Oversmoothing regularization with $\ell^1$-penalty term
Oversmoothing regularization with $\ell^1$-penalty term
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DOI：
10.3934/math.2019.4.1223
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发表时间：
2019-08
期刊：
影响因子：
2.2
通讯作者：
D. Gerth;B. Hofmann
中科院分区：
文献类型：
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作者：
D. Gerth;B. Hofmann
在Tikhonov型正则化中，罚函数通常被解释为携带关于未知真解的先验信息.本文考虑了相应的先验信息太强，使得罚函数过光滑的情况，这意味着它的值对于真解是无穷大的.在过平滑惩罚的情况下，正则解的收敛性和收敛速度的断言很难得到，只有在Hilbert尺度下才有令人信服的结果。我们尝试将此设置扩展到$\ell^1 $-正则化，当解仅在$\ell^2 $中时。不幸的是，我们不得不将我们的研究限制在具有对角结构的有界线性算子的情况下，即映射到可分希尔伯特空间。但对于这种情况，我们能够制定和证明的收敛定理，我们支持的数值例子。
In Tikhonov-type regularization for ill-posed problems with noisy data, the penalty functionalis typically interpreted to carry a-priori information about the unknown true solution.We consider in this paper the case that the corresponding a-priori information is too strong such that thepenalty functional is oversmoothing, which means that its value is infinite for the true solution. In the case of oversmoothing penalties, convergence and convergence rate assertions for the regularized solutions are difficult toderive, only for the Hilbert scale setting convincing results have been published. We attempt to extend this setting to $\ell^1$-regularization when the solutions are only in $\ell^2$. Unfortunately, we have to restrict our studies to the case of bounded linear operators with diagonal structure, mapping between $\ell^2$and a separable Hilbert space. But for this subcase, we are able to formulateand to prove a convergence theorem, which we support with numerical examples.