Helmholtz型方程柯西问题的修正Lavrentiev正则化方法

MODIFIED LAVRENTIEV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF HELMHOLTZ-TYPE EQUATION

本文研究了带非齐次Dirichlet及Neumann数据的一类Helmholtz型方程柯西问题.文章在解的先验假设下建立问题的条件稳定性结果,利用修正Lavrentiev正则化方法克服其不适定性,并结合正则化参数的先验与后验选取规则获得了正则化解的收敛性结果,相应的数值实验结果验证了所提方法是稳定可行的,推广了已有文献在Helmholtz型方程柯西问题正则化理论与算法方面的相关研究结果.

In this paper,a Cauchy problem of Helmholtz-type equation with nonhomogeneous Dirichlet and Neumann datum is researched.We establish the result of conditional stability under an a-priori assumption for exact solution.A modified Lavrentiev regularization method is used to overcome its ill-posedness,and under an a-priori and an a-posteriori selection rule for the regularization parameter we obtain the convergence result for the regularized solution,the corresponding results of numerical experiments verify that the proposed method is stable and workable,this work is an extension on the related research results of existing literature in the aspect of regularization theory and algorithm for Cauchy problem of Helmholtz-type equation.