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A new regularization method for a parameter identification problem in a non-linear partial differential equation
  • M Thamban Nair,
  • Samprita Roy
M Thamban Nair
Indian Institute of Technology Madras
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Samprita Roy
Indian Institute of Technology Madras
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Abstract

We consider a parameter identification problem associated with a quasi-linear elliptic Neumann boundary value problem involving a parameter function \(a(\cdot)\) and the solution \(u(\cdot)\), where the problem is to identify \(a(\cdot)\) on an interval \(I:=g(\Gamma)\) from the knowledge of the solution \(u(\cdot)\) as \(g\) on \(\Gamma\), where \(\Gamma\) is a given curve on the boundary of the domain \(\Omega\subseteq{\mathbb{R}}^{3}\) 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 the recent past.