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.
翻译:我们调查了Maz'ya和Kozlov(见[KM1]、[KM2])为解决以部分差异方程为模型的反面问题建议的迭代方法(见[KM1]、[KM2]),我们考虑了椭圆、双曲和抛物线型的线性进化问题,分析后方法的每种迭代方法都包括解决一个很好的问题(分别是边界价值问题或初始价值问题),迭代方法被描述为同系物操作者的权力,如[KM2]所述,我们利用光谱理论为算法提供了替代的趋同证据,我们通过使用光谱理论和下述事实提供了替代的趋同证据,即这些线性操作者的线性部分没有耗尽额外的功能分析特性(见[L1,2]),还考虑了噪音数据的问题,根据关于问题数据的先期假设,对趋同率进行了估计。