稀疏观测数据下热传导方程逆时反问题的数值反演

Numerical Solution of Backward Problem for Heat Conduction Equation with Sparse Observation Data

探讨一类带第二类边界条件的一维热传导方程逆时问题,首先利用分离变量法推导了反问题的积分表达形式,然后基于解析延拓技术,证明了基于稀疏附加数据下反问题的唯一性,并对反问题的不适定性进行说明,接着利用线性叠加原理及有限元插值技术,给出了该逆时反演问题对应的离散反演方程组形式,借助于Tikhonov正则化方法和正则化参数选取的广义交叉验证准则,设计出了该逆时反演问题的直接反演算法,最后通过数值算例说明所设计的直接反演算法是有效的.

In this paper,we consider the backward problem for one-dimensional heat conduction equations with the second boundary condition.First,the integral expression of the backward problem is deduced by the separation variable method.Then the uniqueness of the backward problem is proved based on analytic continuation technique,and the ill-poseedness of the backward problem is explained.Next,the discrete liner algebra equations corresponding to the backward problem are obtained by the finite element interpolation and the principle of linear superposition.Based on Tikhonov regularization method and generalized cross validation principle,a direct inversion algorithm for the backward problem is designed.Finally,several numerical examples are given to illustrate the effciencies of the proposed inversion algorithm.