本项目旨在研究频域声波逆散射问题中一类特定逆源问题的重构理论和快速算法。我们的研究内容和目标是：考虑以下问题，（1） 实际应用中很多常见的声源，比如点源是Delta函数，但它并不属于Hilbert空间，属于更大的Radon测度空间，是个Banach空间。如何重构这类奇异的常用声源是有意义的。已知的研究结果表明，使用单个频率，无法唯一的重构这类声源。该类问题本质上是一个非单射的、紧算子的求逆问题，非常有挑战。我们将研究如何利用偏微分方程和分析的理论来讨论该类逆源问题在Banach空间中的存在性及其性质。（2）在数值上，将发展和利用新的一阶凸优化算法，利用正则化变分模型，在合适的函数空间中，来高效的求解该问题。进一步研究数值解的数学性质和物理意义，分析算法的效率，收敛性和收敛速度，为实际应用提供理论支持和算法工具。项目的研究成果将对油藏勘探，无损探测等众多科学技术的发展具有指导意义。

This project aims at the reconstructions of certain kinds of acoustic sources for the inverse scattering problems in the frequency domain, mainly focused on the theoretical analysis and development of fast algorithms. We will consider the following problems. First, our motivation is based on the fact that a lot frequently used acoustic sources do not belong to Hilbert spaces, including the point sources which are Delta functions. Instead, they belong to Radon measure space, which is a Banach space. However, how to reconstruct such kinds of acoustic sources is unclear currently. Actually, reconstructing this kind of singular but frequently used source is meaningful. Existing results tell that this kind of sources could not be reconstructed uniquely, and the reconstructing is essentially calculating the inverse of compact operators without injection, which is very difficult and challenging. We will study the mathematical existence and the properties of reconstructed solutions in appropriate Banach spaces, with the help of the PDE theory and analysis techniques. Second, we will develop fast algorithms for efficient numerical computations in appropriate functional spaces, with the newly developed first order algorithms from convex optimization. Furthermore, we will study the properties and physical meaning of the numerical solutions, and analysis the convergence properties of the iteration algorithms, which could provide some theoretical supports and efficient algorithms for real applications. The results of this research program probably could bring out some benefits for lots of science and technology fields including oil exploration, non-invasive detecting and so on.