In this paper, we study the stochastic convergence of regularized solutions for backward heat conduction problems. These problems are recognized as ill-posed due to the exponential decay of eigenvalues associated with the forward problems. We derive an error estimate for the least-squares regularized minimization problem within the framework of stochastic convergence. Our analysis reveals that the optimal error of the Tikhonov-type least-squares optimization problem depends on the noise level, the number of sensors, and the underlying ground truth. Moreover, we propose a self-adaptive algorithm to identify the optimal regularization parameter for the optimization problem without requiring knowledge of the noise level or any other prior information, which will be very practical in applications. We present numerical examples to demonstrate the accuracy and efficiency of our proposed method. These numerical results show that our method is efficient in solving backward heat conduction problems.

翻译：本文研究了反向热传导问题正则化解的随机收敛性。由于正向问题特征值呈指数衰减，此类问题被公认为不适定。我们在随机收敛框架下推导了最小二乘正则化极小化问题的误差估计。分析表明，Tikhonov型最小二乘优化问题的最优误差依赖于噪声水平、传感器数量及真实解。此外，我们提出了一种自适应算法，无需知晓噪声水平或任何先验信息即可确定优化问题的最优正则化参数，这在实际应用中极具实用性。通过数值算例验证了所提方法的准确性与高效性，数值结果表明该方法能有效求解反向热传导问题。