A New Regularization Method for a Parameter Identification Problem in a Non-Linear Partial Differential Equation

Abstract

We consider a parameter identification problem associated with a quasilinear elliptic Neumann boundary value problem involving a parameter function $a(·)$ and the solution $u(·),$ where the problem is to identify $a(·)$ on an interval $I:=g(Γ)$ from the knowledge of the solution $u(·)$ as $g$ on $Γ,$ where Γ is a given curve on the boundary of the domain $Ω⊆\mathbb{R}^3$ of the problem and $g$ is a continuous function. The inverse problem is formulated as a problem of solving an operator equation involving a compact operator depending on the data, and for obtaining stable approximate solutions under noisy data, a new regularization method is considered. The derived error estimates are similar to, and in certain cases better than, the classical Tikhonov regularization considered in the literature in recent past.