Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions

Abstract

In this paper we study the problem of identifying the solution x † of linear ill-posed problems Ax = y with non-negative and self-adjoint operators A on a Hilbert space X where instead of exact data y noisy data y δ ∈ X are given satisfying y − y δ ≤ δ with known noise level δ. Regularized approximations x δ α are obtained by the method of Lavrentiev regularization, that is, x δ α is the solution of the singularly perturbed operator equation Ax + αx = y δ , and the regularization parameter α is chosen either a priori or a posteriori by the rule of Raus. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximations provide order optimal error bounds on the set M. Our results cover the special case of finitely smoothing operators A and extend recent results for infinitely smoothing operators. In addition, we generalize our results to the method of iterated Lavrentiev regularization of order m and discuss a special ill-posed problem arising in inverse heat conduction.

Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Problems

“…Assumption 2.4 is known as a general source condition, and it is similar to that considered in [14] by Nair and Tautenhahn for the linear case and it is also analogous to the source condition considered recently by the current authors [13]. It includes both the well-known source conditions, namely, the Hölder-type source condition, that is, ϕ(λ) = λ µ , and the logarithmic source condition, that is, ϕ(λ) = [log(1/λ)] −µ .…”

Parameter choice in the Lavrentiev regularization of the data completion problem

“…Notice that, contrary to [15] where the so-called general source conditions are added, we do not assume any further regularity on λ (more than H 1/2 00 (Γ I )). The general theory of linear ill-posed problems suggests that a sufficient condition for the Lavrentiev solution λ ǫ ̺ to converge toward λ is to fix ̺ = ̺(ǫ) such that (see [8])…”

“…In effective computations of ill-posed problems, the regularization parameter is most often chosen a posteriori using various approaches (see [8,15]). We set our preference to the discrepancy principle attributed to Morozov.…”

“…The most widespread regularizations are those of Tikhonov type, based on the least-squares (see [18,8]). Given that we are rather involved in the Kohn-Vogelius method (see [11]), and seen the symmetry of our variational Cauchy problem, the Lavrentiev regularization seems better suited and is then applied (see [13,15]). Some partial results about the convergence of the method have been established in [3], when Cauchy data are noisy and hence no existence is guaranteed.…”

Quadratic optimization in ill-posed problems

“…The condition that b ∈ R( √ A) is nothing else than the general source condition (see [23,16]). In the regularization theory, such an assumption is made on the exact solution while it appears naturally on the data in our analysis.…”