In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and we establish the stochastic convergence and optimal finite element convergence rates of these solutions under pointwise measurement data with random noise. The regularization error estimates and the finite element error estimates are derived with 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.

　　Publication: 

　　SIAM Journal on Numerical Analysis, Vol. 60, Iss. 210, pp: 751-780. 

　　Author: 

　　Zhiming Chen 

　　LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences and School of Mathematical Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China 

　　Email: zmchen@lsec.cc.ac.cn  

　　Wenlong Zhang 

　　Department of Mathematics, Southern University of Science and Technology (SUSTech), Shenzhen, Guangdong Province, People's Republic of China 

　　Email: zhangwl@sustech.edu.cn  

　　Jun Zou 

　　Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong 

　　Email: zou@math.cuhk.edu.hk