Results 1  10
of
1,063
Finite elements in computational electromagnetism
, 2002
"... This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinateindependent statement of Maxwell’s equations in the calculus of differential forms. Th ..."
Abstract

Cited by 137 (7 self)
 Add to MetaCart
This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinateindependent statement of Maxwell’s equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete dif
A LevelSet Approach for Inverse Problems Involving Obstacles
, 1996
"... . An approach for solving inverse problems involving obstacles is proposed. The approach uses a levelset method which has been shown to be effective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry. We develop two computational metho ..."
Abstract

Cited by 95 (2 self)
 Add to MetaCart
(Show Context)
. An approach for solving inverse problems involving obstacles is proposed. The approach uses a levelset method which has been shown to be effective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry. We develop two computational methods based on this idea. One method results in a nonlinear timedependent partial differential equation for the levelset function whose evolution minimizes the residual in the data fit. The second method is an optimization that generates a sequence of levelset functions that reduces the residual. The methods are illustrated in two applications: a deconvolution problem and a diffraction screen reconstruction problem. Keywords: Inverse problems, levelset method, HamiltonJacobi equations, surface evolution, optimization, deconvolution, diffraction. 1. Inverse problems involving obstacles There is a host of inverse problems wherein the desired unknown is a region in IR 2 or IR 3 . ...
Adjoint Recovery of Superconvergent Functionals from Approximate Solutions of Partial Differential Equations
, 1998
"... Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that ..."
Abstract

Cited by 86 (12 self)
 Add to MetaCart
(Show Context)
Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasionedimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multidimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations. Key words. PDEs, adjoint equations, error analysis, superconvergence AMS subject classifications. 65G99, 76N15 PII. S0036144598349423
The Factorization Method for Inverse Problems
, 2007
"... since Christian Hülsmeyer showed in 1904 that one could use radio waves to detect metallic objects at a distance (the range of the first apparatus was 3000 meters), the race has been on to tease out ever more information from scattered waves. Within months of his first detection demonstration, Hül ..."
Abstract

Cited by 69 (5 self)
 Add to MetaCart
(Show Context)
since Christian Hülsmeyer showed in 1904 that one could use radio waves to detect metallic objects at a distance (the range of the first apparatus was 3000 meters), the race has been on to tease out ever more information from scattered waves. Within months of his first detection demonstration, Hülsmeyer devised a way to determine the distance to the object. At that rate of improvement one might have extrapolated to unimaginable twentyfirstcentury capabilities. Unfortunately, what has proved to be unimaginable is the difficulty of doing much more than the original device had already accomplished. It would seem that some forms of bionic vision have gone the way of rocket backpacks – that is, until recently. The newest book by Andreas Kirsch with coauthor Natalia Grinberg, The Factorization Method for Inverse Problems, collects over a decade of work by Kirsch and collaborators on a simple method for shape identification in inverse scattering. This book belongs to the next generation of monographs on inverse scattering following the now standard works of Colton and Kress [2] (Inverse Acoustic and Electromagnetic Scattering Theory (1998)) and Isakov [7] (Inverse Problems for Partial Differential Equations). Kirsch’s factorization method arose from experimentation with noniterative inverse scattering methods that avoid the computational expense of calculating the solution to the forward problem at each iteration. Noniterative methods attack headon the inverse problem of determining the scatterer from measured
A hybrid numericalasymptotic boundary integral method for highfrequency acoustic scattering
 NUMERISCHE MATHEMATIK
"... ..."
A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
Abstract

Cited by 48 (16 self)
 Add to MetaCart
(Show Context)
Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
The interior transmission problem
 Inverse Probl. Imaging
"... Abstract. The interior transmission problem is a boundary value problem that plays a basic role in inverse scattering theory but unfortunately does not seem to be included in any existing theory in partial differential equations.This paper presents old and new results for the interior transmission p ..."
Abstract

Cited by 48 (13 self)
 Add to MetaCart
(Show Context)
Abstract. The interior transmission problem is a boundary value problem that plays a basic role in inverse scattering theory but unfortunately does not seem to be included in any existing theory in partial differential equations.This paper presents old and new results for the interior transmission problem,in particular its relation to inverse scattering theory and new results on the spectral theory associated with this class of boundary value problems. 1.
Transmission eigenvalues
 SIAM Journal on Mathematical Analysis
"... Abstract. The scattering of a timeharmonic plane wave in an inhomogeneous medium is modeled by the scattering problem for the Helmholtz equation. A transmission eigenvalue is a wavenumber at which the scattering operator has a nontrivial kernel or cokernel. Because many sampling methods for locati ..."
Abstract

Cited by 47 (6 self)
 Add to MetaCart
(Show Context)
Abstract. The scattering of a timeharmonic plane wave in an inhomogeneous medium is modeled by the scattering problem for the Helmholtz equation. A transmission eigenvalue is a wavenumber at which the scattering operator has a nontrivial kernel or cokernel. Because many sampling methods for locating scatterers succeed only at wavenumbers that are not transmission eigenvalues, they have been studied for some time. Nevertheless, the existence of transmission eigenvalues has previously been proved only for radial scatterers. In this paper, we prove existence for scatterers without radial symmetry. 1. Introduction. The
Solving timeharmonic scattering problems based on the pole condition: Convergence of the PML method
, 2001
"... In this paper we study the PML method for Helmholtztype scattering problems with radially symmetric potential. The PML method consists in surrounding the computational domain by a Perfectly Matched sponge Layer. We prove that the approximate solution obtained by the PML method converges exponential ..."
Abstract

Cited by 47 (9 self)
 Add to MetaCart
In this paper we study the PML method for Helmholtztype scattering problems with radially symmetric potential. The PML method consists in surrounding the computational domain by a Perfectly Matched sponge Layer. We prove that the approximate solution obtained by the PML method converges exponentially fast to the true solution in the computational domain as the thickness of the sponge layer tends to infinity. This is a generalization of results by Lassas and Somersalo based on boundary integral equation techniques. Here we use techniques based on the pole condition instead. This makes it possible to treat problems without an explicitly known fundamental solution
Implementation of a boundary element method for high frequency scattering by convex polygons
 ADVANCES IN BOUNDARY INTEGRAL METHODS (PROCEEDINGS OF THE 5TH UK CONFERENCE ON BOUNDARY INTEGRAL METHODS
"... In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. H ..."
Abstract

Cited by 46 (20 self)
 Add to MetaCart
In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretisation of a wellknown second kind combinedlayerpotential integral equation. We provide a proof that this equation and its adjoint are wellposed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.