SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2276-2307, October 2024.
Abstract. The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output least-squares method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in [math] for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results.

中文翻译：

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来自两个观测值的扩散系数和电位系数的数值重建：解耦恢复和误差估计


SIAM 数值分析杂志，第 62 卷，第 5 期，第 2276-2307 页，2024 年 10 月。

抽象。本文的重点是利用解的两个内部测量，同时重建椭圆/抛物线方程中存在的扩散系数和电位系数。构建解耦算法以按顺序恢复这两个参数。在第一步中，我们实现了一个简单的重新表述，从而产生了一个确定扩散系数的标准问题。然后，通过使用输出最小二乘法和有限元离散化，无需了解电位，即可数值恢复该系数。在第二步中，采用与第一步类似的方法，采用先前恢复的扩散系数来重建电位系数。我们的方法受到建设性条件稳定性的刺激，我们在 [数学] 中为恢复的扩散系数和势系数提供了严格的先验误差估计。为了得出这些估计值，我们开发了加权能量论点和合适的正性条件。这些估计值为根据噪声水平选择正则化参数和离散化网格大小提供了有益的指导。提出了一些数值实验来证明数值方案的准确性并支持我们的理论结果。