A Two Dimensional Backward Heat Problem With Statistical Discrete Data
  In this paper, we focus on the backward heat problem of finding the function $\theta(x,y)=u(x,y,0)$ such that \[ {l l l} u_t - a(t)(u_{xx} + u_{yy}) & = f(x,y,t), & \qquad (x,y,t) \in \Omega\times (0,T), u(x,y,T) & = h(x,y), & \qquad (x,y) \in\bar{\Omega}. \] where $\Omega = (0,\pi) \times (0,\pi)$ 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) + \vartheta\xi_{ij}(t), d_{ij} &=& h(x_i,y_j) + \sigma_{ij}\epsilon_{ij}, where $\xi_{ij}(t)$ are Brownian motions, $\epsilon_{ij}\sim \mathcal{N}(0,1)$, $(x_i,y_j)$ are grid points of $\Omega$ and $\sigma_{ij}, \vartheta$ are unknown positive constants. The noises $\xi_{ij}(t), \epsilon_{ij}$ are mutually independent. From the known data $g_{ij}(t)$ and $d_{ij}$, we can recovery the initial temperature $\theta(x,y)$. 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. 
Nguyen Dang Minh, To Duc Khanh, Nguyen Huy Tuan and Dang Duc Trong
math.AP