Mathematical Sciecnes: Boundary Variations and Analytic Continuation in Electromagnetic and Acoustic Scattering

9622555 Reitich这个项目处理电磁和声散射问题中的分析和计算问题。这涉及到一种新的方法来解决这类问题，基于高阶边界摄动和解析延拓技术，这是由首席研究员在衍射光栅的背景下提出的。在许多情况下，数值算法产生的结果比经典方法给出的结果精度大大提高。提出的研究涉及该方法在电磁学和声学等当前感兴趣的其他具有挑战性的领域的新应用，以及进一步研究和改进所得算法的数值性质。本文将研究各种结构下的正散射和逆散射问题，包括电大有界体和海洋波导中的散射。特别地，将处理三维立方体的电磁散射和粗糙表面海洋中的声音传播的基准问题。根据一些初步研究，预计该方法在这些领域的性能将与它在以前的实现中所展示的质量相当。预测入射波遇到障碍物或界面时散射的电磁场或声场形状的能力，长期以来被认为在许多科学和工程学科中具有重要意义。的确，我们“看到”的许多东西——无论是通过可见光还是x射线、无线电还是微波——或“听到”的许多东西都是通过各种现象的复杂组合到达我们这里的，其中散射在大多数情况下是一个基本因素。因此，在过去的几十年里，对光波和声波如何传播和衍射的更好理解已经导致了各种领域的实质性进展；这些包括通信、监测、地震剖面、断层扫描和目标探测，仅举几例。在这些进步中，数学建模发挥了重要作用，随着计算机的出现，计算科学也发挥了重要作用。然而，当前和未来的需求要求开发更有效、更准确和更可靠的算法来处理复杂的几何形状和介质。事实上，在这种情况下（例如处理人体组织或海底地形时）散射的分辨率通常需要计算高振荡场。当使用大多数经典算法时，这些振荡转化为高计算成本，因为这些方法试图捕获每个时间和空间点上的场的变化。因此，复杂性与所研究领域无关的替代方法变得非常可取。目前的项目涉及一种这样的方法，该方法是由首席研究员在微光学设备的背景下成功开发的。提出的研究涉及这种新方法在电磁学和声学等当前感兴趣的其他具有挑战性的领域的新应用，在这些领域，它有望提供有价值的计算工具。* * *

9622555 Reitich This project deals with analytical and computational issues in electromagnetic and acoustic scattering problems. It relates to a new method for the solution of such problems, based on high-order boundary perturbation and analytic continuation techniques, that was introduced by the principal investigator in the context of diffraction gratings. There, the numerical algorithms produced, in many cases, results with substantially improved accuracy over that given by classical approaches. The proposed research concerns new applications of the method in other challenging areas of current interest in electromagnetics and acoustics and the further study and improvement of the numerical properties of the resulting algorithms. Both the forward and inverse scattering problems will be investigated for a variety of configurations, including scattering by electrically large bounded bodies and in ocean waveguides. In particular, the benchmark problems of electromagnetic scattering by three-dimensional cubes and of sound propagation in oceans with rough surfaces will be treated. Based on some preliminary studies, it is expected that the performance of the method in these areas will be of a quality comparable to the one it exhibited in prior implementations. %%% The ability to predict the shape of the electromagnetic or acoustic field scattered as an incident wave encounters an obstacle or interface has been long recognized as having major implications in a great number of scientific and engineering disciplines. Indeed, much of what we "see" --be it through visible light or x-rays, radio or microwaves-- or "hear" reaches us through a complicated combination of phenomena among which scattering is, in most cases, an essential element. As such, a better understanding of how light and sound waves propagate and diffract has led, in the last few decades, to substantial advances in a variety of fields; these include communications, monitoring, s eismic profiling, tomography and target detection, to name just but a few. An important role in these advances was the one played by mathematical modeling and, with the advent of computers, that of computational science. The present and future needs, however, demand the development of more efficient, accurate and reliable algorithms to deal with complex geometries and media. Indeed, the resolution of the scattering in such situations (such as when dealing with human tissue or undersea topography) often entails calculating a highly oscillatory field. When using most classical algorithms these oscillations translate into high computational costs, as these methods attempt to capture the variations of the field at each point in time and space. Therefore, alternative approaches whose complexity does not correlate to that of the field under study become very desirable. The present project relates to one such approach, that was successfully developed by the principal investigator in the context of micro-optical devices. The proposed research concerns new applications of this novel method in other challenging areas of current interest in electromagnetics and acoustics where it is expected to provide a valuable computational tool. ***