Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations

Abstract

We investigate the convergence rates for Tikhonov regularization of the problem of identifying (1) the coefficient q ∊ L∊ fty(Ω) in the Dirichlet problem −div(q∇u) = f in Ω, u = 0 on ∂Ω, and (2) the coefficient a ∊ L∊ fty(Ω) in the Dirichlet problem −Δu + au = f in Ω, u = 0 on ∂Ω, when u is imprecisely given by zδ ∊ H10(Ω), , We regularize these problems by correspondingly minimizing the strictly convex functionals and where U(q) (U(a)) is the solution of the first (second) Dirichlet problem, ρ > 0 is the regularization parameter and q* (or a*) is an a priori estimate of q (or a). We prove that these functionals attain a unique global minimizer on the admissible sets. Further, we give very simple source conditions without the smallness requirement on the source functions which provide the convergence rate for the regularized solutions.