论文标题
从终端观察中鉴定扩散方程中电势：分析和离散近似
Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation
论文作者
论文摘要
本文的目的是研究（子）扩散方程中从过度含量的最终时间数据中恢复（子）扩散方程中的空间依赖电位。我们构建一个单调操作员，其固定点之一是未知的电位。通过使用操作员的单调性和固定点参数，可以在理论上验证识别的唯一性。此外，我们在问题数据的某些合适条件下显示了希尔伯特空间中的有条件稳定性。接下来，通过在时间和有限差异方法中使用盖金金有限元方法开发完全分散的方案，然后将固定点迭代应用于重建电势。我们通过收缩映射定理证明了迭代算法的线性收敛，并对重建电位进行了彻底的误差分析。我们派生的\ textsl {先验}错误估计提供了一个指南，可以根据噪声级别选择离散参数。该分析在很大程度上取决于直接问题以及上述条件稳定性的一些合适的非标准误差估计。提供数值实验来说明和补充我们的理论分析。
The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived \textsl{a priori} error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis.
