基于变分伴随的扩散方程系数反问题的数值反演算法
An Inversion Algorithm for Inverse Coefficient Problems in the Diffusion Equation Based on Variational Adjoint Method
DOI: 10.12677/AAM.2018.74057, PDF,    国家自然科学基金支持
作者: 王桢东, 李功胜, 王迎美：山东理工大学数学与统计学院，山东 淄博
关键词: 扩散方程；反问题；伴随方法；变分恒等式；数值反演；Diffusion Equation； Inverse Problem； Adjoint Method； Variational Identity； Numerical Inversion
摘要: 对于扩散方程的逆时问题与确定时间–空间依赖源项的反问题，提出一种基于变分伴随方法的数值反演算法。借助正问题的伴随问题,构建联系已知数据与未知量的变分恒等式，通过适当选择伴随方程的解进而得到关于未知量的线性方程组，结合正则化方法，获得了反问题的数值解。几个算例结果表明了这种方法的有效性。
Abstract: A numerical inversion algorithm is proposed based on the variational adjoint method for solving the backward problem and inverse source problem in the diffusion equation. With the help of the adjoint problem, a variational identity connecting the known data with the unknown is derived with which a linear system for the unknown is set forth by suitably choosing the solution of the adjoint equation. Numerical solutions to the inverse problems are obtained by solving the linear system with Tikhonov regularization, and numerical examples are presented to demonstrate the effectiveness of the inversion algorithm.
文章引用：王桢东, 李功胜, 王迎美. 基于变分伴随的扩散方程系数反问题的数值反演算法[J]. 应用数学进展, 2018, 7(4): 466-475. https://doi.org/10.12677/AAM.2018.74057

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