A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations
type: Item
instance of: scholarly article
title: A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations
zbMATH Open document ID: 1226.65060
author: Huibin Chang
author: Danping Yang
published in: Journal of Computational and Applied Mathematics
publication date: 2011-08-02T00:00:00Z
review text: This paper is devoted to the study of the convergence rate of the proposed algorithms (called SA-GP algorithms) for both of the continuous models and the discrete model, respectively. The following distributed optimal control problem is investigated: \[ \min_{u \in U_{ad}} Y (u,y)\text{ such that }- \Delta y = u+f\text{ in }\Omega\tag{1} \] (a bounded open domain in \(\mathbb R^d\) for \(1 \leq d \leq 3\) with a Lipschitz boundary), where \(U_{ad}\) is a closed convex subset of \(U(\Omega)(=L_2(\Omega))\), \(u\) is the control variable, \(y \in V (=H^1_0 (\Omega))\) is the state variable and \(y_d\) is the desired state given in \(L_2(\Omega)\). The weak form of the optimal control problem (1) is characterized by \[ \min_{u \in U_{ad}} Y (u,y)\text{ such that }a (y,w) = (u+f,w), \forall w \in V \] and the equivalent optimality condition is \[ \begin{cases} a(y,w) = (u+f,w), \forall w \in V \\ a(q,p) = (y+y_d,q), \forall q \in V \\ (p + \alpha u, v-u) \geq 0, \forall v \in U_{ad}, \end{cases} \] where the bilinear form is of the type \(a(w,v) = \int_\Omega \nabla w. \nabla v d \Omega, \forall v,w \in H^1_0 (\Omega)\) (\(p\) is co-state variable). A finite element discrete scheme of the optimal control problem (1) \[ \min_{v_h \in U_{ad} \cap U^h} Y (v_h,z_h)\text{ such that }a (z_h,w_h) = (v_h+f,w_h), \forall w_h \in V^h, \] is introduced. The equivalent optimality conditions are as follows: \[ \begin{cases} a(y_h,w_h) = (u_h+f,w_h), \forall w_h \in V^h \\ a(p_h,q_h) = (y_h-y_d,q_h), \forall q_h \in V^h \\ \int_\Omega (u_h + p_h) (v_h - u_h) d \Omega \geq 0, \forall v_h \in U_{ad} \cap U^h.\end{cases} \] To solve the optimality system (1) by a domain decomposition method the authors adopt a decoupled method: the outer iterations are used to solve the inequality with respect to the control variable \(u\), the inner iterations are used to treat partial differential equations with respect to the state and co-state variables. There are many choices of iterative solvers i…
reviewed by: Jan Lovíšek
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49M27
zbMATH DE Number: 5934909
zbMATH Keywords: distributed optimal control
zbMATH Keywords: elliptic partial differential equation
zbMATH Keywords: domain decomposition
zbMATH Keywords: Schwarz alternating method
zbMATH Keywords: gradient projection algorithm
zbMATH Keywords: geometric convergence rate
zbMATH Keywords: bilinear form
zbMATH Keywords: projection operator
zbMATH Keywords: algorithms
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: j.cam.2011.04.037
OpenAlex ID: W2151982596
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DOI: 10.1016/J.CAM.2011.04.037
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description: scientific article; zbMATH DE number 5934909