Fibre Bragg Gratings have become widespread measurement devices in engineering and other fields of application. In all but a few cases, the relation between cause and effect is simplified to a proportional model. However, at its mathematical core lies a nonlinear inverse problem which appears not to have received much attention in the literature. In this paper, we present this core problem to the mathematical community and provide a first report on opportunities and limitations of a regularization approach. In particular, we show that difficulties arise from non-uniqueness and the absence of established parameter selection rules for nonlinear inverse problems with multiple regularization parameters. Nevertheless, the paper takes a first step toward extracting more information from a single FBG measurement.

中文翻译：

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光纤布拉格光栅光谱反演的正则化

光纤布拉格光栅已成为工程和其他应用领域中广泛使用的测量设备。除少数情况外，在所有情况下，因果关系都被简化为比例模型。然而，其数学核心是一个非线性逆问题，该问题在文献中似乎并未受到太多关注。在本文中，我们向数学界提出了这个核心问题，并提供了关于正则化方法的机会和局限性的第一份报告。特别是，我们表明困难源于非唯一性和缺乏针对具有多个正则化参数的非线性逆问题的既定参数选择规则。尽管如此，本文迈出了从单个 FBG 测量中提取更多信息的第一步。