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Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations

  • Alexander S. Leonov ORCID logo EMAIL logo , Alexander N. Sharov and Anatoly G. Yagola
Published/Copyright: June 30, 2021

Abstract

A fast algorithm for calculating the gradient of the Tikhonov functional is proposed for solving inverse coefficient problems for linear partial differential equations of a general form by the regularization method. The algorithm is designed for problems with discretized differential operators that linearly depend on the desired coefficients. When discretizing the problem and calculating the gradient, it is possible to use the finite element method. As an illustration, we consider the solution of two inverse problems of elastography using the finite element method: finding the distribution of Young’s modulus in biological tissue from data on its compression and a similar problem of determining the characteristics of local oncological inclusions, which have a special parametric form.


Dedicated to the 70th anniversary of Michael V. Klibanov, our dear colleague


Award Identifier / Grant number: 19-51-53005-NSFC-a

Award Identifier / Grant number: 02.a03.21.0005

Award Identifier / Grant number: 27.08.2013

Funding statement: The work was supported by the Russian Foundation for Basic Research (grant no. 19-51-53005-NSFC-a) and the Program of Competitiveness Increase of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); contract no. 02.a03.21.0005, 27.08.2013.

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Received: 2020-08-20
Revised: 2021-02-26
Accepted: 2021-05-20
Published Online: 2021-06-30
Published in Print: 2022-02-01

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