arXiv subjects

Peter Monk

Publications and source records attributed to Peter Monk.

27 records · Page 2Linked to original sources

arXiv2018

Near field linear sampling method for an inverse problem in an electromagnetic waveguide

We consider the problem of determining the shape and location of an unknown penetrable object in a perfectly conducting electromagnetic waveguide. The inverse problem is posed in the frequency domain and uses multistatic data in the near field. In particular, we assume that we are given measurements of the electric scattered field due to point sources on a cross-section of the waveguide and measured on the same cross-section, which is away from the scatterer but not in the far field. The problem is solved by using the Linear Sampling Method (LSM) and we also discuss the generalized LSM. We start by giving a brief discussion of the direct problem and its associated interior transmission problem. Then, we adapt and analyze the LSM to deal with the inverse problem. This extends the work on the LSM for perfectly conducting scatterers in a waveguide by one of us (Yang) to the detection of penetrable objects. We provide several useful results concerning reciprocity and the density of fields due to single layer potentials. We also prove the standard results for the LSM in the waveguide context. Finally we give numerical results to show the performance of the method for simple shapes.

math.NA↗

arXiv2018

Eigenvalue problems in inverse electromagnetic scattering theory

The inverse electromagnetic scattering problem for anisotropic media in general does not have a unique solution. A possible approach to this problem is through the use of appropriate "target signatures," i.e. eigenvalues associated with the direct scattering problem that are accessible to measurement from a knowledge of the scattering data. In this paper we shall consider three different sets of eigenvalues that can be used as target signatures: 1) eigenvalues of the electric far field operator, 2) transmission eigenvalues and 3) Stekloff eigenvalues.

math.AP↗

arXiv2018

A Trefftz Discontinuous Galerkin Method for Time Harmonic Waves with Generalized Impedance Boundary Conditions

We show how a Trefftz Discontinuous Galerkin (TDG) method for the displacement form of the Helmholtz equation can be used to approximate problems having a generalized impedance boundary condition (GIBC) involving surface derivatives of the solution. Such boundary conditions arise naturally when modeling scattering from a scatterer with a thin coating. The thin coating can then be approximated by a GIBC. A second place GIBCs arise is as higher order absorbing boundary conditions. This paper also covers both cases. Because the TDG scheme has discontinuous elements, we propose to couple it to a surface discretization of the GIBC using continuous finite elements. We prove convergence of the resulting scheme and demonstrate it with two numerical examples.

math.NA↗

arXiv2016

A Plane Wave Discontinuous Galerkin Method with a Dirichlet-to-Neumann Boundary Condition for the Scattering Problem in Acoustics

We consider the numerical solution of an acoustic scattering problem by the Plane Wave Discontinuous Galerkin Method (PWDG) in the exterior of a bounded domain in $\mathbb{R}^2$. In order to apply the PWDG method, we introduce an artificial boundary to truncate the domain, and we impose a non-local Dirichlet-to-Neumann (DtN) boundary conditions on the artificial curve. To define the method, we introduce new consistent numerical fluxes that incorporate the truncated series of the DtN map. Error estimates with respect to the truncation order of the DtN map, and with respect to mesh width are derived. Numerical results suggest that the accuracy of the PWDG method for the acoustic scattering problem can be significantly improved by using DtN boundary conditions.

math.NA↗

arXiv2015

Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves

The Trefftz Discontinuous Galerkin (TDG) method is a technique for approximating the Helmholtz equation (or other linear wave equations) using piecewise defined local solutions of the equation to approximate the global solution. When coefficients in the equation (for example, the refractive index) are piecewise constant it is common to use plane waves on each element. However when the coefficients are smooth functions of position, plane waves are no longer directly applicable. In this paper we show how Generalized Plane Waves (GPWs) can be used in a modified TDG scheme to approximate the solution for piecewise smooth coefficients. GPWs are approximate solutions to the equation that reduce to plane waves when the medium through which the wave propagates is constant. We shall show how to modify the TDG sesquilinear form to allow us to prove convergence of the GPW based version. The new scheme retains the high order convergence of the original TDG scheme (when the solution is smooth) and also retains the same number of degrees of freedom per element (corresponding to the directions of the GPWs). Unfortunately it looses the advantage that only skeleton integrals need to be performed. Besides proving convergence, we provide numerical examples to test our theory.

math.NA↗

arXiv2015

Time Dependent Scattering from a Grating

Computing the electromagnetic field due to a periodic grating is critical for assessing the performance of thin film solar voltaic devices. In this paper we investigate the computation of these fields in the time domain (similar problems also arise in simulating antennas). Assuming a translation invariant periodic grating this reduces to solving the wave equation in a periodic domain. Materials used in practical devices have frequency dependent coefficients, and we provide a first proof of existence and uniqueness for a general class of such materials. Using Convolution Quadrature we can then prove time stepping error estimates. We end with some preliminary numerical results that demonstrate the convergence and stability of the scheme.

math.NA↗

arXiv2015

A tent pitching scheme motivated by Friedrichs theory

Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space-time finite element scheme of the tent pitching type, with spaces that conform to the continuity property, is designed. Numerical results for a model one-dimensional wave propagation problem are presented.

math.NA↗

arXiv2014

The Time Domain Lippmann-Schwinger Equation and Convolution Quadrature

We consider time domain acoustic scattering from a penetrable medium with a variable sound speed. This problem can be reduced to solving a time domain volume Lippmann-Schwinger integral equation. Using convolution quadrature in time and trigonometric collocation in space we can compute an approximate solution. We prove that the time domain Lippmann-Schwinger equation has a unique solution and prove conditional convergence and error estimates for the fully discrete solution for smooth sound speeds. Preliminary numerical results show that the method behaves well even for discontinuous sound speeds.

math.NA↗

arXiv2014

Residual based adaptivity and PWDG methods for the Helmholtz equation

We present a study of two residual a posteriori error indicators for the Plane Wave Discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular we study the h-version of PWDG in which the number of plane wave directions per element is kept fixed. First we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. Numerical results demonstrate an improvement in the efficiency of the indicators.

math.NA↗