Convergence of a spectral regularization of a time-reversed reaction-diffusion problem with high-order Sobolev-Gevrey smoothness

TL;DR

This paper investigates a spectral regularization method for a time-reversed reaction-diffusion problem with high-order smoothness, demonstrating logarithmic convergence rates and validating the approach through variational source conditions.

Contribution

It introduces a novel spectral regularization technique for nonlinear reaction-diffusion inverse problems with high-order Sobolev-Gevrey smoothness, including convergence analysis and a semi-discrete iterative scheme.

Findings

Logarithmic convergence rate in high-order Sobolev norm.

Validation of variational source condition for nonlinear PDEs.

Convergence analysis of semi-discrete iterative regularization.

Abstract

The present paper analyzes a spectral regularization of a time-reversed reaction-diffusion problem with globally and locally Lipschitz nonlinearities. This type of inverse and ill-posed problems arises in a variety of real-world applications concerning heat conduction and tumour source localization. In accordance with the weak solvability result for the forward problem, we focus on the inverse problem with high-order Sobolev-Gevrey smoothness and with Sobolev measurements. As expected from the well-known results for the linear case, we prove that this nonlinear spectral regularization possesses a logarithmic rate of convergence in a high-order Sobolev norm. The proof can be done by the verification of variational source condition; this way validates such a fine strategy in the framework of inverse problems for nonlinear partial differential equations. Ultimately, we study a semi-discrete version of the regularization method for a class of reaction-diffusion problems with non-degenerate nonlinearity. The convergence of this iterative scheme is also investigated.