2021/04/06 by Zhiming Chen, Wenlong Zhang, Chen, Zhiming +3 · 3 citations
Decision Sciences · Engineering · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design #Structural Health Monitoring Techniques
paper · pdf · doi:10.48550/arxiv.2104.02352
openalex publication_date 2021/04/06 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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.