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

Finite element methods for coefficient identification in an elliptic equation

“…from aquifer analysis. For surveys on the subject, we refer the reader to [1][2][3][4][5][6][7][8][9] and the references therein.…”

“…Concerning such a type of observations, we refer the reader to [7][8][9][10][11][12][14][15][16], where detailed discussions about the ill-posedness of the identification problem and interpolations of discrete measurements of the solution u resulting the data z δ satisfying (1.4) are given. For solving the above identification problem, in [4][5][6][7] we followed Knowles [16,18] and Zou [17] in minimizing the so-called energy functional…”

“…Here, u(q) is the unique solution of (1.1)-(1.3) which depends non-linearly on q. We note that in contrast to the least squares approach, this functional is convex, [4][5][6][7] therefore it is much easier to find its global minimizers. We then applied different kinds of Tikhonov regularization (regularization by L 2 -norm, by total variation and by total variation in combination with L 2 -norm) to the above minimization problem and proved several convergence rates of the methods as the noise level tends to zero without assuming the "small enough" condition of the source functions.…”

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

“…The last problems arise from different contexts of applied sciences, e.g., from aquifer analysis. For surveys on the subject, we refer the reader to [4,5,9,13,[16][17][18]22,26,28,31,34,35,40] and the references therein. To our knowledge, Baumeister and Kunisch [5] were the first who investigated the above inverse problem and that for the Dirichlet problem.…”

“…Further, their source conditions are hard to check and require high regularity of the sought coefficient (see [14,15]). Recently, we [18] followed Zou [41] and Knowles [28] in using the so-called convex energy functionals instead of least squares ones, and then applied Tikhonov regularization to our one-coefficient estimation problems. 1 We got convergence rates of this approach under very simple source conditions without assuming the "small enough condition" which is popular in the theory of nonlinear ill-posed problems (see e.g.…”