一类抛物型方程控制系数反演的数值算法

Numerical Algorithm for Recovering a Control Coefficient in a Parabolic Equation

研究了第3类边界条件下抛物型方程控制系数的反演问题。基于附加的非局部条件和有限差分的Crank-Nicolson思想方法来确定系数函数,构造了控制系数的迭代反演算法。后经进一步简化后,得到了控制系数的一个显式的反演格式。为克服计算带来的不适定性,引入磨光正则化算法对带有误差的测量数据进行去噪处理,最后获得了相对稳定的控制系数反演结果。数值实验表明了所提出的数值方法能有效地重建出控制系数。

In this paper, the inversion of control coefficients for parabolic equations with the third kind of boundary conditions is studied. Based on the additional nonlocal conditions and the Crank Nicolson thought method of finite difference, the coefficient function is determined, and an iterative inversion algorithm for control coefficients is constructed. After further simplification, an explicit inversion format of control coefficients is obtained. In order to overcome the uncertainty caused by the calculation, we introduce the smoothing regularization algorithm to denoise the measurement data with errors, and finally obtain a relatively stable inversion result of the control coefficient. Numerical experiments show that the proposed numerical method can effectively reconstruct the control coefficients.