边界符号固定的对比度反散射的因式分解方法及单调性界
A Factorization Method and Monotonicity Bounds in Inverse Medium Scattering for Contrasts with Fixed Sign on the Boundary
摘要：
利用两个远场算子差的一种特殊分解，推广了逆介质散射的分解方法。虽然因式分解方法到目前为止主要用于识别散射体的支持的形状，我们表明，基于狄利克雷到诺依曼算子的因式分解可以用来计算边界上的介质的支持的边界上的数值。为此，我们从内部-外部对偶性中推广了思想，以获得单调性原理，该原理允许特定逆散射问题的替代唯一性证明（例如，当介质内存在障碍物时）。这个单调性原理确实是我们最重要的技术工具：它进一步直接表明，介质的对比度函数的边界值由相应的远场算子唯一确定。我们的远场算子的特殊因式分解还意味着，因式分解方法严格地表征了非均匀介质的支持，如果对比度函数在其支持的边界上仅取正值或负值，则与其支持内的对比度值无关。最后，单调性原理产生一个简单的算法来计算这些边界值的上界和下界，假设支持的对比度是已知的。数值实验表明，所得到的数值算法的可行性。
英文摘要：
We generalize the factorization method for inverse medium scattering using a particular factorization of the difference of two far field operators. Whilst the factorization method been used so far mainly to identify the shape of a scatterer's support, we show that factorizations based on Dirichlet-to-Neumann operators can be used to compute bounds for numerical values of the medium on the boundary of its support. To this end, we generalize ideas from inside-outside duality to obtain a monotonicity principle that allows for alternative uniqueness proofs for particular inverse scattering problems (e.g., when obstacles are present inside the medium). This monotonicity principle indeed is our most important technical tool: It further directly shows that the boundary values of the medium's contrast function are uniquely determined by the corresponding far field operator. Our particular factorization of far field operators additionally implies that the factorization method rigorously characterizes the support of an inhomogeneous medium if the contrast function takes merely positive or negative values on the boundary of its support, independent of the contrast's values inside its support. Finally, the monotonicity principle yields a simple algorithm to compute upper and lower bounds for these boundary values, assuming the support of the contrast is known. Numerical experiments show feasibility of a resulting numerical algorithm.
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