一类抛物型方程逆时反演的正则化方法

在文献[9]的基础上,采用修正泛函含有一个导数的项作为惩罚项,这样保证方程的解具有一定的光滑性。为了克服反问题的不适定性,利用正则化思想,把原问题分解为一系列适定的正问题和一个病态线性代数方程组。利用无条件稳定的Crank-Nicolson有限差分格式求解正问题和用截断奇异值分解法求解病态线性方程组。数值结果验证了正则化方法的可行性和有效性。

Regularization Method of Backward Problem in Time for a Class of Parabolic Equation

In this paper on the basis of artical [ 9 ] , a modified functional containing a differential item as the penalty item is adopted, which ensures adequate smoothness of solutions. In order to overcome ill-posedness of inverse problem, employing regularization idea of inverse problem, the original problem is divided into a sequence of well-posed forward problems and an ill-conditioned system of linear algebraic equations in the artical. Using unconditional and stable Crank-Nicholson difference perform solves the corresponding forward problem and the ill-conditioned systems of linear algebraic equation is solved by using truncated singular value decomposition. The numerical result manifests that the regularization method is feasible and effective.