一种同时回收不同性质非均匀包裹体的直接取样方法
A Direct Sampling Method for Simultaneously Recovering Inhomogeneous Inclusions of Different Nature
摘要：
在这项工作中，我们研究了一类椭圆反问题，并旨在同时恢复多个非均匀夹杂物所产生的两个不同的物理参数，使用非常有限的边界柯西数据收集只有在一个或两个测量事件。我们提出了一种新的快速，稳定和高度并行的直接采样方法（DSM）的同时重建过程。构造了两组探测函数和索引函数，并分析了它们的性质。为了识别和解耦具有不同物理性质的多个非齐次包含，我们引入了互概正交性的概念，推广了经典DSM中关于具有相同物理性质的非齐次包含的概正交性的重要概念.在这个新概念的帮助下，我们开发了一个可靠的策略来区分两种不同类型的非均匀夹杂物与噪声数据收集在一个或两个测量事件。我们通过选择合适的边界注入量进一步改善解耦效果。数值实验表明了该方法的鲁棒性和有效性。
英文摘要：
In this work, we investigate a class of elliptic inverse problems and aim to simultaneously recover multiple inhomogeneous inclusions arising from two different physical parameters, using very limited boundary Cauchy data collected only at one or two measurement events. We propose a new fast, stable and highly parallelable direct sampling method (DSM) for the simultaneous reconstruction process. Two groups of probing and index functions are constructed, and their desired properties are analyzed. In order to identify and decouple the multiple inhomogeneous inclusions of different physical nature, we introduce a new concept of mutually almost orthogonality property that generalizes the important concept of almost orthogonality property in classical DSMs for inhomogeneous inclusions of same physical nature. With the help of this new concept, we develop a reliable strategy to distinguish two different types of inhomogeneous inclusions with noisy data collected at one or two measurement events. We further improve the decoupling effect by choosing an appropriate boundary influx. Numerical experiments are presented to illustrate the robustness and efficiency of the proposed method.
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