Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T00:20:19.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs

Published online by Cambridge University Press:  28 May 2015

Nary Kim  and
Hyung-Chun Lee
Nary Kim*
Affiliation:
Department of Mathematics, Ajou University, Suwon, Korea 443-749
Hyung-Chun Lee*
Affiliation:
Department of Mathematics, Ajou University, Suwon, Korea 443-749
*
Corresponding author. Email Address: gyory@naver.com
Corresponding author. Email Address: hclee@ajou.ac.kr
Get access

Abstract

In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

Keywords

49A2249B2265M5565N30Sparse grid collocationstochastic partial differential equationdistributed controlfinite element methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 2 , May 2014 , pp. 166 - 188
Copyright
Copyright © Global-Science Press 2014

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. Meth. Appl. Mech. Engrg. 191, 4093–4122 (2002).Google Scholar
[3]Babuska, I., Liu, K. and Tempone, R., Solving stochastic partial differential equations based on the experimental data, Math. Models Meth. Appl. Sci. 13, 415–444 (2003).Google Scholar
[4]Babuska, I., Nobile, F. and Temone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45, 1005–1034 (2007).CrossRefGoogle Scholar
[5]Babuska, I., Nobile, F. and Temone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Review 52, 317–355 (2010).Google Scholar
[6]Babuska, I., Tempone, R. and Zouraris, G. E., Galerkin finite element approximations of stochastic elliptic partial differential equations, Siam J. Numer. Anal. 42, 800–825 (2004).Google Scholar
[7]Babuska, I., Tempone, R. and 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, 1251–1294 (2005).CrossRefGoogle Scholar
[8]Barthelmann, V., Novak, E. and Ritter, K., High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12, 273–288 (2000).CrossRefGoogle Scholar
[9]Brenner, S.C. and Scott, L.R., The Mathematical Theory of Finite Element Methods, Second Edition. Springer, 2002.CrossRefGoogle Scholar
[10]Brezzi, F., Rappaz, J. and Raviart, P., Finite-dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math. 36, 1–25 (1980).Google Scholar
[11]Crouzeix, M. and Rappaz, J., On Numerical Approximation in Bi urcation Theory. Masson, Paris, 1990.Google Scholar
[12]Deb, M.K., Babuska, I. and Oden, J.T., Solution of stochastic partial di erential equations using Galerkin finite element techniques, Comput. Meth. Appl. Mech. Engrg. 190, 6359–6372 (2001).Google Scholar
[13]!Debussche, A., Fuhman, M. and Tessitore, G., Optimal control of a stochastic heat equation with boundary-noise and boundary-control, ESAIM:COCV 13, 178–205 (2007).Google Scholar
[14]Frauenfelder, P., Schwab, C. and Todor, R.A., Finite elements or elliptic problems with stochastic coe cients, Comput. Meth. Appl. Mech. Engrg. 194, 205–228 (2005).Google Scholar
[15]Ghanem, R.G. and Spanos, P.D., Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, 1991.Google Scholar
[16]Girault, V. and Raviart, P., Finite Element Methods or Navier-Stokes Equations. Springer, Berlin, 1986.Google Scholar
[17]Gunzburger, M.D., Lee, H.-C. and Lee, J., Error estimates of stochastic optimal Neumann bound-arycontrol problems., SIAM J. Numer. Anal. 49, 1532–1552 (2011).CrossRefGoogle Scholar
[18]Lions, J.L., Optimal Control of Systems governed by Partial Di erential Equations. Springer, New York, 1971.Google Scholar
[19]Luo, W., Wiener chaos expansion and numerical solutions of stochastic partial di erential equations, Ph.D. thesis, California institute of Technology, Pasadena, California 2006.Google Scholar
[20]Nobile, F., Tempone, R. and Webster, C.G., A sparse grid stochastic collocation method for partial di erential equations with random input data, SIAM J. Numer. Anal. 46, 2309–2345 (2008).Google Scholar
[21]Nobile, F., Tempone, R. and Webster, C.G., An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal. 46, 2411–2442 (2008).CrossRefGoogle Scholar
[22]Schwab, C. and Todor, R.A., Sparse finite elements for elliptic problems with stochastic loading, Numer. Math. 95, 707–734 (2003).Google Scholar
[23]Smolyak, S.A., Quadrature and interpolation formulas for tensor products ofcertain classes of functions, Dokl. Akad. Nauk SSSR 4, 240–243 (1963).Google Scholar
[24]Xiu, D. and Hesthaven, J.S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput. 27, 1118–1139 (2005).Google Scholar
[25]Xiu, D., Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys. 5, 242–272 (2009).Google Scholar