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An adaptive uniaxial perfectly matched layer method for time harmonic Maxwell scattering problems
"... Abstract. In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the timeharmonic scattering problems in twolayered media. The exterior region of the scatterer is divided into two half spaces by an infinite plane, on two sides of which the wave number takes different ..."
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Abstract. In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the timeharmonic scattering problems in twolayered media. The exterior region of the scatterer is divided into two half spaces by an infinite plane, on two sides of which the wave number takes different values. We surround the computational domain where the scattering field is interested by a PML layer with the uniaxial medium property. By imposing homogenous boundary condition on the outer boundary of the PML layer, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational domain as either the PML absorbing coefficient or the thickness of the PML layer tends to infinity. 1. Introduction. We
Numerical solution of an inverse medium scattering problem for Maxwell's . . .
 JOURNAL OF COMPUTATIONAL PHYSICS
, 2009
"... ... bounded domain in R³. In this paper, wellposedness of the variational problem for the direct scattering is examined. An energy estimate for the scattered field is obtained on which the Born approximation is based. A regularized recursive linearization method for the inverse medium scattering, w ..."
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Cited by 9 (4 self)
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... bounded domain in R³. In this paper, wellposedness of the variational problem for the direct scattering is examined. An energy estimate for the scattered field is obtained on which the Born approximation is based. A regularized recursive linearization method for the inverse medium scattering, which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field, is developed. The algorithm requires only singlefrequency data. Using an initial guess from the Born approxima
An exact bounded perfectly matched layer for timeharmonic scattering problems
 SIAM J Sci Comput
"... Abstract. The aim of this paper is to introduce an “exact ” bounded perfectly matched layer (PML) for the scalar Helmholtz equation. This PML is based on using a non integrable absorbing function. “Exactness ” must be understood in the sense that this technique allows exact recovering of the solutio ..."
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Abstract. The aim of this paper is to introduce an “exact ” bounded perfectly matched layer (PML) for the scalar Helmholtz equation. This PML is based on using a non integrable absorbing function. “Exactness ” must be understood in the sense that this technique allows exact recovering of the solution to timeharmonic scattering problems in unbounded domains. In spite of the singularity of the absorbing function, the coupled fluid/PML problem is well posed when the solution is sought in an adequate weighted Sobolev space. The resulting variational formulation can be numerically dealt with standard finite elements. The high accuracy of this approach is numerically demonstrated as compared with a classical PML technique. Key words. Perfectly matched layer, timeharmonic scattering, Helmholtz equation AMS subject classifications. 65N30 65N99 76Q05 1. Introduction. The
CONVERGENCE OF THE TIMEDOMAIN PERFECTLY MATCHED LAYER METHOD FOR ACOUSTIC SCATTERING PROBLEMS
, 2009
"... In this paper we establish the stability and convergence of the timedomain perfectly matched layer (PML) method for solving the acoustic scattering problems. We first prove the wellposedness and the stability of the timedependent acoustic scattering problem with the DirichlettoNeumann boundar ..."
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In this paper we establish the stability and convergence of the timedomain perfectly matched layer (PML) method for solving the acoustic scattering problems. We first prove the wellposedness and the stability of the timedependent acoustic scattering problem with the DirichlettoNeumann boundary condition. Next we show the wellposedness of the unsplitfield PML method for the acoustic scattering problems. Then we prove the exponential convergence of the nonsplitting PML method in terms of the thickness and medium property of the artificial PML layer. The proof depends on a stability result of the PML system for constant medium property and an exponential decay estimate of the modified Bessel functions.
An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures
 Mathematics of Compu20 posted on May 7, 2009, PII S
"... Abstract. An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medi ..."
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Abstract. An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method. 1.
A SOURCE TRANSFER DOMAIN DECOMPOSITION METHOD FOR HELMHOLTZ EQUATIONS IN UNBOUNDED DOMAIN
"... Abstract. We propose and study a domain decomposition method for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problems. The method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer wh ..."
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Abstract. We propose and study a domain decomposition method for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problems. The method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The convergence of the method is proved forthe case of constant wave number based on the analysis of the fundamental solution of the PML equation. The method can be used as an efficient preconditioner in the preconditioned GMRES method for solving discrete Helmholtz equations with constant and heterogeneous wave numbers. Numerical examples are included. Key words. Helmholtz equation, high frequency waves, PML, source transfer. 1. Introduction. We
AN ADAPTIVE FINITE ELEMENT METHOD FOR THE DIFFRACTION GRATING PROBLEM WITH TRANSPARENT BOUNDARY CONDITION∗
"... Abstract. The diffraction grating problem is modeled by a boundary value problem governed by a Helmholtz equation with transparent boundary conditions. An a posteriori error estimate is derived when the truncation of the nonlocal boundary operators takes place. To overcome the difficulty caused by t ..."
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Abstract. The diffraction grating problem is modeled by a boundary value problem governed by a Helmholtz equation with transparent boundary conditions. An a posteriori error estimate is derived when the truncation of the nonlocal boundary operators takes place. To overcome the difficulty caused by the fact that the truncated DirichlettoNeumann (DtN) mapping does not converge to the original DtN mapping in its operator norm, a duality argument without assuming more regularity than the weak solution is applied. The a posteriori error estimate consists of two parts, the finite element discretization error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter. Based on the a posteriori error control, a finite element adaptive strategy is established for the diffraction grating problem, such that the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive algorithm. Key words. Helmholtz equation, transparent boundary condition, a posteriori error estimates, adaptive algorithm, diffractive optics
LONGTIME STABILITY AND CONVERGENCE OF THE UNIAXIAL PERFECTLY MATCHED LAYER METHOD FOR TIMEDOMAIN ACOUSTIC SCATTERING PROBLEMS
"... Abstract. The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the timedom ..."
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Abstract. The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the timedomain scattering problem based on the Laplace transform and the complex coordinate stretching in the frequency domain. We prove the longtime stability of the initialboundary value problem of the uniaxial PML system for piecewise constant medium property and show the exponential convergence of the timedomain uniaxial PML method. Our analysis shows that for fixed PML absorbing medium property, any error of the timedomain PML method can be achieved by enlarging the thickness of the PML layer as lnT for large T> 0. Numerical experiments are included to illustrate the efficiency of the PML method.
Reverse time migration for extended obstacles: electromagnetic waves, Inverse Problem
"... Abstract. We propose a new single frequency reverse time migration (RTM) algorithm for imaging extended targets using electromagnetic waves. The imaging functional is defined as the imaginary part of the crosscorrelation of the Green function for Helmholtz equation and the backpropagated electroma ..."
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Abstract. We propose a new single frequency reverse time migration (RTM) algorithm for imaging extended targets using electromagnetic waves. The imaging functional is defined as the imaginary part of the crosscorrelation of the Green function for Helmholtz equation and the backpropagated electromagnetic field. The resolution of our RTM method for both penetrable and nonpenetrable extended targets is studied by virtue of HelmholtzKirchhoff identity for the timeharmonic Maxwell equation. The analysis implies that our imaging functional is always positive and thus may have better stability properties. Numerical examples are provided to demonstrate the powerful imaging quality and confirm our theoretical results. ar X iv