Fast solvers for frequency-domain wave-scattering problems and applications

在许多应用领域都需要对波动现象进行计算，例如雷达和电信设备的模型都需要对电磁波进行计算，而地震和医学成像算法的实现则分别使用声波、弹性波和电磁波来获取地球地下和人体的信息。在这些技术的设计和实现中，自然产生了波传播的计算机模型。医学成像技术人员使用人体物质组成如何散射入射电磁波的计算机模型，以解决根据观察到的散射波场重建人体内部构成的“逆问题”。同样，地震学家使用计算机模型来模拟地下物质的特性如何影响弹性波的传播，以便根据观测到的弹性波的回声来重建地球的地下特性。这项技术非常有用，例如在医学领域，它意味着我们可以经常诊断健康问题，而不需要更多的侵入性技术。在地震学的案例中，它使一些看似不可能的事情成为可能——因为从物理上讲，仅仅通过钻孔来探索地球所有的地下属性是不可能的。然而，这种波现象的快速和精确的计算机模拟是复杂和昂贵的（就计算机时间而言），主要（但不是唯一）是因为波的高度振荡性质和它们所通过的复杂介质。因此，迫切需要新的方法来加快这种模型的速度，而这项任务是本研究的主要焦点。该项目将设计并在数学上证明实现这些计算机波模型的新系列快速方法，并将通过两个软件平台使新方法可用，这些软件平台可供广泛的科学家使用，也可以在一个额外的专业高性能计算库中使用。除了设计新的建模方法（在今天的多处理器计算机上运行良好），该项目还将涉及与两家公司的直接合作——斯伦贝谢（一个对地震学感兴趣的项目合作伙伴）和ABB（对电磁计算感兴趣）——以及两个学术团体，一个在地球科学，一个在电磁学。

The computation of wave phenomena is widely needed in many application areas, for example models of radar and telecommunications devices require the computation of electromagnetic waves while the implementations of seismic and medical imaging algorithms use acoustic, elastic, and electromagnetic waves to obtain information about the earth's subsurface and the human body respectively. Computer models of the propagation of waves arise naturally in the design and implementation of these technologies. Medical imaging technicians use computer models of how the material composition of the human body scatters incoming electromagnetic waves in order to solve the "inverse problem'' of reconstructing the internal makeup of a human being from an observed scattered wave field. Similarly, seismologists use computer models of how the material properties of the earth's subsurface affects the transmission of elastic waves in order to reconstruct the earth's subsurface properties from observed echoes of elastic wavesThis technology is hugely useful, for example in the medical context it means we can often diagnose health problems without a need for more invasive techniques. In the seismology case it makes something seemingly impossible become possible - since it is never physically possible to explore all of the earth's subsurface properties by simply boring holes. However the fast and accurate computer modelling of such wave phenomena is complicated and costly (in terms of computer time), principally (but not solely) because of the highly oscillatory nature of the waves and the complicated media through which they pass. Thus there is a strong need for new methods that speed up such models and that task is a principal focus of this research. This project will devise and mathematically justify new families of fast methods for implementing these computer wave models, and will make the new methods available through two software platforms which are accessible to a wide range of scientists as well as in an additional specialist high performance computing library. As well as devising new methods for modelling (which work well on today's multiprocessor computers), the project will also involve direct collaboration with two companies - Schlumberger (a Project Partner, interested in seismology) and ABB (interested in electromagnetic computations) - as well as two academic groups, one in geosciences and one in electromagnetics.