Abstract
In this talk, we will present the Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to the optimal control problems arising from elliptic partial differential equations (PDE) under an $H^{-1}$ regularization form. This problem reads as for a given state $y$ governed by a stationary heat conduction equation, we wish to drive the solution of this PDE to a desired state $\hat y$ through a control $u$. The goal is to find the optimal control $u_*$ which minimizes the discrepancy between these states (i.e. original state $y$ and desired state $\hat y$). We first use the Lagrange multiplier approach to derive a forward-backward system. Thanks to the $H^{-1}$ regularization, we show that this forward-backward system can be simplified into one single second order PDE related to the state $y$, which avoids to solve a coupled BiLaplacian problem. This simplified problem can then be solved by using DN and NN methods. Finally, we provide the convergence analysis for these two methods along with some numerical results.
卢 柳 䃅
Postdoctoral Fellow in Applied Mathematics
My research interests include Numerical Analysis, Mathematical Biology, Scientific Computing, Optimization and Control.