A high-order, fast algorithm for scattering calculation in two dimensions

AbstractWe present a high-order, fast, iterative solver for the direct scattering calculation for the Helmholtz equation in two dimensions. Our algorithm solves the scattering problem formulated as the Lippmann-Schwinger integral equation for compactly supported, smoothly vanishing scatterers. There are two main components to this algorithm. First, the integral equation is discretized with quadratures based on high-order corrected trapezoidal rules for the logarithmic singularity present in the kernel of the integral equation. Second, on the uniform mesh required for the trapezoidal rule we rewrite the discretized integral operator as a composition of two linear operators: a discrete convolution followed by a diagonal multiplication; therefore, the application of these operators to an arbitrary vector, required by an iterative method for the solution of the discretized linear system, will cost N2log(N) for a N-by-N mesh, with the help of FFT. We will demonstrate the performance of the algorithm for scatterers of complex structures and at large wave numbers. For numerical implementations, GMRES iterations will be used, and corrected trapezoidal rules up to order 20 will be tested

Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.Comment: 18 page

An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper

Application of the inhomogeneous Lippmann-Schwinger equation to inverse scattering problems

In this paper we present a hybrid approach to numerically solve two-dimensional electromagnetic inverse scattering problems, whereby the unknown scatterer is hosted by a possibly inhomogeneous background. The approach is `hybrid' in that it merges a qualitative and a quantitative method to optimize the way of exploiting the a priori information on the background within the inversion procedure, thus improving the quality of the reconstruction and reducing the data amount necessary for a satisfactory result. In the qualitative step, this a priori knowledge is utilized to implement the linear sampling method in its near-field formulation for an inhomogeneous background, in order to identify the region where the scatterer is located. On the other hand, the same a priori information is also encoded in the quantitative step by extending and applying the contrast source inversion method to what we call the `inhomogeneous Lippmann-Schwinger equation': the latter is a generalization of the classical Lippmann-Schwinger equation to the case of an inhomogeneous background, and in our paper is deduced from the differential formulation of the direct scattering problem to provide the reconstruction algorithm with an appropriate theoretical basis. Then, the point values of the refractive index are computed only in the region identified by the linear sampling method at the previous step. The effectiveness of this hybrid approach is supported by numerical simulations presented at the end of the paper.Comment: accepted in SIAM Journal on Applied Mathematic