A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations
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 in the outer loops and the inner loops. Here the gradient projection algorithm as outer iteration and the Schwarz alternating procedure as inner iteration are adopted. In order to prove the convergence of the algorithm, the authors introduce the projection operators. Main result: If \( \langle u^*, y^*, p^*\rangle \in U_{ad} \times V \times V\) is given by the algorithm SA-GP for \(0 \leq K < \infty\), then there exist some constants \([c_1, c_2, \lambda_1, \lambda_2] (|\lambda_1| <1, |\lambda_2| <1)\) such that \[ \|u_K - u^* \|_0 + |y_k - y^*|_1 + |p_K - p^*|_1 \leq c_1 \lambda^K_1 + c_2 \lambda^K_2.\tag{2} \] (The parameters in (2) are very important in both practical applications and theoretical analysis.) A precise proof of the convergence rate of the \(L^2\)-error of the control variable is proposed. Some numerical experiments are performed to verify the theoretical results.
scientific article; zbMATH DE number 5934909
scientific article