Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements

TL;DR

This paper develops a modified quasi-reversibility method using a filter regularized operator and finite element analysis to solve unstable terminal value parabolic problems with noisy data, providing convergence and regularity results.

Contribution

It introduces a novel quasi-reversibility approach with filter regularization for nonlinear parabolic problems, analyzed within a variational framework suitable for finite element methods.

Findings

Convergence rates in $L^2$-$H^1$ are established for smooth solutions.

Error estimates depend on domain dimension and noise level.

Backward uniqueness for the quasi-linear system is proved.

Abstract

This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the so-called filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semi-linear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo-Galerkin method to study the weak solvability of the approximate problem. Relying on the energy-like analysis, we provide detailed convergence rates in $L^2$-$H^1$ of the proposed method when the true solution is sufficiently smooth. Depending on the dimensions of the domain, we obtain an error estimate in $L^{r}$ for some $r>2$. Proof of the backward uniqueness for the quasi-linear system is also depicted in this work. To prove the regularity assumptions acceptable, several physical applications are discussed.