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

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An inverse boundary value problem for the p-Laplacian : A linearization approach. / Hannukainen, Antti; Hyvönen, Nuutti; Mustonen, Lauri.

In: Inverse Problems, Vol. 35, No. 3, 034001, 28.01.2019.

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@article{7b1d8b3ce3be43649a5f645e6a68c753,
title = "An inverse boundary value problem for the p-Laplacian: A linearization approach",
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 H{\"o}lder continuous conductivity coefficient to the solution of the Neumann problem, is Fr{\'e}chet differentiable, excluding the degenerate case T = 0 that corresponds to the classical (weighted) -Laplace equation.",
keywords = "Bayesian inversion, inverse boundary value problem, linearization, p-Laplacian",
author = "Antti Hannukainen and Nuutti Hyv{\"o}nen and Lauri Mustonen",
year = "2019",
month = "1",
day = "28",
doi = "10.1088/1361-6420/aaf2df",
language = "English",
volume = "35",
journal = "Inverse Problems",
issn = "0266-5611",
publisher = "IOP Publishing Ltd.",
number = "3",

}

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TY - JOUR

T1 - An inverse boundary value problem for the p-Laplacian

T2 - A linearization approach

AU - Hannukainen, Antti

AU - Hyvönen, Nuutti

AU - Mustonen, Lauri

PY - 2019/1/28

Y1 - 2019/1/28

N2 - 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.

AB - 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.

KW - Bayesian inversion

KW - inverse boundary value problem

KW - linearization

KW - p-Laplacian

UR - http://www.scopus.com/inward/record.url?scp=85064439591&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/aaf2df

DO - 10.1088/1361-6420/aaf2df

M3 - Article

AN - SCOPUS:85064439591

VL - 35

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 3

M1 - 034001

ER -

ID: 38958905