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-square 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 $L^2(\Omega)$ for the recovered diffusion and potential coefficients. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in $L^2(\Omega)$ 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.

翻译：本文聚焦于同时重构椭圆/抛物型方程中的扩散系数与势系数，利用两个内部测量数据。我们构建了一个解耦算法，用于逐步恢复这两个参数。第一步，通过直接的重新公式化，将问题转化为标准的扩散系数识别问题。该系数可在未知势系数的情况下，通过输出最小二乘法结合有限元离散进行数值恢复。第二步，利用第一步恢复的扩散系数，采用类似方法重构势系数。本方法受构造性条件稳定性的启发，并为恢复的扩散系数和势系数提供了在$L^2(\Omega)$空间中的严格先验误差估计。为推导这些估计，我们发展了加权能量论证及合适的正性条件。这些估计为根据噪声水平选择正则化参数和离散网格尺寸提供了有益指导。数值实验验证了数值方案的精度，并支持了理论结果。