Convergence Estimates of Regularized Solutions to Inverse Space-Dependent Source Problems with Time-Dependent Boundary Measurement

SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1369-1397, August 2025.
Abstract. In this work, we investigate the Tikhonov-type regularized solutions and their finite element solutions to the inverse space-dependent source problem from boundary measurement data. First, with the classical source condition, we establish the convergence of regularized solutions and their finite element solutions under the standard [math] norm. The error estimates present explicit dependence on the critical parameters like noise level, regularization parameter, mesh size, and time step size. Next, based on a proposed weak norm, we get the stability of Lipschitz type for the inverse problem, and then the first order convergence of regularized solutions can be derived in the sense of weak norm. We get this convergence without any source condition. Moreover, this work is carried out for the discrete data. We suppose that the observation points are discrete and the pointwise measurement data come with independent sub-Gaussian random noises. Then we give the stochastic convergence of regularized solutions and propose an efficient iterative algorithm to determine the optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithms.