Convergence rates of Tikhonov regularizations for elliptic and parabolic inverse radiativity problems

Abstract

We shall study in this paper the convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions in general dimensional spaces. The conditional stability estimates are first 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 conditional stability estimates. We shall finally derive the reasonable convergence rates, which explicitly reveals the relation between the regularity of the radiativities and the convergence rates results.

Variational source conditions for inverse Robin and flux problems by partial measurements

“…Recently, the validity of VSC was shown for the ill-posed backward nonlinear Maxwell's equations in [10], by means of the semigroup theory and extrapolation of Hilbert spaces. The authors of this work verified VSC and obtained a Hölder type convergence rate in [7,9,11] for the Tikhonov regularized solutions of the inverse elliptic and parabolic radiativity problems. It is important to emphasize that all the above results were established under the full measurement data.…”

“…An inverse elliptic Robin problem by the boundary data was taken to be an application for the novel theory in [25], but the special source condition and strong nonlinearity condition may not be easy to verify in general dimensional spaces. We emphasize that all the above results were achieved under the full data in the whole domain (see, e.g., [9,15,19]) or on the entire boundary (see, e.g., [31,25]).…”

“…But the parameters to be identified may not be accessible over the entire boundary in most applications. The authors derived recently in [9] some reasonable convergence rates of the Tikhonov regularizations for elliptic and parabolic inverse radiativity problems in the framework of the variational source condition. Even if the radiativities may not be known on the boundary, the variational source conditions can still be rigorously verified, and the convergence results also revealed the explicit relation between the regularity of the radiativities and the convergence rates.…”

Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion

“…We end this section by establishing the following Sobolev norm estimate for a product of two functions over a Lipschitz domain (see e.g. [6]) and the Aubin-Dubinskii lemma ( see e.g. [1]).…”

“…under the parameter choice α = O(δ 2−θ ). 6. Discussions on possible extensions of the nonlinearity F .…”