Liu Tao, Xue Runqi, Liu Chao, Qi Yunfei
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066000, China.
Eighth Geological Bridage of Hebei Geology, Mineral Resources Exploration Bureau, Qinhuangdao 066000, China.
Entropy (Basel). 2021 Nov 9;23(11):1480. doi: 10.3390/e23111480.
The main difficulty posed by the parameter inversion of partial differential equations lies in the presence of numerous local minima in the cost function. Inversion fails to converge to the global minimum point unless the initial estimate is close to the exact solution. Constraints can improve the convergence of the method, but ordinary iterative methods will still become trapped in local minima if the initial guess is far away from the exact solution. In order to overcome this drawback fully, this paper designs a homotopy strategy that makes natural use of constraints. Furthermore, due to the ill-posedness of inverse problem, the standard Tikhonov regularization is incorporated. The efficiency of the method is illustrated by solving the coefficient inversion of the saturation equation in the two-phase porous media.
偏微分方程参数反演所面临的主要困难在于代价函数中存在大量局部极小值。除非初始估计接近精确解，否则反演无法收敛到全局最小点。约束条件可以改善该方法的收敛性，但如果初始猜测远离精确解，普通迭代方法仍会陷入局部极小值。为了充分克服这一缺点，本文设计了一种自然利用约束条件的同伦策略。此外，由于反问题的不适定性，引入了标准的蒂霍诺夫正则化。通过求解两相多孔介质中饱和度方程的系数反演来说明该方法的有效性。
