Convergence Rates of Tikhonov Regularizations for Elliptic and Parabolic Inverse Radiativity Problems

TL;DR

This paper analyzes the convergence rates of Tikhonov regularization methods for recovering radiative properties in elliptic and parabolic inverse problems, establishing stability estimates and introducing new variational source conditions.

Contribution

It introduces new variational source conditions for high-dimensional inverse radiativity problems and rigorously verifies them under Lipschitz stability, achieving reliable convergence rates.

Findings

Lipschitz stability estimates are derived for the inverse problems.

New variational source conditions are proposed and verified.

Convergence rates are established under these conditions.

Abstract

We shall study in this paper the Lipschitz type stabilities and convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions. The Lipschitz type stability estimates are derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions for the considered inverse radiativity problems in high dimensional spaces, some new variational source conditions are proposed. The conditions are rigorously verified in general dimensional spaces under the Lipschitz type stability estimates and the reasonable convergence rates are achieved.