Recovering Initial Values and a Random Source Simultaneously for a Damped Wave Equation

ABSTRACT This paper introduces an inverse problem for a stochastic damped wave equation, in which the source is driven by a fractional Brownian motion. The well‐posedness of direct problem is obtained by analyzing regularity of the solution for the equivalent stochastic initial value problem in frequency domain. The inverse problem involves recovering two initial values and a random source simultaneously. It is shown that the mean of final observations at two different moments uniquely determines the initial values. Additionally, for inversion of random source, it is demonstrated that its sine modulus can be uniquely determined by the variance of final observations. Based on this sine modulus, recovering the unknown random source can be transformed into a well‐known phase retrieval problem, which can be successfully solved using the methodology of phaselift. Finally, numerical examples are presented to demonstrate the effectiveness of this method.