Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T22:58:38.745Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stochastic Galerkin Method for Stochastic Control Problems

Published online by Cambridge University Press:  03 June 2015

Hyung-Chun Lee  and
Jangwoon Lee
Hyung-Chun Lee*
Affiliation:
Department of Mathematics, Ajou University, Suwon, Korea 443-749
Jangwoon Lee*
Affiliation:
Department of Mathematics, University of Mary Washington, Fredericksburg, VA 22401, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

In an interdisciplinary field on mathematics and physics, we examine a physical problem, fluid flow in porous media, which is represented by a stochastic partial differential equation (SPDE). We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature. We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem. Finally, we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates.

Keywords

65M5565N30Stochastic Galerkin methodsstochastic partial differential equationdistributed controlfinite element methods
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013 , pp. 77 - 106
Copyright
Copyright © Global Science Press Limited 2013

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]Adams, R., Sobolev Spaces, Academic, New York, 1975.Google Scholar
[2]Babuska, I. and Chatzipantelidis, P., On solving elliptic stochastic partial differential equations, Comput. Method Appl. Mech. Engrg. 191, pp. 4093–4122, 2002.Google Scholar
[3]Babuska, I., Liu, K., Tempone, R., Solving stochastic partial differential equations based on the experimental data, Math. Models Methods Appl. Sci. 13, no. 3, pp. 415–444, 2003.CrossRefGoogle Scholar
[4]Babuska, I., Tempone, R., Zouraris, G. E., Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal. Vol. 42, no. 2, pp. 800–825, 2004.Google Scholar
[5]Babuska, I., Tempone, R., Zouraris, G. E., Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation, Comput. Method Appl. Mech. Engrg. 194, pp. 1251–1294, 2005.Google Scholar
[6]Bonaccorsi, S., Confortola, F., and Mastrogiacomo, E., Optimal control of stochastic differential equations with dynamical boundary conditions, J. Math. Anal. Appl. 344, pp 667–681, 2008CrossRefGoogle Scholar
[7]Bieri, M. and Schwab, C., Sparse high order FEM for elliptic sPDEs, Comput. Methods Appl. Mech. Engrg., vol. 198, Issues 13-14, pp. 1149–1170, 2009.Google Scholar
[8]Brenner, S. C., Scott, L. R., The mathematical theory of finite element methods, Second Edition, Springer, 2002.Google Scholar
[9]Brezzi, F., Rappaz, J., Raviart, P., Finite-dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math., 36 (1980), pp. 1–25Google Scholar
[10]Cutland, N. J. and Grzesiak, K., Optimal Control for Two-Dimensional Stochastic Navier-Stokes Equations, Appl Math Optim 55:61–91, 2007.Google Scholar
[11]Chrysafinos, K., Moving mesh finite element methods for an optimal control problem for the advection-diffusion equation, Journal of Scientific Computing, Vol. 25, No. 3, pp:401–421, 2005.Google Scholar
[12]Courant, R. and Hilbert, D., Methods of mathematical physics, Interscience, New York, 1953.Google Scholar
[13]Crouzeix, M., Rappaz, J., On numerical approximation in bifurcation theory, Masson, Parix, 1990.Google Scholar
[14]Deb, M. K., Babuska, I., Oden, J. T., Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. Method Appl. Mech. Engrg. 190, pp. 6359–6372, 2001.Google Scholar
[15]Debussche, A., Fuhman, M., and Tessitore, G., Optimal Control of a Stochastic Heat Equation with Boundary-noise and Boundary-control,ESAIM:COCV, 13, pp 178–205, 2007.Google Scholar
[16]Da Prato, G., Debussche, A., Control of the Stochastic Burgers Model of Turbulent, SIAM J. Control Optim. Vol. 37, No. 4, pp. 1123–1149.Google Scholar
[17]Doostan, A., Ghanem, R. G., and Red-Horse, J., Stochastic model reduction for chaos representations, Comput. Methods Appl. Mech. Engrg., vol. 196, Issues 37-40, pp. 3951–3966, 2007.Google Scholar
[18]Evans, L. C., Partial differential equations, AMS, 1998.Google Scholar
[19]Gockenbach, M. S., Understanding and implementing the finite element method, SIAM, 2006.Google Scholar
[20]Loeve, M., Probability Theory Vol. 1 + 2, Springer, New York, 1978.Google Scholar
[21]Frauenfelder, P., Schwab, C., Todor, R. A., Finite elements for elliptic problems with stochastic coefficients, Comput. Method Appl. Mech. Engrg. 194, pp. 205–228, 2005.Google Scholar
[22]Galvis, J. and Sarkis, M., Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity, SIAM J. Numer. Anal. Vol. 47, no. 5, pp. 3624–3651, 2009.Google Scholar
[23]Ghanem, R. G., Spanos, P. D., Stochastic finite elements: Aspectral approach, Springer-Verlag, 1991.Google Scholar
[24]Girault, V., Raviart, P., Finite element methods for Navier-Stokes equations, Springer, Berlin, 1986.Google Scholar
[25]Gockenbach, M. S., Understanding and implementing the finite element method, Siam, 2006.Google Scholar
[26]Gunzburger, M. D., Hou, L. S., Svobodny, T., Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls., Math. Comp. 57, pp. 123–151, 1991.Google Scholar
[27]Gunzburger, M. D., Hou, L. S., Svobodny, T., Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Cirichlet controls., RAIRO Model. Math. Anal. Numer. 25, pp. 711–748, 1991.CrossRefGoogle Scholar
[28]Gunzburger, M. D., Hou, L. S., Finite-dimensional approximation of a class of constrained nonlinear optimal control problems., SIAM J. Control. Optim., Vol. 34, no. 3, pp. 1001–1043, May 1996.Google Scholar
[29]Gunzburger, M. D., Lee, H.-C., and Lee, J., Error estimates of stochastic optimal Neumann boundary control problems., SIAM J. Numer. Anal. Vol. 49, No. 4, pp. 1532–1552, 2011.Google Scholar
[30]Holden, H., B.|Øksendal, Ubøe, J., Zhang, T.Stochastic partial differential equations; A modeling, white noise functional approach, Birkhaäuser, 1996.Google Scholar
[31]Hou, L. S., Lavindran, S. S., A penalized neumann control approach for solving an optimal dirichlet control problem for the navier-stokes equations, SIAM J. Control. Optim., Vol. 36, no. 5, pp. 1795–1814, September 1998.Google Scholar
[32]Hou, L. S., Lee, J., A Robin-Robin non-overlapping Domain Decomposition Method for An Elliptic Boundary Control Problem, Int. J. Numer. Anal. Model. Vol. 8, No. 3, pp 443–465, 2011.Google Scholar
[33]Hou, L. S., Lee, J., and Manouzi, H.Finite Element Approximations of Stochastic Optimal Control Problems Constrained by Stochastic Elliptic PDEs, J. Math. Anal. Appl. Vol. 384, No. 1, pp 87–103, 2011.Google Scholar
[34]Lagness, J. E., General boundary value problems for differential equations of Sobolev type, SIAM J. Math. Anal., 3, pp. 105–119, 1972.Google Scholar
[35]Lions, J. L., Magenes, E., Non-homogeneous boundary value problems and applications Vol. 1, Springer, New York, 1972.Google Scholar
[36]Luo, W., Wiener chaos expansion and numerical solutions of stochastic partial differential equations, Ph.D. thesis, California institute of technology, Pasadena, California 2006.Google Scholar
[37]Manouzi, H., A finite element approximation of linear stochastic PDEs driven by multiplicative white noise, Int. J. Comput. Math., Vol. 85, no. 3-4, pp. 527–546, 2008.Google Scholar
[38]Manouzi, H. and Hou, L. S., An optimal control problem for stochastic linear PDE’s driven by a Gaussian white noise, Numerical Mathematics and Advanced Applications, Springer Berlin Heidelberg, pp. 629–636, 2008.Google Scholar
[39]Rosseel, E. and Wells, G. N., Optimal control with stochastic PDE constraints and uncertain controls, Comput. Methods Appl. Mech. Engrg., Vol. 213-216, pp. 152–167, 2012.Google Scholar
[40]Schwab, C., Todor, R. A., Sparse finite elements for elliptic problems with stochastic loading, Numer. Math., pp. 707–734, 2003.Google Scholar
[41]Stefanou, G., The stochastic finite element method: Past, present, and future, Comput. Methods Appl. Mech. Engrg., vol. 198, Issues 9-12, pp. 1031–1051, 2009.CrossRefGoogle Scholar