构造成像中一类双曲型方程不适定问题的数值解法（Ⅱ）逼近问题的适定性与解的收敛性

NUMERICAL SOLUTION METHOD FOR A CLASS OF ILL-POSED PROBLEM OF HYPERBOLIC EQUATION ABOUT STRUCTURE IMAGE(Ⅱ): PROPERLY POSEDNESS OF THE APPROXIMATE PROBLEM AND CONVERGENCE OF ITS SOLUTION

地球物理中一类构造成像问题可以用双曲型偏微分方程不适定问题来描述，这时表征地质构造的地下反射波场由这类不适定问题的解来表示．作者已撰文讨论了这类不适定问题的提法并提出了数值求解这类基本问题的逼近问题．在前文基础上进一步讨论了逼近问题的适定性，即逼近问题的解的存在性、唯一性及对初始数据的连续依赖性．证明了逼近问题的解对原始基本问题的解的收敛性，即当近似的初始数据收敛于精确的初始数据时，近似数据下逼近问题的解收敛于精确数据下原始基本问题的解．从而将不可能直接求解的不适定问题的数值解问题化归为可以直接求解的适定性的逼近问题的数值解问题，后者可以用有限差分法或有限元法实现数值求解．

A class of structure image problem in geophysics may be described by ill-posed problem of hyperbolic partial differential equation.In this case, the solution of this ill-posed problem represents the reflected wave field which characterizes geological structure.In the paper[1],the formulation of this ill-posed problem and its approximate problem were proposed. In this paper, the properly-posedness of the approximate problem is discussed on the basis of paper [1].It means that the solution of the approximate problem is existential,unique and continuous about the initial data.Furthermore,the convergence of the solution of the approximate problem Is also proved.It means that the solution of the approximate problem with the approximate initial data converges to the solution of the original basic problem with the exactinitial data, when the approximate initial data converges to the exact initial data.Conse quently,the numerical solution problem of ill-posed basic problem is reduced to the numerical solution problem of the properly-posed approximate problem.The latter can be implemented by use of finite difference method or finite element method.