Convergence Rates For Tikhonov Regularization of Coefficient Identification Problems in Robin-Boundary Equation

Abstract

This paper investigates the convergence rate for Tikhonov regularization of the problem of identifying the coefficient $a \in L^{\infty}(\Omega)$ in the Robin-boundary equation $-\mathrm{div}(a\nabla u)-bu=f,~ x \in \Omega \subset \mathbb R^M,~ M \geq 1$ and $u=0,~ x ~on~ \partial\Omega$, where $f(x)\in L^{\infty}(\Omega)$. Assume we only know the imprecise values of $u$ in the subset $\Omega_1 \subset \Omega$ given by $z^{\delta} \in {H}^1(\Omega_1)$, satisfies $\|u-z^{\delta}\|_{H^1(\Omega_1)}\leq \delta$. We assume $u$ satisfy the following boundary conditions on $\partial\Omega_1$: \begin{align*} \nabla u \cdot \vec{n}+\gamma u =0~on~\partial\Omega_1, \end{align*} where $\vec{n}$ is the normal vector of $\partial\Omega_1$ and $\gamma>0$ is a constant. We regularize this problem by correspondingly minimizing the strictly convex functional: \begin{align*} \min \limits_{a \in \mathbb A} &\frac12 \int_{\Omega_1} a | {\nabla(U(a)-z^\delta)}|^2 +\frac12\int_{\partial\Omega_1} a\gamma [U(a)-z^\delta]^2-\frac12 \int_{\Omega_1} b [U(a)-z^\delta]^2\\ &+ \rho \| a-a^* \|^2_{L^2(\Omega)}, \end{align*} where $U(a)$ is a map for $a$ to the solution of the Robin-boundary problem, $\rho > 0$ is the regularization parameter and $a^*$ is a priori estimate of $a$. We prove that the functional attain a unique global minimizer on the admissible set. Further, we give very simple source condition without the smallness requirement on the source function which provide the convergence rate $O(\sqrt{\delta})$ for the regularized solution.