论文标题
关于解决由PDE建模的不适性问题的迭代方法
On iterative methods for solving ill-posed problems modeled by PDE's
论文作者
论文摘要
我们研究了Maz'ya和Kozlov提出的迭代方法（请参阅[KM1]，[KM2]），以解决由部分微分方程建模的错误的反问题。我们考虑椭圆形，双曲线和抛物线类型的线性进化问题。分析方法的每次迭代都在于解决良好的问题（分别为边界价值问题或初始值问题）的解决方案。如[KM2]中，迭代被描述为仿射操作员的能力。我们通过使用光谱理论为算法提供了替代性收敛证明，并且这些仿射操作员的线性部分具有非膨胀性，具有其他功能分析特性（请参见[LE1,2]）。同样，考虑到噪声数据的问题，并根据问题数据的先验规律性假设获得了收敛率的估计。
We investigate the iterative methods proposed by Maz'ya and Kozlov (see [KM1], [KM2]) for solving ill-posed inverse problems modeled by partial differential equations. We consider linear evolutionary problems of elliptic, hyperbolic and parabolic types. Each iteration of the analyzed methods consists in the solution of a well posed problem (boundary value problem or initial value problem respectively). The iterations are described as powers of affine operators, as in [KM2]. We give alternative convergence proofs for the algorithms by using spectral theory and the fact that the linear parts of these affine operators are non-expansive with additional functional analytical properties (see [Le1,2]). Also problems with noisy data are considered and estimates for the convergence rate are obtained under a priori regularity assumptions on the problem data.
