This paper studies an inverse problem of recovering an unknown diffusion coefficient in a parabolic equation. We adopt a total variation regularization method to deal with the ill-posedness. This method has the advantage to solve problems that the solution is non-smooth or discontinuous. By transforming the problem into an optimal control problem, we derive a necessary condition of the control functional. Through some prior estimates of the direct problem, the uniqueness and stability of the minimizer are obtained. In the numerical part, a Gauss–Jacobi iteration scheme is used to deal with the non-linear term. Some numerical examples are presented to illustrate the performance of the proposed algorithm.

中文翻译：

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抛物线方程中未知扩散系数恢复反问题的全变分正则化方法

本文研究了一个在抛物线方程中恢复未知扩散系数的反问题。我们采用全变分正则化方法来处理不适定性。这种方法对于解决解不平滑或不连续的问题具有优势。通过将问题转化为最优控制问题，我们推导出控制泛函的必要条件。通过对直接问题的一些先验估计，得到了最小化器的唯一性和稳定性。在数值部分，使用 Gauss-Jacobi 迭代方案来处理非线性项。给出了一些数值例子来说明所提出算法的性能。