Approximation of penalty terms in Tikhonov functionals—theory and applications in inverse problems
Approximation of penalty terms in Tikhonov functionals—theory and applications in inverse problems
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DOI：
10.1088/0266-5611/30/7/075005
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发表时间：
2014-06
期刊：
影响因子：
2.1
通讯作者：
R. Strehlow;K. Kazimierski
中科院分区：
文献类型：
--
作者：
R. Strehlow;K. Kazimierski
最小化非光滑泛函的一种可行方法是用某种光滑的泛函来代替它，这导致了一个很容易用标准方法处理的代理最小化问题。这类问题的一个突出例子是含有稀疏强制惩罚项的Tikhonov型泛函。近年来，它受到了极大的关注，但其有效的最小化仍然具有挑战性。本文考虑一般的Tikhonov型泛函，并在较温和的条件下，证明了它们的极小元关于用适当的近似替换惩罚项的稳定性。特别是，我们考虑了可分离惩罚条款的情况。最后，我们将所提出的策略应用于逆介质问题，并给出了数值结果，表明了该方法的有效性。
One feasible way to minimize a non-smooth functional is to replace it by some smoothed version, which leads to a surrogate minimization problem that is easily treated by standard means. A prominent example of such a problem is given by a Tikhonov-type functional incorporating a sparsity-enforcing penalty term. It has received enormous attention in recent years, yet its efficient minimization remains challenging. In this paper we consider general Tikhonov-type functionals and show, under mild conditions, the stability of their minimizer with respect to the replacement of the penalty term with an appropriate approximation. In particular, we consider the case of separable penalty terms. Finally, we apply the proposed strategy to the inverse medium problem and demonstrate numerical results that indicate the efficiency of the approach.