Symm 积分方程的正则化 Gmres 方法∗

摘要

Symme 积分方程是 Hadamard 意义下的不适定问题，在位势理论中具有重要意义。本文提出了数值求解 Symme 积分方程的正则化 Gmres 方法，首先将 Symm 积分方程进行离散，其次利用正则化方法—参数化信赖域法，将离散后的方程转化为一个适定方程，最后通过 Gmres 算法得到其数值解。该方法与一般的正则化方法相比，克服了正则参数选取的困难。数值结果显示，在数据出现噪声的情况下，正则化 Gmres 方法能有效地求得 Symm 积分方程的数值解，表明该方法的可行性和有效性。%The Symm integral equation is typically ill-posed in the sense of Hadamard, and has an important significance in potential theory. A regularization Gmres method is presented to reconstruct the solution of the Symm integral equation in the article. Firsly, we present the discrete form of the Symm integral equation. Then, we transform the discrete equation into a well-posed equation by using the regularization method-parameterized trust region method. Finally, we obtain the numerical solution of the Symm integral equation by applying Gmres method. Compared with the general regularization methods, the regularization Gmres method overcomes the difficulties in selecting regularization parameter. In the numerical simulation, different methods are compared with regularization Gmres method, and the latter can recon-struct the numerical solution of the Symm integral equation efficiently under the conditions of noise or corrupted data, and the results show that the method is feasible and effective.