标题： 矩，时间反演和热方程的源识别 显示英文标题

标题： Moments, Time-Inversion and Source Identification for the Heat Equation

摘要： 我们解决热方程的初始源识别问题，这是一个显著不适定的反问题，其特征是指数不稳定性。 脱离经典的Tikhonov正则化，我们提出了一种基于热流矩分析的新方法，将问题转化为更稳定的逆矩公式。 通过将测量的终端时间矩通过其控制的ODE系统向后演化，我们恢复了初始分布的矩。 然后通过求解一个凸优化问题来重建源，该问题在这些矩约束下最小化测度的总变分。 这种公式自然促进了稀疏性，产生由Dirac测度之和组成的原子解。 与现有方法相比，我们的基于矩的方法将终端时间的指数误差增长降低为多项式增长。 我们提供了关于恢复的初始分布的显式误差估计，涉及矩阶数、终端时间和测量误差。 此外，我们开发了高效的数值离散化方案，并通过全面的数值实验展示了我们方法的显著稳定性提升。 显示英文摘要

摘要： We address the initial source identification problem for the heat equation, a notably ill-posed inverse problem characterized by exponential instability. Departing from classical Tikhonov regularization, we propose a novel approach based on moment analysis of the heat flow, transforming the problem into a more stable inverse moment formulation. By evolving the measured terminal time moments backward through their governing ODE system, we recover the moments of the initial distribution. We then reconstruct the source by solving a convex optimization problem that minimizes the total variation of a measure subject to these moment constraints. This formulation naturally promotes sparsity, yielding atomic solutions that are sums of Dirac measures. Compared to existing methods, our moment-based approach reduces exponential error growth to polynomial growth with respect to the terminal time. We provide explicit error estimates on the recovered initial distributions in terms of moment order, terminal time, and measurement errors. In addition, we develop efficient numerical discretization schemes and demonstrate significant stability improvements of our approach through comprehensive numerical experiments.