## Abstract

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

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

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

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

where $({X}_{i},{Y}_{j})$ 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.

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