Error analysis of finite element approximations of the optimal control problem for stochastic Stokes equations with additive white noise

Multigrid Method for Optimal Control Problem Constrained by Stochastic Stokes Equations with Noise

“…where we denote the Lagrange multipliers λ ∈ H 1 0 (Ω), q ∈ L 2 0 (Ω). Thus, we have the optimality system given by [8]:…”

“…Then, by Choi and Lee [8], we have the following estimates (for u h and p h , where h is the mesh size):…”

“…For our control problem, we do not need an exact solution. Therefore, we took the desired velocity [8] where…”

“…Particularly, control problem constrined by Stokes and/or Navier-Stokes equations play an important role in many real-world applications such as data assimilation for ocean flows, weather models, and flow problems. Recently, there has been a growing interest in mathematical analysis and simulations of stochastic PDEs (e.g., see [3][4][5][6][7][8]). The motivation of our work is the stochastic optimal control problem with PDEs as constraints (with random inputs) (see [8][9][10][11] and the references therein).…”

“…In particular, we propose a full multigrid method with three-factor coarsening strategy to solve numerically the distributed optimal control problem constrained by stochastic Stokes equations (driven by the white noise perturbed forcing term). The scope of this paper is to develop an efficient multigrid algorithm without the need of any preconditioner [26], which results in fewer computations (CPU time) for the control problems with stochastic models, such as the one in [8]. For this, we take advantage of our previous works on multigrid methods for distributed optimal control problem governed by Stokes equations [27,28].…”