on the use of perfectly matched layers at corners for scattering problems with sign changing coefficients we investigate in a 2d setting the scattering of time harmonic electromagnetic waves by a plasmonic device represented as a non dissipative bounded and penetrable obstacle with a negative permittivity using the t coercivity approach we first prove that the problem is well posed in the classical framework h loc 1 if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device when the latter has corners for values inside the critical interval unusual strong singularities for the electromagnetic field can appear in that case well posedness is obtained by imposing a radiation condition at the corners to select the outgoing black hole plasmonic wave that is the one which carries energy towards the corners a simple and systematic criterion is given to define what is the outgoing solution finally we propose an original numerical method based on the use of perfectly matched layers at the corners we emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods

a simple scheme for adapting time frequency representations we propose a simple efficient technique for continuously adapting time frequency and time scale representations over time the procedure computes a short time quality measure of the representation for a range of values of a free parameter and estimates the optimal parameter value maximizing the quality measure via interpolation

t coercivity and continuous galerkin methods application to transmission problems with sign changing coefficients to solve variational indefinite problems one uses classically the banach ne as babuska theory here we study an alternate theory to solve those problems t coercivity moreover we prove that one can use this theory to solve the approximate problems which provides an alternative to the celebrated fortin lemma we apply this theory to solve the indefinite problem text div sigma nabla u f set in h 1 0 with sigma exhibiting a sign change

perfectly matched layers for diffraction gratings in inhomogeneous media stability and error estimates we study the scattering of acoustic or electromagnetic waves at diffraction gratings in inhomogeneous media the refractive index stabilizes to 1 as the distance to the grating increases outgoing solutions are characterized by means of the limiting absorption principle we prove the unique solvability of the problem with perfectly matched layer of finite length further we show that solutions of the latter problem approximate outgoing solutions of the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity contribution of eigenvalues and resonances to the error of approximation is clarified

hermitian boundary conditions at a dirichlet singularity the marletta rozenblum model in domains b with smoothly varying boundary conditions points where wavefunctions are required to vanish were recently identified as dirichlet singularities d points where the hamiltonian h does not define discrete eigenvalues and a scattering phase is undetermined berry and dennis 2008 j phys a math theor 41 135203 this is explained marletta and rozenblum 2009 j phys a math theor 42 125204 by the observation illustrated with an exactly solvable separable model that a d point requires the specification of an additional parameter defining a family of self adjoint extensions of h here the underlying theory is presented in an elementary way and a d point is identified as a leak through which current can flow into or out of b hermiticity seals the leak ensuring that no current flows though the d point as well as across the boundary of b the solvable model is examined in detail for bound states where b is a semidisk and for wave reflections where b is a half plane the quantization condition for a nonseparable billiard is obtained explicitly

a high resolution data adaptive time frequency representation a data adaptive time frequency representation is developed that overcomes some limitations of the short time fourier transform while avoiding the cross terms that make the wigner distribution and other bilinear representations difficult to interpret the adaptive time frequency representations uses gaussian basis functions but varies their time width and chirp rate with time and frequency to achieve high signal concentration everywhere a measure of local signal concentration allows fully automated determination of the optimal basis parameters the adaptive method is computationally expensive but may provide much better performance than any currently known technique

an adaptive optimal kernel time frequency representation time frequency representations with fixed windows or kernels figure prominently in many applications but perform well only for limited classes of signals representations with signal dependent kernels can overcome this limitation however while they often perform well most existing schemes are block oriented techniques unsuitable for on line implementation or for tracking signal components with characteristics that change with time the time frequency representation developed in the present paper based on a signal dependent radially gaussian kernel that adapts over time surmounts these difficulties the method employs a short time ambiguity function both for kernel optimization and as an intermediate step in computing constant time slices of the representation careful algorithm design provides reasonably efficient computation and allows on line implementation certain enhancements such as cone kernel constraints and approximate retention of marginals are easily incorporated with little additional computation while somewhat more expensive than fixed kernel representations this new technique often provides much better performance several examples illustrate its behavior on synthetic and real world signals

numerical solution to the time dependent maxwell equations in axisymmetric singular domains the singular complement method in this paper we present a method to solve numerically the axisymmetric time dependent maxwell equations in a singular domain in math methods appl sci 25 2002 49 math methods appl sci 26 2003 861 the mathematical tools and an in depth study of the problems posed in the meridian half plane were exposed the numerical method and experiments based on this theory are now described here it is also the generalization to axisymmetric problems of the singular complement method that we developed to solve maxwell equations in 2d singular domains see c r acad sci paris t 330 2000 391 it is based on a splitting of the space of solutions in a regular subspace and a singular one derived from the singular solutions of the laplace problem numerical examples are finally given to illustrate our purpose in particular they show how the singular complement method captures the singular part of the solution

development and analysis of both finite element and fourth order in space finite difference methods for an equivalent berenger s pml model abstract this paper deals with an equivalent berenger s perfectly matched layer pml model we first develop a finite element scheme using edge elements to solve this model we prove a discrete stability of this method which inherits the stability obtained in the continuous case then we propose a fourth order in space finite difference scheme for solving this pml model numerical stability similar to the continuous stability and the optimal error estimate are established for the difference scheme here only second order time discretizations are considered for both schemes finally numerical results are presented to justify our analysis and demonstrate the effectiveness of this pml model for absorbing impinging waves

a posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients we perform the a posteriori error analysis of residual type of transmission problem with sign changing coefficients according to bonnet bendhia et al 2010 9 if the contrast is large enough the continuous problem can be transformed into a coercive one we further show that a similar property holds for the discrete problem for any regular meshes extending the framework from bonnet bendhia et al 9 the reliability and efficiency of the proposed estimator are confirmed by some numerical tests

computationally lag invariant recursive spectrum estimators this paper is concerned with fourier based nonparametric spectral estimation methods for time varying environments it presents a class of interactive spectrum estimators in which the number of computations required to update the estimate at every data sample is independent of the number of the autocorrelation lags employed to produce this estimate these estimators are termed lag invariant and form an important class due to their computational simplicity because of the lag invariant property resolution enhancement is achieved without additional cost in computations per frequency sample the proposed class of recursive spectral estimators is based on specific selection of the lagged product windows used in the autocorrelation function estimation in this paper the conditions necessary to produce the lag invariance property are delineated

a singular field method for maxwell s equations numerical aspects for 2d magnetostatics the present paper deals with the solution of maxwell type problems by means of nodal h1 conforming finite elements in a nonconvex piecewise regular domain surrounded by a perfect conductor such a discretization cannot in general approximate the singular behavior of the electromagnetic field near reentrant corners or edges the singular field method consists of adding to the finite element discretization space some particular fields which take into account the singular behavior the latter are deduced from the singular functions associated with the scalar laplace operator the theoretical justification of this approach as well as the analysis of the convergence of the approximation are presented for a very simple model problem arising from magnetostatics in a translation invariant setting but the study can be easily extended to numerous maxwell type problems the numerical implementation of both variants is studied for a domain containing a single reentrant corner

extracting edge flux intensity functions for the laplacian the solution to the laplace operator in three dimensional domains in the vicinity of straight edges is presented as an asymptotic expansion involving eigenpairs with their coefficients called edge flux intensity functions efifs the eigenpairs are identical to their two dimensional counterparts over a plane perpendicular to the edge extraction of efifs however cannot be obtained in a straightforward manner over this two dimensional plane a method based on l 2 projection and richardson extrapolation is presented for point wise extraction of efifs from p finite element solutions and illustrated by examples a similar but more efficient method based on energy projection for extracting efifs is proposed

the use of cone shaped kernels for generalized time frequency representations of nonstationary signals generalized time frequency representations gtfrs which use cone shaped kernels for nonstationary signal analysis are presented the cone shaped kernels are formulated for the gtfrs to produce good resolution simultaneously in time and frequency specifically for a gtfr with a cone shaped kernel finite time support is maintained in the time dimension along with an enhanced spectrum in the frequency dimension and the cross terms are smoothed out experimental results on simulated data and real speech show the advantages of the gtfrs with the cone shaped kernels through comparisons to the spectrogram and the pseudo wigner distribution

alias free generalized discrete time time frequency distributions a definition of generalized discrete time time frequency distribution that utilizes all of the outer product terms from a data sequence so that one can avoid aliasing is introduced the new approach provides 1 proper implementation of the discrete time spectrogram 2 correct evaluation of the instantaneous frequency of the underlying continuous time signal and 3 correct frequency marginal the formulation provides a unified framework for implementing members of cohen s class which was formulated in the continuous time domain some requirements for the discrete time kernel in the new approach are discussed in association with desirable distribution properties some experimental results are provided to illustrate the features of the proposed method

time harmonic wave diffraction problems in materials with sign shifting coefficients some electromagnetic materials present in a given frequency range an effective dielectric permittivity and or magnetic permeability which are negative we are interested in the reunion of such a negative material and a classical one more precisely we consider here a scalar model problem for the simulation of a wave transmission between two such materials this model is governed by a helmholtz equation with a weight function in the d principal part which takes positive and negative real values introducing additional unknowns we have already proposed in bonnet ben dhia et al 2006 1 some new variational formulations of this problem which are of fredholm type provided the absolute value of the contrast of permittivities is large enough and therefore suitable for a finite element discretization we prove here that under similar conditions on the contrast the natural variational formulation of the problem although not coercive plus compact is nonetheless suitable for a finite element discretization this leads to a numerical approach which is straightforward less costly than the previous ones and very accurate

a signal dependent time frequency representation fast algorithm for optimal kernel design a time frequency representation based on an optimal signal dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time frequency distributions that the best kernel and distribution depend on the signal to be analyzed the optimization formulation for the signal dependent kernel results in a linear program with a unique feature a tree structure that summarizes a set of constraints on the kernel the authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest for a kernel with q variables the running time of the algorithm is o q log q which is several orders of magnitude less than any other known method for solving this class of linear program this efficiency enables the computation of the signal dependent optimal kernel time frequency representation at a cost that is on the same order as a fixed kernel distribution an important property of the optimal kernel is that it takes on essentially only the values of 1 and 0

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

mesh requirements for the finite element approximation of problems with sign changing coefficients transmission problems with sign changing coefficients occur in electromagnetic theory in the presence of negative materials surrounded by classical materials for general geometries establishing fredholmness of these transmission problems is well understood thanks to the t coercivity approach moreover for a plane interface there exist meshing rules that guarantee an optimal convergence rate for the finite element approximation we propose here a new treatment at the corners of the interface which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates this treatment relies on the use of simple geometrical transforms to define the meshes numerical results illustrate the importance of this new design

mathematical analysis of ziolkowski s pml model with application for wave propagation in metamaterials abstract in this paper we investigate one perfectly matched layer pml model proposed by ziolkowski in 1999 various schemes for solving this pml model have been developed and have been shown to be effective in absorbing outgoing waves when the wave propagation in unbounded domain problem is reduced to a bounded domain problem however a rigorous analysis of this model is lacking in this paper we establish the stability of this pml model and propose a fully discrete finite element scheme to solve this model with edge elements discrete stability and optimal error estimate of this scheme are proved numerical results justifying the analysis and demonstrating the effectiveness of this pml model are presented

modified cohen lee time frequency distributions and instantaneous bandwidth of multicomponent signals cohen 1989 1995 has introduced and extensively studied and developed the concept of the instantaneous bandwidth of a signal specifically instantaneous bandwidth is interpreted as the spread in frequency about the instantaneous frequency which is itself interpreted as the average frequency at each time this view stems from a joint time frequency distribution tfd analysis of the signal where instantaneous frequency and instantaneous bandwidth are taken to be the first two conditional spectral moments respectively of the distribution however the traditional definition of instantaneous frequency namely as the derivative of the phase of the signal is not consistent with this interpretation and new definitions have therefore been proposed previously we show that similar problems arise with the cohen lee 1988 1989 instantaneous bandwidth of a signal and propose a new formulation for the instantaneous bandwidth that is consistent with its interpretation as the conditional standard deviation in frequency of a tfd we give the kernel constraints for a distribution to yield this new result which is a modification of the kernel proposed by cohen and lee these new kernel constraints yield a modified cohen lee tfd whose first two conditional moments are interpretable as the average frequency and bandwidth at each time respectively

numerical solution to the time dependent maxwell equations in two dimensional singular domains in this paper we present a method to solve numerically the time dependent maxwell equations in nonsmooth and nonconvex domains indeed the solution is not of regularity h1 in space in general moreover the space of h1 regular fields is not dense in the space of solutions thus an h1 conforming finite element method can fail even with mesh refinement the situation is different than in the case of the laplace problem or of the lame system for which mesh refinement or the addition of conforming singular functions work to cope with this difficulty the singular complement method is introduced this method consists of adding some well chosen test functions these functions are derived from the singular solutions of the laplace problem also the scm preserves the interesting features of the original method easiness of implementation low memory requirements small cost in terms of the cpu time to ascertain its validity some concrete problems are solved numerically