]Stoll, M., Wathen, A.: All-at-once solution of time-dependent PDE-constrained optimization problems. Technical Report, University of Oxford, (2010)Google Scholar
[67]Strang, G., Fix, G.: An Analysis of the Finite Element Method 2nd Edition, 2nd edn. Wellesley-Cambridge (2008)Google Scholar
[68]Takacs, S., Zulehner, W.: Convergence analysis of multigrid methods with collective point smoothers for optimal control problems. Computing and Visualization in Science 14, 131–141 (2011)Google Scholar
[69]Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Amer Mathematical Society (2010)Google Scholar
[70]Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems. SIAM Philadelphia (2011)Google Scholar
[71]Van Der Vorst, H.A.: BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–634 (1992).CrossRefGoogle Scholar
[72]Wathen, A.J., Rees, T.: Chebyshev semi-iteration in preconditioning for problems including the mass matrix. Electronic Transactions in Numerical Analysis 34, 125–135 (2008)Google Scholar
[73]Wachsmuth, D. and Wachsmuth, G.: Necessary conditions for convergence rates of regularizations of optimal control problems, RICAM Report 4 (2012)Google Scholar
[74]Wathen, A.J.: Preconditioning and convergence in the right norm. International Journal of Computer Mathematics, 84 (2007), pp. 1199–1209.Google Scholar
[75]Wesseling, P.: An introduction to multigrid methods. Pure and Applied Mathematics (New York), John Wiley & Sons Ltd., Chichester, 1992.Google Scholar
[76]Wilson, J. and Patwari, N. and Vasquez, F.G.: Regularization methods for radio tomographic imaging. 2009 Virginia Tech Symposium on Wireless Personal Communications (2009)Google Scholar
[77]Zulehner, W.: Analysis of iterative methods for saddle point problems: a unified approach. Math. Comp 71(238), 479–505 (2002)Google Scholar