  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.
