Convergence rates for Tikhonov regularization of a two-coefficient identification problem in an elliptic boundary value problem

Abstract

We investigate the convergence rates for Tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the Neumann problem for the elliptic equation ${{-{\rm div}(q \nabla u) + au = f \;{\rm in}\; \Omega, q{\partial u}/{\partial n} = g}}$ on the boundary ${{\partial\Omega, \Omega \subset \mathbb{R}^d, d \geq 1}}$ , when u is imprecisely given by ${{{z^\delta} \in {H^1}(\Omega), \|u-z^\delta\|_{H^1(\Omega)}\le\delta, \delta > 0}}$. We regularize this problem by minimizing the strictly convex functional of (q, a)

$$\begin{array}{lll}\int\limits_{\Omega}\left(q| \nabla (U(q,a)-z^{\delta})|^2 + a(U(q,a)-z^{\delta})^2\right)dx\\\quad+\rho\left(\|q-q^*\|^2_{L^2(\Omega)} + \|a-a^*\|^2_{L^2(\Omega)}\right)\end{array}$$

over the admissible set K, where ρ > 0 is the regularization parameter and (q*, a*) is an a priori estimate of the true pair (q, a) which is identified, and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem. We obtain the convergence rate ${{{\mathcal {O}}(\sqrt{\delta})}}$, as δ → 0 and ρ ~ δ, for the regularized solutions under the simple and weak source condition

$${\rm there\;is\;a\;function}\;w^* \in V^*\;{\rm such\;that}\;{U^\prime (q^ \dagger, a^\dagger)}^*w^* = (q^\dagger - q^*, a^\dagger - a^*)$$

with ${{(q^\dagger, a^\dagger)}}$ being the (q*, a*)-minimum norm solution of the coefficient identification problem, U′(·, ·) the Fréchet derivative of U(·, ·), V the Sobolev space on which the boundary value problem is considered. Our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill-posed problems. Furthermore, some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces.