一种用于多尺度目标的新型采样方法：在固定频率下的散射振幅
A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency
AI总结：
本文提出一种基于散射振幅的新采样方法，用于逆声学散射问题中的形状和位置重建。该方法通过矩阵乘法实现，具有简单易用的优点。通过远场算子的分解，建立了用于表征底层散射体的inf准则，并展示了所提指标函数在散射体内部和外部的性质。
摘要：
本文提出了一种利用散射振幅进行形状和位置重建的新型采样方法，用于逆声学散射问题。仅涉及矩阵乘法运算，因此该新采样方法非常容易和简单地实现。借助远场算子的分解，我们建立了用于表征底层散射体的inf准则。这一结果随后被用来给出所提指示泛函在散射体内部采样点的下界。对于位于散射体外部的采样点，我们显示当采样点远离散射体边界时，指标泛函衰减类似于贝塞尔函数。我们还展示了所提指标泛函连续依赖于散射振幅，这进一步表明该新采样方法对数据误差具有极高的稳定性。与经典采样方法如线性采样方法或分解方法不同，从数值角度来看，所提指标函数在其底层目标边界附近取最大值，并且当采样点远离边界时，其衰减类似于贝塞尔函数。数值模拟还表明，所提出的采样方法能够处理多尺度情况，即使不同组件彼此靠近。
英文摘要：
A sampling method by using scattering amplitude is proposed for shape and location reconstruction in inverse acoustic scattering problems. Only matrix multiplication is involved in the computation, thus the novel sampling method is very easy and simple to implement. With the help of the factorization of the far field operator, we establish an inf-criterion for characterization of underlying scatterers. This result is then used to give a lower bound of the proposed indicator functional for sampling points inside the scatterers. While for the sampling points outside the scatterers, we show that the indicator functional decays like the bessel functions as the sampling point goes away from the boundary of the scatterers. We also show that the proposed indicator functional continuously dependents on the scattering amplitude, this further implies that the novel sampling method is extremely stable with respect to errors in the data. Different to the classical sampling method such as the linear sampling method or the factorization method, from the numerical point of view, the novel indicator takes its maximum near the boundary of the underlying target and decays like the bessel functions as the sampling points go away from the boundary. The numerical simulations also show that the proposed sampling method can deal with multiple multiscale case, even the different components are close to each other.
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