Non-quadratic Penalization in Generalized Local Regularization for Linear and Nonlinear Inverse Problems

具有非二次惩罚项的变分方法（如与全变分或稀疏性约束相关的变分方法）是解决应用逆问题的非光滑解的尖锐特征的非常有效的求解方法。不幸的是，这种全局定义的方法通常伴随着大量的计算成本，这种情况由于潜在问题的规模（例如，大型成像数据集）而变得更糟。利用他们在广义局部正则化方法方面的专业知识，研究者和她的同事开发了不适定逆问题的局部非二次正则化方法，作为一种减少计算成本的方法，同时仍然保持解决方案的尖锐分辨率。这些新方法的应用包括图像去模糊、非线性自卷积、盲反卷积、分数积分/微分、逆热传导和反向热传导等问题。研究者和她的同事开发的方法导致更快，更少的计算密集型技术，用于在广泛的应用中精确重建图像，从卫星图像采集，生物医学成像（CT扫描，PET扫描等）到地球物理勘探。改进后的方法可以在更短的时间内分析更多的数据，并且总体费用更低。除了成像应用之外，这些新方法也适用于确定结构和材料中隐藏的裂缝和弱点的问题，以及大气中臭氧水平的检测，以及纳米结构分析中出现的问题。

Variational methods with non-quadratic penalty terms (such as those associated with total variation or sparsity constraints) are extremely effective solution methods for the resolution of sharp features of nonsmooth solutions to applied inverse problems. Unfortunately, such globally-defined methods are often associated with large computational costs, a situation made even worse by the scale of the underlying problem (e.g., large imaging data sets). Using their expertise in methods of generalized local regularization, the investigator and her colleagues develop localized non-quadratic regularization methods for ill-posed inverse problems as a way to reduce computational costs while still maintaining the sharp resolution of solutions. Applications of these new methods include problems of image deblurring, nonlinear autoconvolution, blind deconvolution, fractional integration/differentiation, and inverse and backward heat conduction. The methods developed by the investigator and her colleagues lead to faster, less computationally-intensive techniques for the accurate reconstruction of pictures in a wide number of applications, from satellite image gathering, biomedical imaging (CT scans, PET scans, etc.), to geophysical exploration. The improved methods enable the analysis of more data in a shorter amount of time, and with less overall expense. In addition to imaging applications, these new methods are also applicable to the problem of determining hidden cracks and weaknesses in structures and materials, and the detection of ozone levels in the atmostphere, as well as to problems arising in the analysis of nano-structures.