Convergence Rates for Identification of Robin Coefficient from Terminal Observations

Abstract This paper deals with the problem of identification of a Robin coefficient (also known as impedance coefficient) in a parabolic PDE from terminal observations of the temperature distributions. The problem is ill-posed in the sense that small perturbation in the observation may lead to a large deviation in the solution. Thus, in order to obtain stable approximations, we employ the Tikhonov-regularization. We propose a weak source condition motivated by the work of Engl and Zou (2000) and obtain a convergence rate of $$O(\delta ^\frac{1}{2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>δ</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> is the noise level of the observed data. The obtained rate is better than some of the previous known rates. Moreover, the advantage of the proposed source condition is that the above mentioned convergence rate is obtained without the need for characterizing the range space of modelling operator and its Fréchet derivative, which is in contrast to the general convergence theory of Tikhonov-regularization for non linear operators.