A Two Dimensional Backward Heat Problem With Statistical Discrete Data

TL;DR

This paper addresses the ill-posed backward heat problem with statistical noisy data, proposing a regularization method using trigonometric expansion to recover initial temperature and analyzing its convergence.

Contribution

It introduces a novel regularization approach combining trigonometric methods with truncated expansion for statistical inverse heat problems.

Findings

Proposed a regularization method for noisy inverse heat problems.

Established convergence rates for the regularized solution.

Validated the method through numerical experiments.

Abstract

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.