Convergence rates for total variation regularization of coefficient identification problems in elliptic equations II

Abstract

AbstractWe investigate the convergence rates for total variation regularization of the problem of identifying (i) the coefficient q in the Neumann problem for the elliptic equation −div(q∇u)=f in Ω, q∂u/∂n=g on ∂Ω, (ii) the coefficient a in the Neumann problem for the elliptic equation −Δu+au=f in Ω, ∂u/∂n=g on ∂Ω, Ω⊂Rd, d⩾1, when u is imprecisely given by zδ∈H1(Ω), ‖u−zδ‖H1(Ω)⩽δ, δ>0. We regularize these problems by correspondingly minimizing the strictly convex functionals12∫Ωq|∇(U(q)−zδ)|2dx+ρ(12‖q‖L2(Ω)2+∫Ω|∇q|), and12∫Ω|∇(U(a)−zδ)|2dx+12∫Ωa(U(a)−zδ)2dx+ρ(12‖a‖L2(Ω)2+∫Ω|∇a|) over admissible sets, where U(q) (U(a)) is the solution of the first (second) Neumann boundary value problem, ρ>0 is the regularization parameter. Taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems, we obtain the convergence rates of them to the solution of the inverse problem in the sense of the Bregman distance and in the L2-norm under relatively simple source conditions without the smallness requirement on the source functions