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 $a \in L^{\infty}(Ω)$ in the Robin-boundary equation $-\mathrm{div}(a\nabla u)-bu=f,~ x \in Ω\subset \mathbb R^M,~ M \geq 1$ and $u=0,~ x ~on~ \partialΩ$, where $f(x)\in L^{\infty}(Ω)$. Assume we only know the imprecise values of $u$ in the subset $Ω_1 \subset Ω$ given by $z^δ \in {H}^1(Ω_1)$, satisfies $\|u-z^δ\|_{H^1(Ω_1)}\leq δ$. We assume $u$ satisfy the following boundary conditions on $\partialΩ_1$: \begin{align*} \nabla u \cdot \vec{n}+γu =0~on~\partialΩ_1, \end{align*} where $\vec{n}$ is the normal vector of $\partialΩ_1$ and $γ>0$ is a constant. We regularize this problem by correspondingly minimizing the strictly convex functional: \begin{align*} \min \limits_{a \in \mathbb A} &\frac12 \int_{Ω_1} a | {\nabla(U(a)-z^δ)}|^2 +\frac12\int_{\partialΩ_1} aγ[U(a)-z^δ]^2-\frac12 \int_{Ω_1} b [U(a)-z^δ]^2\\ &+ ρ\| a-a^* \|^2_{L^2(Ω)}, \end{align*} where $U(a)$ is a map for $a$ to the solution of the Robin-boundary problem, $ρ> 0$ 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 $O(\sqrtδ)$ for the regularized solution.