正则化预条件方法在矩量法中的应用

计算电磁学中矩量法产生的系统矩阵是病态矩阵,使用迭代方法求解时很难收敛,即使采用现有的预条件技术也经常不收敛.本文借用不适定问题求解中的正则化方法的概念,提出采用正则化矩阵作为矩量法中矩阵方程的一个预条件矩阵.这种预条件方法可以直接改善原矩阵的特征值分布,而且不需要额外的空间来存储预条件矩阵.此外,本文提出通过正则化矩阵方程的L曲线的二阶导数的最大值点来确定正则化参数,使得预条件矩阵方程求解的效率最高.数值实验表明,对于高阶矩量法求解电场积分方程或者磁场积分方程时分别产生的矩阵方程,采用常见的预条件迭代方法求解时收敛很慢,但是采用本文的预条件迭代方法却可以较快地收敛.

Application of the Regularization Preconditioning to the Moment Methods

The system matrices generated by the moment methods are ill-conditioned matrices,which make the iterative methods hardly converge,even accelerated by the existing preconditioning techniques.This paper applies the concept of regularization methods of ill-posed problems,and introduces the so-called regularization matrix as a precondidoner.This preconditioner can shift the eigenvalues of the system matrix directly.And it needs no additional memory to store the preconditioner.Furthermore,this paper proposes to determine the optimized regularization parameter by finding the maximum value of the second derivative of the L-curve of the regularized matrix.Numerical experiments show that the proposed method can converge relatively fast for some matrix equation generated by the electric field integral equation (EFIE) or the magnetic field integral equation (MFIE) solved with the higher order moment method,while iterative methods with the existing preconditioners may converge slowly.