TY - JOUR

T1 - Numerical solving of an inverse problem of a hyperbolic heat equation with small parameter

AU - Akindinov, G. D.

AU - Matyukhin, V. V.

AU - Krivorotko, O. I.

N1 - The research was supported by Russian Science Foundation (project No. 21-71-30005), https://rscf.ru/en/project/21-71-30005/. Публикация для корректировки.

PY - 2023

Y1 - 2023

N2 - In this paper we describe an algorithm of numerical solving of an inverse problem on a hyperbolic heat equation with additional second time derivative with a small parameter. The problem in this case is finding an initial distribution with given final distribution. This algorithm allows finding a solution to the problem for any admissible given precision. Algorithm allows evading difficulties analogous to the case of heat equation with inverted time. Furthermore, it allows finding an optimal grid size by learning on a relatively big grid size and small amount of iterations of a gradient method and later extrapolates to the required grid size using Richardson’s method. This algorithm allows finding an adequate estimate of Lipschitz constant for the gradient of the target functional. Finally, this algorithm may easily be applied to the problems with similar structure, for example in solving equations for plasma, social processes and various biological problems. The theoretical novelty of the paper consists in the developing of an optimal procedure of finding of the required grid size using Richardson extrapolations for optimization problems with inexact gradient in ill-posed problems.

AB - In this paper we describe an algorithm of numerical solving of an inverse problem on a hyperbolic heat equation with additional second time derivative with a small parameter. The problem in this case is finding an initial distribution with given final distribution. This algorithm allows finding a solution to the problem for any admissible given precision. Algorithm allows evading difficulties analogous to the case of heat equation with inverted time. Furthermore, it allows finding an optimal grid size by learning on a relatively big grid size and small amount of iterations of a gradient method and later extrapolates to the required grid size using Richardson’s method. This algorithm allows finding an adequate estimate of Lipschitz constant for the gradient of the target functional. Finally, this algorithm may easily be applied to the problems with similar structure, for example in solving equations for plasma, social processes and various biological problems. The theoretical novelty of the paper consists in the developing of an optimal procedure of finding of the required grid size using Richardson extrapolations for optimization problems with inexact gradient in ill-posed problems.

KW - Richardson method

KW - hyperbolic heat equation

KW - inexact gradient

KW - inverse and ill-posed problems

KW - regularization

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85165887177&origin=inward&txGid=453d0ba6c8bf2cb25a165209d811cdf3

UR - https://www.elibrary.ru/item.asp?id=53767542

UR - https://www.mendeley.com/catalogue/c5a2a7e0-9138-366e-8a18-33f7241e508d/

U2 - 10.20537/2076-7633-2023-15-2-245-258

DO - 10.20537/2076-7633-2023-15-2-245-258

M3 - Article

VL - 15

SP - 245

EP - 258

JO - Computer Research and Modeling

JF - Computer Research and Modeling

SN - 2076-7633

IS - 2

ER -