A Two Dimensional Backward Heat Problem With Statistical Discrete Data

In this paper, we focus on the backward heat problem of finding the function $θ(x,y)=u(x,y,0)$ such that \[ {l l l} u_t - a(t)(u_{xx} + u_{yy}) & = f(x,y,t), & \qquad (x,y,t) \in Ω\times (0,T), u(x,y,T) & = h(x,y), & \qquad (x,y) \in\barΩ. \] where $Ω= (0,π) \times (0,π)$ and the heat transfer coefficient $a(t)$ is known. In our problem, the source $f = f(x,y,t)$ and the final data $h(x,y)$ are unknown. We only know random noise data $g_{ij}(t)$ and $d_{ij}$ satisfying the regression models g_{ij}(t) &=& f(x_i,y_j,t) + \varthetaξ_{ij}(t), d_{ij} &=& h(x_i,y_j) + σ_{ij}ε_{ij}, where $ξ_{ij}(t)$ are Brownian motions, $ε_{ij}\sim \mathcal{N}(0,1)$, $(x_i,y_j)$ are grid points of $Ω$ and $σ_{ij}, \vartheta$ are unknown positive constants. The noises $ξ_{ij}(t), ε_{ij}$ are mutually independent. From the known data $g_{ij}(t)$ and $d_{ij}$, we can recovery the initial temperature $θ(x,y)$. However, the result thus obtained is not stable and the problem is severely ill--posed. To regularize the instable solution, we use the trigonometric method in nonparametric regression associated with the truncated expansion method. In addition, convergence rate is also investigated numerically.

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