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

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.…”

“…For more detailed discussions about this, we refer the reader to our papers. [4][5][6][7] In this paper, we apply the finite element method to this optimization problem and prove some convergence results and error estimates of the method. We note that several researchers, such as Falk [19], Kaltenbacher and Schöberl [14], Kohn and Lowe [8], Kärkkäinen [20], Zou [17], and Wang and Zou [9], used the finite element method to the above coefficient identification problem.…”

Finite element approximation of source term identification with TV-regularization

“…Finite element analysis of the problem of simultaneously identifying the source term and coefficients from distributed observations can be found in [45]. For the treatment of coefficient identification problems employing T V -regularization techniques we refer to [15,18,26,27,28,47].…”