Abstract
In this talk, we will present some non-overlapping Domain Decomposition Methods (DDM) applied to the optimal control problems arising from elliptic partial differential equations (PDE). 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. Instead of the commonly used $L^2$ regularization, we show that this forward-backward system can be simplified into one single second order PDE related to the state $y$ by applying an $H^{-1}$ regularization. This avoids solving a coupled BiLaplacian problem. The simplified problem can then be solved with DDM. We provide the convergence analysis for Dirichlet-Neumann and Neumann-Neumann 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.