This paper investigates an inverse random source problem for stochastic evolution equations, including stochastic heat and wave equations. The unknown source is modeled as g(x)f(t)Ẇ(t), where g is the spatial component of the source, f is the unknown time-dependent component, and Ẇ denotes the time-dependent Gaussian white noise. The research commences with the establishment of the well-posedness of the corresponding stochastic direct problem. Under suitable regularity conditions, the existence of stochastic strong solutions for both the stochastic heat and wave equations is demonstrated. For the inverse problem, the objective is to uniquely recover the strength |f(t)| of the time-dependent component of the source from the boundary flux on a nonempty open subset. The uniqueness of the recovery for both the stochastic heat and wave equations is proven, and several numerical examples are given to verify the theoretical results.

Publication:

INVERSE PROBLEMS

http://dx.doi.org/10.1088/1361-6420/ae871a

Author:

Xu Wang

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

Email: wangxu@lsec.cc.ac.cn

Guanlin Yang

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

Author to whom any correspondence should be addressed.

Email: yangguanlin@lsec.cc.ac.cn

Zhidong Zhang

School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, People’s Republic of China

Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510000, Guangdong, People’s Republic of China

Email: zhangzhidong@mail.sysu.edu.cn