We consider in this work an inverse acoustic scattering problem when only phaseless data is available. The inverse problem is highly nonlinear and ill-posed due to the lack of the phase information. Solving inverse scattering problems with phaseless data is important in applications as the collection of physically acceptable phased data is usually difficult and expensive. A novel direct sampling method (DSM) will be developed to effectively estimate the locations and geometric shapes of the unknown scatterers from phaseless data generated by a very limited number of incident waves. With a careful theoretical analysis of the behavior of the index function and some representative numerical examples, the new DSM is shown to be computationally efficient, easy to implement, robust to large noise, and does not require any prior knowledge of the unknown scatterers. Furthermore, to fully exploit the index functions obtained from the DSM, we also propose to integrate the DSM with a deep learning technique (DSM-DL) to achieve high-quality reconstructions. Several challenging and representative numerical experiments are carried out to demonstrate the accuracy and robustness of reconstructions by DSM-DL. The DSM-DL networks trained by phased data are further theoretically and numerically shown to be able to solve problems with phaseless data. Additionally, our numerical experiments also show the DSM-DL can solve inverse scattering problems with mixed types of scatterers, which renders its applications in many important practical scenarios.

翻译：本文研究仅有无相位数据时的逆声波散射问题。由于缺乏相位信息，该逆问题具有高度非线性和不适定性。利用无相位数据求解逆散射问题在实际应用中具有重要意义，因为获取物理上可接受的含相位数据通常困难且成本高昂。本文将提出一种新型直接采样方法，能够利用极少量入射波生成的无相位数据，有效估计未知散射体的位置与几何形状。通过对指标函数行为的严谨理论分析及若干代表性数值算例，证明该新型直接采样方法具有计算高效、易于实现、对强噪声鲁棒性强且无需任何先验散射体知识的优势。进一步地，为充分利用直接采样方法获得的指标函数，我们提出将直接采样方法与深度学习技术相融合，以实现高质量重构。通过多个具有挑战性的代表性数值实验，验证了直接采样-深度学习融合方法重构结果的准确性与鲁棒性。理论分析与数值实验均表明，通过含相位数据训练的直接采样-深度学习网络能够有效求解无相位数据问题。此外，数值实验还证明该方法可处理混合类型散射体的逆散射问题，这使其在众多重要实际应用场景中具有广阔前景。