利用终端观测数据重构扩散方程中的势函数  

A Reconstructed Potential Function in the Diffusion Equation Based on Terminal Observation Data

作　　者：王清艳 周子融 杨柳[1] WANG Qingyan;ZHOU Zirong;YANG Liu(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,兰州730070

出　　处：《兰州交通大学学报》2024年第3期94-103,共10页Journal of Lanzhou Jiaotong University

基　　金：国家自然科学基金(61663018,11961042);甘肃省自然科学基金(22JR5RA341)。

摘　　要：利用终端时刻观测数据,研究扩散方程与空间相关势函数的重构问题。以一维空间域扩散模型为例,研究了一种重构势函数的单调性方法,对于多维空间域扩散模型的问题,该方法同样适用。在理论分析方面,首先推导出扩散方程正问题解的极值原理以及正则性估计。然后根据扩散方程构造一个有界算子,并证明其单调性,进而利用算子的单调性和不动点迭代,证明了势函数重构的唯一性。最后,基于算子半群理论,在终端时刻T足够大的条件下,证明了重构势函数在Hilbert空间中的条件稳定性。在数值实验方面,基于理论分析设计了合适的迭代算法,选取3个典型的数值算例进行数值实验,实验结果表明该算法是稳定有效的,且验证了单调性、唯一性、稳定性等理论结果的准确性。通过理论分析与数值实验进行研究可得,重构扩散方程势函数的单调性方法是可行的。The reconstruction of spatial correlation potential function in diffusion equation is studied by using terminal time observation data.Taking the one-dimensional spatial domain diffusion model as an example,a monotonicity method for reconstructing the potential function is studied.This method is also applicable to problems involving multi-dimensional spatial diffusion models.In terms of theoretical analysis,the extremum principle and regularity es-timation of the solution for the forward problem of the diffusion equation are first derived.Then a bounded operator is constructed according to the diffusion equation,and its monotonicity is proved.Furthermore,the uniqueness of the reconstructed potential function is proved by using the monotonicity of the operator and the fixed point iteration.Fi-nally,based on the operator semigroup theory,the stability of the reconstructed potential function in Hilbert space is verified under the condition that the terminal time is large enough.In terms of numerical experiments,an appropriate iterative algorithm is designed based on theoretical analysis.Three typical numerical examples are selected for numer-ical experiments.The experimental results show that the algorithm is stable and effective,and the accuracy of theo-retical results such as monotonicity,uniqueness and stability is verified.Through theoretical analysis and numerical experiments,it is feasible to reconstruct the monotonicity method of the potential function of the diffusion equation.

关 键 词：反问题 扩散方程 势函数 半群 单调性 唯一性 稳定性 

分 类 号：O175.26[理学—数学]