Abstract
A multigrid and sparse-grid computational approach to solving nonlinear elliptic optimal control problems with random coefficients is presented. The proposed scheme combines multigrid methods with sparse-grids collocation techniques. Within this framework the influence of randomness of problem’s coefficients on the control provided by the optimal control theory is investigated. Numerical results of computation of stochastic optimal control solutions and formulation of mean control functions are presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Babuška I., Nobile F., Tempone R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)
Bebernes J., Eberly D.: Mathematical Problems from Combustion Theory. Springer, New York (1988)
Borzì A.: High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems. J. Comput. Appl. Math. 200, 67–85 (2007)
Borzì A.: Smoothers for control- and state-constrained optimal control problems. Comput. Vis. Sci. 11, 59–66 (2008)
Borzì A., Borzì G.: An efficient algebraic multigrid method for solving optimality systems. Comput. Vis. Sci. 7(3/4), 183–188 (2004)
Borzì A., Kunisch K.: The numerical solution of the steady state solid fuel ignition model and its optimal control. SIAM J. Sci. Comput. 22(1), 263–284 (2000)
Borzì A., Schulz V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361–395 (2009)
Borzì A., von Winckel G.: Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J. Sci. Comput. 31, 2172–2192 (2009)
Brandt A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)
Bungartz H.-J., Griebel M.: Sparse grids. Acta Numerica 13, 147–269 (2004)
Elman H., Furnival D.: Solving the stochastic steady-state diffusion problem using multigrid. IMA J. Numer. Anal. 27, 675–688 (2007)
Ganapathysubramanian B., Zabaras N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007)
Gasca M., Sauer T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12, 377–410 (2000)
Ghanem R.G., Spanos P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)
Hackbusch W.: Multi-grid Methods and Applications. Springer, New York (1985)
Huang S.P., Mahadevan S., Rebba R.: Collocation-based stochastic finite element analysis for random field problems. Probabilistic Eng. Mech. 22, 194–205 (2007)
Ito K., Kunisch K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)
Klimke A., Wohlmuth B.: Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Trans. Math. Softw. 31, 561–579 (2005)
Lions J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Lions J.-L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)
Loeve M.: Probability Theory. Vols. I & II, IV edn. Springer, New York (1978)
Marzouk Y.M., Najm H.N., Rahn L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224, 560–586 (2007)
Nobile F., Tempone R., Webster C.: A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)
Putko M.M., Newman P.A., Taylor A.C. III, Green L.L.: Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives. J. Fluids Eng. 124, 60–69 (2002)
Schulz V., Schillings C.: On the nature and treatment of uncertainties in aerodynamic design. AIAA J. 47, 646–654 (2009)
Schwab Ch, Todor R.A.: Sparse finite elements for stochastic elliptic problems higher order moments. Computing 71, 43–63 (2003)
Seynaeve B., Rosseel E., Nicolaï B., Vandewalle S.: Fourier mode analysis of multigrid methods for partial differential equations with random coefficients. J. Comput. Phys. 224, 132–149 (2007)
Trottenberg U., Oosterlee C., Schüller A.: Multigrid. Academic Press, London (2001)
Xiu D., Hesthaven J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)
Xiu D., Karniadakis G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)
Zabaras N., Ganapathysubramanian B.: A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach. J. Comput. Phys. 227, 4697–4735 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the Austrian Science Fund FWF project P18136-N13 “Quantum optimal control of semiconductor nanostructures” and F3205-N18 “Fast Multigrid Methods for Inverse Problems”.
Rights and permissions
About this article
Cite this article
Borzì, A. Multigrid and sparse-grid schemes for elliptic control problems with random coefficients. Comput. Visual Sci. 13, 153–160 (2010). https://doi.org/10.1007/s00791-010-0134-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-010-0134-4