标题： 随机多项时间分数阶扩散波方程的反源问题 显示英文标题

标题： An inverse source problem for the stochastic multi-term time-fractional diffusion-wave equation

摘要： 在本文中，我们研究了由分数布朗运动驱动的多项时间分数阶扩散波方程相关的直接和反向随机源问题。 关于直接问题，建立了适定性，并表征了方程解的正则性。 在反问题的背景下，研究了随机源确定的唯一性和不稳定性。 此外，基于边界数据的方差，提供了随机源中扩散系数的无相位傅里叶模式的重构公式。 为了从其无相位傅里叶模式重构时变源函数，采用结合谱截断正则化技术的PhaseLift方法来解决相位恢复问题。 通过一系列数值实验验证了所提出方法的有效性。 显示英文摘要

摘要： In this paper, we study both the direct and inverse random source problems associated with the multi-term time-fractional diffusion-wave equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is established and the regularity of the solution is characterized for the equation. In the context of the inverse problem, the uniqueness and instability are investigated on the determination of the random source. Furthermore, a reconstruction formula is provided for the phaseless Fourier modes of the diffusion coefficient in the random source, based on the variance of the boundary data. To reconstruct the time-dependent source function from its phaseless Fourier modes, the PhaseLift method, combined with a spectral cut-off regularization technique, is employed to tackle the phase retrieval problem. The effectiveness of the proposed method is demonstrated through a series of numerical experiments.