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

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 concern-ing heat conduction and tumour source localization. In accordance with the weak solvability result for the forward problem, we focus on the inverse problem with the high-order Sobolev-Gevrey smoothness and with Sobolev measurements. As ex-pected from the well-known results for the linear case, we prove that this nonlinear spectral regularization possesses a logarithmic rate of convergence in the high-order Sobolev norm. The proof can be done by the verification of variational source con-dition; this way interestingly validates such a fine strategy in the framework of inverse problems for nonlinear partial differential equations. Ultimately, we design an iteration-based version of the proposed method for a class of reaction-diffusion problems with non-degenerate nonlinearity, taking into account the conventional time discretization. The convergence of this iterative scheme is also investigated.(c) 2022 Elsevier Inc. All rights reserved.