Stochastic Convergence of Regularized Solutions and Their Finite Element Approximations to Inverse Source Problems

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