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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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# A two-dimensional backward heat problem with statistical discrete data

Nguyen Dang Minh
• Corresponding author
• Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, 227 Nguyen Van Cu, Dist. 5, Ho Chi Minh City, Vietnam
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• Other articles by this author:
/ Khanh To Duc
/ Nguyen Huy Tuan
• Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, 227 Nguyen Van Cu, Dist. 5, Ho Chi Minh City, Vietnam
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• Other articles by this author:
/ Dang Duc Trong
• Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, 227 Nguyen Van Cu, Dist. 5, Ho Chi Minh City, Vietnam
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Published Online: 2017-05-05 | DOI: https://doi.org/10.1515/jiip-2016-0038

## Abstract

We focus on the nonhomogeneous backward heat problem of finding the initial temperature $\theta =\theta \left(x,y\right)=u\left(x,y,0\right)$ such that

$\left\{\begin{array}{cccc}\hfill {u}_{t}-a\left(t\right)\left({u}_{xx}+{u}_{yy}\right)& =f\left(x,y,t\right),\hfill & \left(x,y,t\right)\hfill & \hfill \in \mathrm{\Omega }×\left(0,T\right),\\ \hfill u\left(x,y,t\right)& =0,\hfill & \left(x,y\right)\hfill & \hfill \in \partial \mathrm{\Omega }×\left(0,T\right),\\ \hfill u\left(x,y,T\right)& =h\left(x,y\right),\hfill & \left(x,y\right)\hfill & \hfill \in \overline{\mathrm{\Omega }},\end{array}$

where $\mathrm{\Omega }=\left(0,\pi \right)×\left(0,\pi \right)$. In the problem, the source $f=f\left(x,y,t\right)$ and the final data $h=h\left(x,y\right)$ are determined through random noise data ${g}_{ij}\left(t\right)$ and ${d}_{ij}$ satisfying the regression models

${g}_{ij}\left(t\right)=f\left({X}_{i},{Y}_{j},t\right)+\vartheta {\xi }_{ij}\left(t\right),$${d}_{ij}=h\left({X}_{i},{Y}_{j}\right)+{\sigma }_{ij}{\epsilon }_{ij},$

where $\left({X}_{i},{Y}_{j}\right)$ are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.

MSC 2010: 35K05; 47A52; 62G08

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Revised: 2017-02-14

Accepted: 2017-02-26

Published Online: 2017-05-05

Published in Print: 2018-02-01

This research was supported by Vietnam National University-Ho Chi Minh City (VNU-HCM) under the Grant number B2017-18-03.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 1, Pages 13–31, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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