  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.
