A numerical algorithm for solving optimization problems with stochastic diffusion equation as a constraint is proposed. First, separation of random and deterministic variables is done via Karhunen-Loeve expansion. Then, the problem is discretized, in spatial part, using the finite element method and the polynomial chaos expansion in the stochastic part of the problem. This process leads to the optimal control problem with a large scale system in its constraint. To overcome these difficulties the adjoint technique for derivative computation to implementation of the optimal control issue in preconditioned Newton’s conjugate gradient method is used. By some numerical simulation, it is shown that this hybrid approach is efficient and simple to implement. 1. Introduction Physical problems, in many cases, can be formulated as optimization problems. These problems are utilized to gain a more widely understanding of physical systems. Typically, these problems depend on some models which, in many cases, are deterministic. Uncertainty might plague everything from modeling assumptions to experimental data. As such, in order to accommodate for these uncertainties, many practitioners have developed stochastic models. In order to make sense of and solve these models, in addition to randomness of the models, some additional theory is required to the resulting optimization problems. This paper proposes an adjoint based approach for solving optimization problems governed by stochastic diffusion equations. In order to deal with the stochastic partial differential equations (SPDE) as a constraint in an optimization problem one may first solve the SPDE. Since the provided systems by this approach are random and nonlinear, such kind of problems are difficult to handle and very challenging. One of the most challenging examples in this area is the control of stochastic diffusion equations with random forcing [1]. Polynomial chaos expansion (PCE) [2] provides a good direction for solution of nonlinear SPDEs numerically. In the presence of the random forcing, PCE seems to be more accurate and efficient numerical method than Monte Carlo simulation. In fact, the PCE can be interpreted as the Fourier expansion in the probability space. Particularly, the aim of this work is the numerical solution for the distributed control problems involving the stochastic diffusion equations in the form: where is the spatial domain, is the boundary of the spatial domain, is probability space, , , is the solution of the SPDE, is the source function, and is the permeability field of the problem. As
References
[1]
M. D. Gunzburger and L. S. Hou, “Finite-dimensional approximation of a class of constrained nonlinear optimal control problems,” SIAM Journal on Control and Optimization, vol. 34, no. 3, pp. 1001–1043, 1996.
[2]
M. Gunzburger, Perspectives in Flow Control and Optimization, SIAM, 1987.
[3]
I. Babu？ka, R. Tempone, and G. E. Zouraris, “Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 12–16, pp. 1251–1294, 2005.
[4]
K. Karhunen, “Uberlineare Methoden in der Wahrscheinlichkeitsrechnung,” Annales Academi？ Scientiarum Fennic？, vol. 37, p. 79, 1947.
[5]
M. Loève, “Fonctions aléatoire de second ordre,” La Revue Scientifique, vol. 84, pp. 195–206, 1946.
[6]
R. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, Berlin, Germany, 1991.
[7]
E. Haber and L. Hanson, “Model problems in PDE-constrained optimization,” Tech. Rep. TR-2007-009, Emory, Atlanta, Ga, USA, 2007.
[8]
M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with Partial Differential Equations, vol. 23 of Mathematical Modelling, Theory and Applications, Springer, Heidelberg, Germany, 2009.
[9]
N. Wiener, “The homogeneous chaos,” The American Journal of Mathematics, vol. 60, pp. 897–938, 1938.
[10]
C. F. van Loan and N. Pitsianis, “Approximation with Kronecker products,” in Linear Algebra for Large Scale and Real-Time Applications, M. S. Moonen, G. H. Golub, and B. L. R. de Moor, Eds., pp. 293–314, Kluwer Academic, Dordrecht, Germany, 1993.
[11]
M. Heinkenschloss, “Numerical solution of implicitly constrained optimization problems,” Tech. Rep. TR08-05, Department of Computational and Applied Mathematics, Rice University, Houston, Tex, USA, 2008.
[12]
C. T. Kelley, Iterative Methods for Optimization, SIAM, Philadelphia, Pa, USA, 1999.
[13]
J. Nocedal and S. J. Wright, Numerical Optimization, Springer, Berlin, Germany, 2nd edition, 2006.
[14]
E. Ullmann, “A kronecker product preconditioner for stochastic galerkin finite eleme nt discretizations,” SIAM Journal on Scientific Computing, vol. 32, no. 2, pp. 923–946, 2010.
[15]
C. E. Powell and E. Ullmann, “Preconditioning stochastic Galerkin saddle point systems,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 5, pp. 2813–2840, 2010.
[16]
J. Sch？berl and W. Zulehner, “Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 3, pp. 752–773, 2007.
