In this paper, a Cauchy problem of non-homogenous stochastic heat equation is considered together with its inverse source problem, where the source term is assumed to be driven by an additive white noise. The Cauchy problem (direct problem) is to determine the displacement of random temperature field, while the considered inverse problem is to reconstruct the statistical properties of the random source, i.e. the mean and variance of the random source. It is proved constructively that the Cauchy problem has a unique mild solution, which is expressed an integral form. Then the inverse random source problem is formulated into two Fredholm integral equations of the first kind, which are typically ill-posed. To obtain stable inverse solutions, the regularized block Kaczmarz method is introduced to solve the two Fredholm integral equations. Finally, numerical experiments are given to show that the proposed method is efficient and robust for reconstructing the statistical properties of the random source.

中文翻译：

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非齐次随机热方程的柯西问题及其在逆随机源问题中的应用

在本文中，考虑了非齐次随机热方程的柯西问题及其逆源问题，其中源项被假定为由加性白噪声驱动。Cauchy问题（直接问题）是确定随机温度场的位移，而所考虑的反问题是重构随机源的统计属性，即随机源的均值和方差。建设性地证明了柯西问题具有唯一的温和解，并用积分形式表示。然后，将逆随机源问题公式化为两个第一类通常是不适定的第一类Fredholm积分方程。为了获得稳定的逆解，引入了正则化块Kaczmarz方法来求解两个Fredholm积分方程。最后，