Integral Equation Methods for Scattering by Inhomogeneous Media: Preconditioning, Efficiency, and High-Order Accuracy

PI建议发展和严格分析二维和三维声散射的新的高效和高精度的积分方程法，较长期的目标是发展三维全电磁散射(非均匀甚至各向异性介质)的方法。这些方法结合了现有的基于快速傅立叶变换的非均匀光滑区域积分方法，以及一种加速的、高精度的积分方案，用于接近散射体不连续的小区域。PI还建议通过两条不同的研究路线来发展预条件策略：(1)通过对简单的分段恒定的径向分层非均匀结构的扰动展开来预条件；(2)通过问题的无限频率限制，即几何光学来预条件。为了寻求一种基于几何光学的预条件，在最近发展的相空间水平集方法的基础上，发展了一种有效的计算非均匀介质中eikonial方程多值解(捕捉不连续界面上的反射和折射)的方法。该方法结合了相变量中的谱表示和空间变量中的间断Galerkin有限元离散。利用数值方法计算复杂材料结构对波浪的散射在广泛的科学和工程技术中起着重要的作用。例如雷达和遥感、医学和生物成像(例如CT扫描、超声波)、激光-等离子体相互作用(例如激光驱动的聚变)、光子晶体设计、无损检测、中子和电子衍射以及勘探地震学。与此同时，实际感兴趣的物质结构的不断增长的大小和复杂性使许多问题超出了最先进的计算方法的能力，因此需要新的和创新的数学方法。PI建议为这类问题开发新的高效和高精度的计算方法，以便能够处理更大和更复杂的散射结构。

The PI proposes to develop and rigorously analyze new efficient andhigh-order accurate integral equation methods for acoustic scattering intwo and three dimensions, with the longer-range goal of developing methodsfor full electromagnetic scattering (by inhomogeneous and even anisotropicmedia) in three dimensions. These methods combine an existing FFT-basedmethod for integrating over smooth regions of the inhomogeneity, with anaccelerated, high-order accurate integration scheme, for small regionsnear scatterer discontinuities. The PI also proposes to developpreconditioning strategies through two distinct lines of investigation:(1) preconditioning by means of a perturbation expansion about a simple,piecewise-constant, radially layered inhomogeneity and (2) preconditioningby means of the infinite frequency limit of the problem, i.e., geometricaloptics. To pursue a preconditioner based on geometrical optics, aneffective method for computing the multi-valued solution to the eikonalequation in inhomogeneous media (capturing the reflection and refractionat discontinuous interfaces) will be developed by building on a recentlydeveloped phase-space-based level set method, which combines a spectralrepresentation in the phase variables with a discontinuous Galerkinfinite-element discretization in the spatial variables.The use of numerical methods to evaluate the scattering of waves bycomplex material structures plays an important role in an enormous rangeof scientific and engineering technologies. Examples include radar andremote sensing, medical and biological imaging (e.g., CT-scans,ultrasound), laser-plasma interactions (e.g., in laser-driven fusion),photonic crystal design, nondestructive testing, neutron and electrondiffraction, and exploration seismology. At the same time, the everincreasing size and complexity of material structures of practicalinterest place many problems beyond the capabilities of state-of-the-artcomputational methods, thus requiring new and innovative mathematicalapproaches. The PI proposes to develop new efficient and highly accuratecomputational methods for such problems to enable the treatment of largerand more complex scattering structures.