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

William Marsh Rice University, Houston TX

Investigators

Abstract

The PI proposes to develop and rigorously analyze new efficient and high-order accurate integral equation methods for acoustic scattering in two and three dimensions, with the longer-range goal of developing methods for full electromagnetic scattering (by inhomogeneous and even anisotropic media) in three dimensions. These methods combine an existing FFT-based method for integrating over smooth regions of the inhomogeneity, with an accelerated, high-order accurate integration scheme, for small regions near scatterer discontinuities. The PI also proposes to develop preconditioning 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) preconditioning by means of the infinite frequency limit of the problem, i.e., geometrical optics. To pursue a preconditioner based on geometrical optics, an effective method for computing the multi-valued solution to the eikonal equation in inhomogeneous media (capturing the reflection and refraction at discontinuous interfaces) will be developed by building on a recently developed phase-space-based level set method, which combines a spectral representation in the phase variables with a discontinuous Galerkin finite-element discretization in the spatial variables. The use of numerical methods to evaluate the scattering of waves by complex material structures plays an important role in an enormous range of scientific and engineering technologies. Examples include radar and remote 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 electron diffraction, and exploration seismology. At the same time, the ever increasing size and complexity of material structures of practical interest place many problems beyond the capabilities of state-of-the-art computational methods, thus requiring new and innovative mathematical approaches. The PI proposes to develop new efficient and highly accurate computational methods for such problems to enable the treatment of larger and more complex scattering structures.