In this paper, a simultaneous inversion problem for the Riesz-Feller space-fractional diffusion equation with inexact operators is investigated, which is to identify the source term and initial value from two over-specified measurements. The problem model is well known to be ill-posed. We propose a regularization method to deal with the inverse problem using the idea of mollification. Under an a priori and an a posteriori parameter choice rules, we derive explicit error estimates between the exact solutions and their regularized approximations in the practical case where both the operators and the data are noisy. Numerical results show that the proposed method is efficient, and the unknown terms are recovered quite well.

中文翻译：

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一种在空间分数扩散方程中同时识别未知来源和初始分布的 mollifier 方法


在本文中，研究了具有不精确算子的 Riesz-Feller 空间分数扩散方程的联相反演问题，即从两个超指定的测量中识别源项和初始值。众所周知，问题模型是病态的。我们提出了一种正则化方法，使用柔和化的思想来处理逆问题。在先验和后验参数选择规则下，在算子和数据都嘈杂的实际情况下，我们在精确解和它们的正则化近似值之间推导出明确的误差估计。数值结果表明，所提方法效率高，未知项恢复良好。