An inverse boundary value problem for the p-Laplacian: A linearization approach

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Abstract

This work tackles an inverse boundary value problem for a p-Laplace type partial differential equation parametrized by a smoothening parameter T ≥ 0. The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve as data. The numerical studies are based on a straightforward linearization of the forward map, and they demonstrate that the accuracy of such an approach depends nontrivially on 1 < p < ∞ and the chosen parametrization for the unknown coefficient. The numerical considerations are complemented by proving that the forward operator, which maps a Hölder continuous conductivity coefficient to the solution of the Neumann problem, is Fréchet differentiable, excluding the degenerate case T = 0 that corresponds to the classical (weighted) -Laplace equation.

Original languageEnglish
Article number034001
JournalInverse Problems
Volume35
Issue number3
DOIs
Publication statusPublished - 28 Jan 2019
MoE publication typeA1 Journal article-refereed

Keywords

  • Bayesian inversion
  • inverse boundary value problem
  • linearization
  • p-Laplacian

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  • Projects

    Centre of Excellence of Inverse Modelling and Imaging

    Ojalammi, A., Perkkiö, L., Hyvönen, N., Hirvi, P., Kuutela, T. & Puska, J.

    01/01/201831/12/2020

    Project: Academy of Finland: Other research funding

    Centre of Excellence of Inverse Modelling and Imaging

    Hannukainen, A., Kuutela, T., Ojalammi, A., Kuortti, J., Puska, J. & Perkkiö, L.

    01/01/201831/12/2020

    Project: Academy of Finland: Other research funding

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