In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting decay behavior of the observation in time, we establish a novel H{\"o}lder type stability estimate for a large terminal time $T$. This is achieved by novel decay estimates of the (fractional) time derivative of the solution. To numerically recover the diffusion coefficient, we employ the standard output least-squares formulation with an $H^1(\Omega)$-seminorm penalty, and discretize the regularized problem by the Galerkin finite element method with continuous piecewise linear finite elements in space and backward Euler convolution quadrature in time. Further, we provide an error analysis of discrete approximations, and prove a convergence rate that matches the stability estimate. The derived $L^2(\Omega)$ error bound depends explicitly on the noise level, regularization parameter and discretization parameter(s), which gives a useful guideline of the \textsl{a priori} choice of discretization parameters with respect to the noise level in practical implementation. The error analysis is achieved using the conditional stability argument and discrete maximum-norm resolvent estimates. Several numerical experiments are also given to illustrate and complement the theoretical analysis.

翻译：本文研究了利用给定的终端数据恢复(次)扩散模型中与空间相关的扩散系数的数值方法,并对该过程进行了严格的数值分析。通过利用观测值的时间衰减特性,我们针对较大的终端时间$T$建立了新型Hölder型稳定性估计。该估计通过解(分数阶)时间导数的全新衰减估计实现。为数值恢复扩散系数,我们采用标准输出最小二乘格式并引入$H^1(\Omega)$半范数惩罚项,利用Galerkin有限元方法（空间上采用连续分片线性有限元）结合时间方向的后向欧拉卷积求积对正则化问题进行离散。进一步,我们给出了离散近似的误差分析,证明了与稳定性估计相匹配的收敛速度。导出的$L^2(\Omega)$误差界显式依赖于噪声水平、正则化参数和离散化参数,为实际实现中根据噪声水平进行离散参数先验选择提供了有用指导。该误差分析基于条件稳定性论证和离散最大模范数预解估计。最后通过若干数值实验验证并补充了理论分析。