论文标题
关于$γ$ - tikhonov功能的$γ$ - 非线性反问题的注释
A note on $Γ$-convergence of Tikhonov functionals for nonlinear inverse problems
论文作者
论文摘要
我们考虑使用Tikhonov功能在BANACH空间中非线性反问题的变异正则化。本文解决了$γ$ - tikhonov功能家族的$γ$ - 融合的问题以及其各自的Intima收敛性的主张。出现此类问题，如果模型不确定性，不准确的远期操作员，前向解决方案和 /或数据的有限维近似等。但是对于应用程序，最重要的是，替换功能的最小化器近似于原始的最小化器。在某些其他条件下，如果近似功能从$γ$ - convergence的意义上收敛到原始功能，则可以满足这种满足。我们推断出不同拓扑中的简单标准，这些标准可以保证$γ$ - 融合以及最小化序列的收敛性。
We consider variational regularization of nonlinear inverse problems in Banach spaces using Tikhonov functionals. This article addresses the problem of $Γ$-convergence of a family of Tikhonov functionals and assertions of the convergence of their respective infima. Such questions arise, if model uncertainties, inaccurate forward operators, finite dimensional approximations of the forward solutions and / or data, etc. make the evaluation of the original functional impossible and, thus, its minimizer not computable. But for applications it is of utmost importance that the minimizer of the replacement functional approximates the original minimizer. Under certain additional conditions this is satisfied if the approximated functionals converge to the original functional in the sense of $Γ$-convergence. We deduce simple criteria in different topologies which guarantee $Γ$-convergence as well as convergence of minimizing sequences.
