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

小行星9622555 这个项目涉及电磁学中的分析和计算问题 和声散射问题。它涉及到一种新的方法来解决这些问题，高阶边界扰动和解析延拓技术的基础上，这是由主要研究者在衍射光栅的背景下介绍。 在那里，数值算法产生的，在许多情况下，结果大大提高了精度比经典的方法。 所提出的研究涉及新的应用程序的方法在其他具有挑战性的领域，目前的兴趣，在电磁学和声学和进一步研究和改进的数值特性的算法。无论是正向还是反向 本课程将研究各种不同结构的散射问题，包括电大尺寸有界物体的散射和海洋波导中的散射。特别是三维立方体的电磁散射和声音的基准问题 将处理在粗糙表面的海洋中的传播。根据一些初步研究，预计该方法在这些领域的性能将具有可比性 它在先前的实现中展示的一个。 %%% 预测当入射波遇到障碍物或界面时散射的电磁场或声场的形状的能力长期以来一直被认为在许多科学和工程学科中具有重大意义。事实上，我们"看到"的大部分东西--无论是通过可见光还是X射线， 无线电或微波-或“听到”通过一系列复杂的现象到达我们，其中散射是，在大多数情况下， 基本元素。因此，在过去的几十年里，对光波和声波如何传播和传播的更好理解导致了各种领域的实质性进展;这些领域包括通信、监测、地震剖面、断层扫描和目标定位。 检测，仅举几例。在这些进步中，数学建模发挥了重要作用， 计算机，计算科学。然而，目前和未来的需求，要求开发更有效，准确和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（如当处理人体组织或海底地形）的散射的分辨率往往需要计算一个高度振荡场。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捉场在时间和空间中的每个点的变化。因此， 复杂性与所研究领域的复杂性无关变得非常可取。本项目涉及一种这样的方法，这是成功地开发了由首席研究员在微光学器件的背景下。该研究涉及新的 这种新方法在电磁学和声学中当前感兴趣的其他具有挑战性的领域中的应用，预计将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. ***