Peter Monk

The Linear Sampling Method in Inverse Electromagnetic Scattering

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in and . Included in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions.

Recent Developments in Inverse Acoustic Scattering Theory

We survey some of the highlights of inverse scattering theory as it has developed over the last 15 years, with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves. Included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects.

The Perfectly Matched Layer in Curvilinear Coordinates

In 1994 Berenger showed how to construct a perfectly matched absorbing layer for the Maxwell system in rectilinear coordinates. This layer absorbs waves of any wavelength and any frequency without reflection and thus can be used to artificially terminate the domain of scattering calculations. In this paper we show how to derive and implement the Berenger layer in curvilinear coordinates (in two space dimensions). We prove that an infinite layer of this type can be used to solve time harmonic scattering problems. We also show that the truncated Berenger problem has a solution except at a discrete set of exceptional frequencies (which might be empty). Finally numerical results show that the curvilinear layer can produce accurate solutions in the time and frequency domain.

A least-squares method for the Helmholtz equation

Abstract We investigate the use of least-squares methods to approximate the Helmholtz equation. The basis used in the discrete method consists of solutions of the Helmholtz equation (either consisting of plane waves or Bessel functions) on each element of a finite element grid. Unlike previous methods of this type, we do not use polynomial based finite elements. The use of small elements (and relatively few basis functions per element) allows us to prove convergence theorems for the method and, to some extent, control the conditioning of the resulting linear sy stem. Numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods.

Optimizing the Perfectly Matched Layer

The Berenger perfectly matched layer (or PML) [4, 3] is used in computational electromagnetism as a “sponge layer” to terminate finite element approximations of scattering problems. An infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction, and the waves decay exponentially in magnitude into the layer. This property is lost when the layer is truncated since the truncation boundary generates a reflected wave. Discretization of the differential equations further changes the reflection coefficient of the layer. In this talk we shall describe the PML in the simple case of a two dimensional electromagnetic problem. The first part of this abstract is expository. We shall show how to derive the layer using a change of variable approach [5, 9, 6]. After this derivation we describe some new analysis to show the effects of truncating and discretizing the PML [9]. We end by showing how to compute an optimal layer, and make some remarks about PML for the full Maxwell system.

A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region II

In a previous paper [SIAM J. Appl. Math. (1985), pp. 1039–1053] we presented a new method for determining the shape of an acoustically soft obstacle from a knowledge of the time-harmonic incident wave and the far field pattern of the scattered wave. The method given there was based on knowing the far field pattern for an interval of values of the square of the wave number k such that this interval contained the first eigenvalue

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

\lambda _1

Analysis of a finite element method for Maxwell's equations

of the interior Dirichlet problem. The purpose of this paper is to extend the methods of our earlier paper to treat the case when the far field pattern is only known for a single value of the wave number such that

A convergence analysis of Yee's scheme on nonuniform grids

k^2

The Linear Sampling Method for Solving the Electromagnetic Inverse Scattering Problem

is not equal to