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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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Volume 26, Issue 1

Issues

A two-dimensional backward heat problem with statistical discrete data

Nguyen Dang Minh
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  • 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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/ 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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/ 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 θ=θ(x,y)=u(x,y,0) such that

{ut-a(t)(uxx+uyy)=f(x,y,t),(x,y,t)Ω×(0,T),u(x,y,t)=0,(x,y)Ω×(0,T),u(x,y,T)=h(x,y),(x,y)Ω¯,

where Ω=(0,π)×(0,π). In the problem, the source f=f(x,y,t) and the final data h=h(x,y) are determined through random noise data gij(t) and dij satisfying the regression models

gij(t)=f(Xi,Yj,t)+ϑξij(t),dij=h(Xi,Yj)+σijεij,

where (Xi,Yj) 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.

Keywords: Backward heat problems; nonhomogeneous heat equation; ill-posed problems; nonparametric regression; statistical inverse problems

MSC 2010: 35K05; 47A52; 62G08

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About the article

Received: 2016-06-17

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, DOI: https://doi.org/10.1515/jiip-2016-0038.

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