In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and establish the stochastic convergence and optimal finite element convergence rates of these solutions, under pointwise measurement data with random noise. Unlike most existing regularization theories, the regularization error estimates are derived without any source conditions, while the error estimates of finite element solutions show their explicit dependence on the noise level, regularization parameter, mesh size, and time step size, which can guide practical choices among these key parameters in real applications. The error estimates also suggest an iterative algorithm for determining an optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the analytical results.

翻译：在这项工作中,我们调查受部分差异方程式制约的反源问题的正规化解决方案及其有限元素解决方案,并根据随机噪音的点度测量数据,确定这些解决方案的随机趋同率和最佳有限元素趋同率。 与大多数现有的正规化理论不同,正规化误差估计数是在没有任何源条件的情况下得出的,而有限元素解决方案的误差估计数表明它们明显依赖噪音水平、正规化参数、网目尺寸和时间级大小,这些误差可以指导这些关键参数在实际应用中的实际选择。 误差估计数还表明确定最佳正规化参数的迭代算法。