Sparsity regularization for parameter identification problems
Sparsity regularization for parameter identification problems
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DOI：
10.1088/0266-5611/28/12/123001
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发表时间：
2012-11
期刊：
影响因子：
2.1
通讯作者：
Bangti Jin;P. Maass
中科院分区：
文献类型：
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作者：
Bangti Jin;P. Maass
近年来，带稀疏促进罚项的正则化格式的研究一直是反问题研究领域的热点之一，其中带1 <$p <$2稀疏促进罚项的Tikhonov泛函得到了广泛的研究.第一次调查集中在正则化性质的极小化这样的泛函与线性算子和迭代计划近似的极小。这些结果很快被转移到非线性算子方程，包括非光滑算子和更一般的函数空间设置。关于正则化性质的最新结果还假设了真实解的稀疏表示以及广义源条件，这产生了一些令人惊讶的最佳收敛速度。在这种情况下，带有BLP稀疏约束的正则化理论相对完整;参见本综述的第一部分。相比之下，发展有效的数值方法来近似极小的吉洪诺夫泛函与稀疏约束的非线性算子仍在进行中。基本的迭代软收缩方法已经在几个方向上得到了扩展，半光滑牛顿法在这一领域也逐渐得到应用。特别是，扩展到更一般的非凸，不可微泛函变分原理导致各种广义迭代计划。我们专注于这样的迭代计划在第二部分的审查。本次调查的一个主要部分是致力于应用稀疏约束正则化技术的偏微分方程，我们认为非线性反问题的原型设置的参数识别问题。参数识别问题表现出不同程度的复杂性，我们的目标是表征这样的问题的层次结构。定义这些逆问题的算子是参数到状态的映射。首先总结了由弱形式微分方程导出的一些一般解析性质，然后详细分析了几个具体的参数识别问题。当然，不可能涵盖所有感兴趣的参数识别问题。特别是，我们不包括与逆散射或非线性层析成像问题，如光学，热声或光声成像相关的问题。此外，我们不审查广泛的文献上密切相关的领域的控制问题的偏微分方程。然而，我们包括一个例子，突出了控制理论和反问题方法之间的差异和相似之处。
The investigation of regularization schemes with sparsity promoting penalty terms has been one of the dominant topics in the field of inverse problems over the last years, and Tikhonov functionals with ℓp-penalty terms for 1 ⩽ p ⩽ 2 have been studied extensively. The first investigations focused on regularization properties of the minimizers of such functionals with linear operators and on iteration schemes for approximating the minimizers. These results were quickly transferred to nonlinear operator equations, including nonsmooth operators and more general function space settings. The latest results on regularization properties additionally assume a sparse representation of the true solution as well as generalized source conditions, which yield some surprising and optimal convergence rates. The regularization theory with ℓp sparsity constraints is relatively complete in this setting; see the first part of this review. In contrast, the development of efficient numerical schemes for approximating minimizers of Tikhonov functionals with sparsity constraints for nonlinear operators is still ongoing. The basic iterated soft shrinkage approach has been extended in several directions and semi-smooth Newton methods are becoming applicable in this field. In particular, the extension to more general non-convex, non-differentiable functionals by variational principles leads to a variety of generalized iteration schemes. We focus on such iteration schemes in the second part of this review. A major part of this survey is devoted to applying sparsity constrained regularization techniques to parameter identification problems for partial differential equations, which we regard as the prototypical setting for nonlinear inverse problems. Parameter identification problems exhibit different levels of complexity and we aim at characterizing a hierarchy of such problems. The operator defining these inverse problems is the parameter-to-state mapping. We first summarize some general analytic properties derived from the weak formulation of the underlying differential equation, and then analyze several concrete parameter identification problems in detail. Naturally, it is not possible to cover all interesting parameter identification problems. In particular we do not include problems related to inverse scattering or nonlinear tomographic problems such as optical, thermo-acoustic or opto-acoustic imaging. Also we do not review the extensive literature on the closely related field of control problems for partial differential equations. However, we include one example which highlights the differences and similarities between control theory and the inverse problems approach in this context.