对流弥散方程线性源项系数反演的变分伴随方法  

Variational Adjoint Method for an Inverse Problem of Determining a Linear Source Coefficient in the Advection Dispersion Equation

作　　者：李功胜[1] 贾现正[1] 孙春龙[1] 杜殿虎[1] 

机构地区：[1]山东理工大学理学院,淄博255049

出　　处：《应用数学学报》2015年第6期1001-1015,共15页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(11371231;10171148)资助项目

摘　　要：应用变分伴随方法研究终值数据条件下一维对流弥散方程中确定空间依赖源项系数的反问题.基于正问题的伴随问题,建立一个联系已知数据与未知系数的变分恒等式,进而验证误差泛函的极小点即为反问题的一个解.进一步,利用变分恒等式及对伴随问题解的控制,证明反问题解的唯一性.最后,应用最佳摄动量算法给出数值反演算例说明该反问题的数值稳定性与唯一性.This article deals with an inverse problem of determining a continuous spacedependent source coefficient in the advection dispersion equation with final observations using variational adjoint method. Since the solution operator from the unknown to the known for the inverse problem is linear, the two-order error functional of the unknown is convex by which existence of the minimum of the error functional is obtained. A variational identity connecting the known data with the unknown is established by controlling the solution of an adjoint problem to the forward problem, with which existence of the solution to the inverse problem is proved by computing the first variation of the error functional. Furthermore, uniqueness of the inverse problem is proved by utilizing a variational identity connecting the changes of the known data with that of the unknown and the denseness of the set of the solutions to the adjoint problem in the square integrable space. Finally, the optimal perturbation algorithm is applied to solve the inverse problem numerically, and two numerical inversions with random noisy data are presented to support the numerical stability and uniqueness of the inverse problem.

关 键 词：对流弥散方程 源项系数 反问题 伴随问题 变分恒等式 存在唯一性 数值反演 

分 类 号：O175.29[理学—数学]