A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs
A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs
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DOI：
10.1088/1361-6420/aaf129
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发表时间：
2018-08
期刊：
影响因子：
2.1
通讯作者：
B. Crestel;G. Stadler;O. Ghattas
中科院分区：
文献类型：
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作者：
B. Crestel;G. Stadler;O. Ghattas
联合反演是指从由单个或多个正演模型控制的系统的观测值中同时推断多个参数场。在许多情况下，这些参数字段反映了单个介质的不同属性，因此在空间上相关或结构上相似。通过施加先验信息的空间相关性，通过一个联合正则化项，我们试图提高重建的参数字段相对于反演每个字段独立。其中一个主要的挑战是设计一个联合正则化功能，传达的空间相关性或结构相似性的字段之间，而在同一时间允许可扩展的和有效的求解器的联合逆问题。我们描述了几个联合正则化，这些目标的动机：一个交叉梯度和归一化的交叉梯度结构相似性项，向量总变差，和联合正则化的基础上的核规范的梯度。通过对三类分段齐次参数场反问题的数值计算，我们得出了向量全变差泛函优于其他方法的结论。除了在所有实验中得到良好的重建外，它还允许可扩展的，有效的求解器用于PDE正演模型所控制的联合逆问题。
Joint inversion refers to the simultaneous inference of multiple parameter fields from observations of systems governed by single or multiple forward models. In many cases these parameter fields reflect different attributes of a single medium and are thus spatially correlated or structurally similar. By imposing prior information on their spatial correlations via a joint regularization term, we seek to improve the reconstruction of the parameter fields relative to inversion for each field independently. One of the main challenges is to devise a joint regularization functional that conveys the spatial correlations or structural similarity between the fields while at the same time permitting scalable and efficient solvers for the joint inverse problem. We describe several joint regularizations that are motivated by these goals: a cross-gradient and a normalized cross-gradient structural similarity term, the vectorial total variation, and a joint regularization based on the nuclear norm of the gradients. Based on numerical results from three classes of inverse problems with piecewise-homogeneous parameter fields, we conclude that the vectorial total variation functional is preferable to the other methods considered. Besides resulting in good reconstructions in all experiments, it allows for scalable, efficient solvers for joint inverse problems governed by PDE forward models.