An optimization-based domain decomposition method for stochastic elliptic partial differential equations is presented. The main idea of the method is a constrained optimization problem for which the minimization of an appropriate functional forces the solutions on the two subdomains to agree on the interface; the constraints are the stochastic partial differential equations. The existence of optimal solutions for the stochastic optimal control problem is shown as is the convergence to the exact solution of the given problem. We prove the existence of a Lagrange multiplier and derive an optimality system from which solutions of the domain decomposition problem may be determined. Finite element approximations to the solutions of the optimality system are defined and analyzed with respect to both spatial and random parameter spaces. Then, the results of some numerical experiments are given to confirm theoretical error estimate results.
Bibliographical noteFunding Information:
The authors thank the anonymous referees for various useful comments that helped us improve the presentation of this paper. We would also like to mention here that the work of Jangwoon Lee was partially supported by the Jepson Fellowship and Sabbatical Leave Programs at the University of Mary Washington; the work of Jeehyun Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2013R1A1A2058848 ).
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All Science Journal Classification (ASJC) codes
- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics