椭圆方程系数识别问题的正则化解

Regularization Solution for Coefficient Identification in Elliptic Equation

研究了一般椭圆型方程系数反问题正则化解的收敛速度,这里向量的维数仅与区域相关.利用Tikhonov正则化方法,将不适定问题转化为最优化问题,并构造相应的能量泛函.由Lax Milgram引理,首先给出了变分等式,并利用变分等式得到解的唯一性;其次,利用源条件和先验估计,获得正则化解的收敛速度.当向量为非散度算子时,原问题可化为非散度型椭圆方程问题,更具有一般性和广泛性.

In this paper,the convergence rate of Tikhonov regularized solution for the identification of coefficients of general elliptic equations is studied,where the dimension of the vector is only related to the region.By using Tikhonov regularization method,the ill posed problem is transformed into an optimization problem,and the corresponding functional is constructed.Based on the lax Milgram lemma,the variational equation is given,and the uniqueness of the solution is obtained by using the variational equation.Secondly,the convergence rate of the regularized solution is obtained by using the source condition and a priori estimation.When the vector is a non divergence operator,the original problem is transformed into a non divergence elliptic equation problem,which would be more general and universal.