inverse medium scattering problems for electromagnetic waves consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in mathbb r 3 in this paper existence and uniqueness of the variational problem for forward scattering are established an energy estimate for the scattered field with a uniform bound with respect to the wavenumber is obtained in the case of low frequency on which the born approximation is based a continuation method for the inverse medium scattering problem which reconstructs the scatterer of an inhomogeneous medium from boundary measurements of the scattered wave is developed the algorithm requires multifrequency scattering data using an initial guess from the born approximation each update is obtained via recursive linearization on the wavenumber k by solving one forward problem and one adjoint problem of maxwell s equations

stability estimates for an inverse scattering problem at high frequencies we consider an inverse scattering problem and its near field approximation at high frequencies we first prove for both problems lipschitz stability results for determining the low frequency component of the potential then we show that in the case of a radial potential supported sufficiently near the boundary infinite resolution can be achieved from measurements of the near field operator in the monotone case

an inverse random source problem for the helmholtz equation this paper is concerned with an inverse random source problem for the one dimensional stochastic helmholtz equation which is to reconstruct the statistical properties of the random source function from boundary measurements of the radiating random electric field although the emphasis of the paper is on the inverse problem we adapt a computationally more efficient approach to study the solution of the direct problem in the context of the scattering model specifically the direct model problem is equivalently formulated into a two point spatially stochastic boundary value problem for which the existence and uniqueness of the pathwise solution is proved in particular an explicit formula is deduced for the solution from an integral representation by solving the two point boundary value problem based on this formula a novel and efficient strategy which is entirely done by using the fast fourier transform is proposed to reconstruct the mean and the variance of the random source function from measurements at one boundary point where the measurements are assumed to be available for many realizations of the source term numerical examples are presented to demonstrate the validity and effectiveness of the proposed method

linearized cauchy data inversion method for two dimensional buried target imaging we propose a novel inversion algorithm to image buried objects in inhomogeneous media from the electromagnetic data on the outer boundary our method is based on exploiting the cauchy data to derive a new born like linearization of the inverse problem the main advantage of this formulation is to avoid the use of the background green function and therefore is computationally more efficient it also provides better accuracy than classical born approximation in the case of stratified media our approach can be coupled with any appropriate continuation method we discuss here the coupling with a continuation method based on the use of approximate transmission conditions the feasibility and robustness of our methodology is validated through numerical experiments for single and multiple targets

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

multidimensional born velocity inversion single wideband point source the inverse scattering problem for an acoustic medium is considered within the approximate direct inversion framework as opposed to iterative methods the direct inversion approach gives an estimate of the medium velocities by operating on the observed scattered field without repeated solutions of the forward problem previous solutions to the multidimensional born velocity inversion problem require either coincident source and receiver arrays or plane wave sources in both cases an array of point sources is required to collect the proper data for inversion in this paper the solution to the single point source problem is derived it is shown that by extrapolating and imaging the observed scattered field appropriately the projections of the velocity function at all angles can be obtained the velocities are then reconstructed by the inverse radon transform method of tomography

stability for the multifrequency inverse medium problem abstract the solution of a multi frequency 1d inverse medium problem consists of recovering the refractive index of a medium from measurements of the scattered waves for multiple frequencies in this paper rigorous stability estimates are derived when the frequency takes value in a bounded interval it is showed that the ill posedness of the inverse medium problem decreases as the width of the frequency interval becomes larger more precisely under certain regularity assumptions on the refractive index the estimates indicate that the power in holder stability is an increasing function of the largest value in the frequency band finally a lipschitz stability estimate is obtained for the observable part of the medium function defined through a truncated trace formula

numerical solution of inverse scattering problems with multi experimental limited aperture data in this paper a regularized homotopy continuation method is presented for numerical solution of nonlinear ill posedness inverse problems two major difficulties for solving these inverse problems by an optimization method are the ill posedness and the presence of many local minima classic iterative methods such as gauss newton or levenberg marquardt algorithms offer fast local convergence but might not be able to compute the global minimum based on a natural concept of multi experimental data a regularized homotopy continuation method is constructed to compute the global minimum as the experimental index t increases continuously the global minimizer can be computed continuously by using local optimization methods by discretizing the continuous homotopy method various recursive linearization algorithms are developed these algorithms are applied to numerical solution of an inverse medium scattering problem which reconstructs the refractive index of an inhomogeneous medium from limited aperture measurements of the far field pattern of the scattered fields it is assumed that the data is measured at multiple frequencies the inverse scattering problem with limited aperture is challenging since without full aperture measurements the ill posedness and nonlinearity of the inverse problem become more severe an efficient regularized iterative linearization method recursive linearization with respect to the wave number is developed for solving the inverse problem the convergence of the iterative method is examined by numerical examples

adequate is better based on the interaction of individuals particle swarm optimization pso is a well recognized algorithm to find optima in search space in its canonical version the trajectory of each particle is usually influenced by the best performer among its neighborhood which thus ignores some useful information from other neighbors to capture information of all the neighbors the fully informed pso is proposed which however may bring redundant information into the search process motivated by both scenarios here we present a particle swarm optimization with limited information which provides each particle adequate information yet avoids the waste of information by means of systematic analysis for the widely used standard test functions it is unveiled that our new algorithm outperforms both canonical pso and fully informed pso especially for multimodal test functions we further investigate the underlying mechanism from a microscopic point of view revealing that moderate velocity moderate diversity and best motion consensus facilitate a good balance between exploration and exploitation which results in the good performance

invariants of particle motion in one dimensional time dependent potentials the author considers one dimensional time dependent hamiltonians of the form h p2 2 v q t it is shown that the potentials determined in an implicit way by the relation phi v a1v2 a2v a3 t q where phi is an arbitrary function and a1 a2 and a3 are arbitrary parameters admit a single valued constant of motion this constant of motion is a higher transcendental function in the momentum

on the construction of approximate solutions for a multidimensional nonlinear heat equation we study three methods based on continuous symmetries to find approximate solutions for the multidimensional nonlinear heat equation delta u delta x0 delta u aun epsilon f u where a and n are arbitrary real constants f is a smooth function and 0 epsilon 1

enhancing network robustness against malicious attacks in a recent work schneider et al proc natl acad sci usa 108 3838 2011 the authors proposed a simple measure for network robustness under malicious attacks on nodes using a greedy algorithm they found that the optimal structure with respect to this quantity is an onion structure in which high degree nodes form a core surrounded by rings of nodes with decreasing degree however in real networks the failure can also occur in links such as dysfunctional power cables and blocked airlines accordingly complementary to the node robustness measurement r n we propose a link robustness index r l we show that solely enhancing r n cannot guarantee the improvement of r l moreover the structure of an r l optimized network is found to be entirely different from that of an onion network in order to design robust networks that are resistant to a more realistic attack condition we propose a hybrid greedy algorithm that takes both the r n and r l into account we validate the robustness of our generated networks against malicious attacks mixed with both nodes and links failure finally some economical constraints for swapping the links in real networks are considered and significant improvement in both aspects of robustness is still achieved

a numerical study on the stability of a class of helmholtz problems this paper concerns the stability of a class of helmholtz problems in rectangular domains a well known application is the electromagnetic scattering from a rectangular cavity embedded in an infinite ground plane error analysis of numerical methods for cavity problems relies heavily on the stability estimates however it is extremely difficult to derive an optimal stability bound with the explicit dependency on wave numbers in this paper a high order finite element approximation is proposed for calculating the stability bound numerical experiments show that the stability depends strongly on wave numbers in extreme case and it is almost independent on the wave numbers in an average sense our numerical results also help to understand the stability of the multi frequency inverse problems

inverse scattering inverse source theory the inverse scattering inverse source problem associated with the inhomogeneous helmholtz wave equation the special case sturm liouville acoustic wave equation and the time independent schrodinger equation is treated to this end the concepts of a reference wave velocity and an associated free reference space green s function spectrum are introduced a modified kirchhoff surface integral containing only the gradient of the real part of this free reference space green s function spectrum and the fields on a measurement surface is formulated yielding an integral equation for the unknown fields and sources in the interior of the closed piecewise smooth surface on which the remotely sensed fields are known a well posed analytic closed form solution of this integral equation for the unknown fields and their laplacians is obtained with the aid of a modified spatial fourier transform in which the reference velocity is continually varied in such a fashion that the ewald sphere shell sweeps to fill

a recursive algorithm for multifrequency acoustic inverse source problems an iterative recursive algorithm is studied for recovering unknown sources of acoustic field with multifrequency measurement data under additional regularity assumptions on source functions the first convergence result toward multifrequency inverse source problems is obtained by assuming the background medium is homogeneous and the measurement data is noise free error estimates are also provided when the observation data is contaminated by noise numerical examples verify the reliability and efficiency of our proposed algorithm

on exact solutions for damped anharmonic oscillators for arbitrary functions f1 f2 and f3 the anharmonic oscillator x f1 t x f2 t x f3 t x3 0 cannot be solved in closed form i e the general solution cannot be expressed as elliptic functions the authors apply the painleve test to obtain the constraint on the functions f1 f2 and f3 for which the equation passes the test the constraint on f1 f2 and f3 i e the differential equation which f1 f2 and f3 obey is discussed and solutions are given

attack vulnerability of complex networks we study the response of complex networks subject to attacks on vertices and edges several existing complex network models as well as real world networks of scientific collaborations and internet traffic are numerically investigated and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph for each case of attacks on vertices and edges four different attacking strategies are used removals by the descending order of the degree and the betweenness centrality calculated for either the initial network or the current network during the removal procedure it is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network suggesting that the network structure changes as important vertices or edges are removed furthermore the correlation between the betweenness centrality and the degree in complex networks is studied

linear stochastic differential equations with boundary conditions we study linear stochastic differential equations with affine boundary conditions the equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution the solution is not adapted to the driving brownian motion and we use the extended stochastic calculus of nualart and pardoux 16 to analyse them we give analytical necessary and sufficient conditions for existence and uniqueness of a solution we establish sufficient conditions for the existence of probability densities using both the malliavin calculus and the co aera formula and give sufficient conditions that the solution be either a markov process or a markov field

a survey of the near field far field inverse scattering inverse source integral equation the kirchhoff direct integration of the scalar wave equation is reviewed and some properties of the kirchhoff surface integral are discussed from the perspective of the inverse scattering inverse source problem a modified kirchhoff surface integral is introduced leading to a fredholm integral equation of the first kind for the unknown sources induced by the incident field inside a volume in terms of the scattered fields on the surface enclosing this volume the properties and physical meaning of this integral equation are discussed a generalization of this integral equation for the vector electromagnetic wave equations is presented

singularity clustering in the duffing oscillator an asymptotic and numerical study is made of the singularity structure in the complex t plane of the duffing oscillator the presence of logarithmic terms in the local psi series expansion of the form t4 ln t leads to a multisheeted singularity structure of great complexity this structure is built recursively from an elemental pattern which takes the form of four armed stars of singularities this construction is deduced analytically from the properties of the mapping z t4 ln t and is confirmed quite accurately numerically a systematic resummation of the psi series in terms of lame functions is developed this series exhibits the same analytic structure at all orders and provides a semi local analytical representation of the solution which is apparently valid even in the chaotic regime

inverse scattering by a continuation method with initial guesses from a direct imaging algorithm a novel continuation method is presented for solving the inverse medium scattering problem of the helmholtz equation which is to reconstruct the shape of the inhomogeneous medium from boundary measurements of the scattered field the boundary data is assumed to be available at multiple frequencies initial guesses are chosen from a direct imaging algorithm multiple signal classification music along with a level set representation at a certain wavenumber where the born approximation may not be valid each update via recursive linearization on the wavenumbers is obtained by solving one forward and one adjoint problem of the helmholtz equation

an inverse source problem for maxwell s equations in magnetoencephalography consider an inverse source problem for maxwell s equations which arises in determining locations of epileptic foci in the living human brain the human brain is modeled as a heterogeneous medium where the electric permittivity magnetic permeability and conductivity may all be functions a current dipole is used to model the epilepsy the inverse source problem in this context is to determine the current dipole from boundary measurements of the fields in this paper a new computational method is introduced for solving the inverse problem the method is based on a low frequency asymptotic analysis of maxwell s equations our method is constructive and has good convergence properties a crucial step of the method is to construct special test functions uniqueness and stability results for the inverse problem are also established the method and results are expected to find applications particularly in magnetoencephalography

the generalized damped cubic equation integrability and general solution the nonlinear ordinary differential equation of the form u f1u f2u f3u3 o representing a generalized damped cubic equation is fully analysed the author finds the general expression for f2 in terms of f1 and f3 that renders the equation integrable also he shows that if f2 satisfies the above mentioned condition the former nonlinear differential equation can always be reduced through adequate changes of variable to the second painleve transcendent

magnetoencephalography in ellipsoidal geometry an exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the 1d case of a sphere and the 2d case of a spheroid where the excitation field is due to an electric dipole within the corresponding homogeneous conductor in the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry the results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid using lame functions and ellipsoidal harmonics the basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior lame functions the laborious task of reducing the results to the spherical geometry is also included

variable background born inversion by wavefield backpropagation the inverse scattering problem for an acoustic medium is formulated by using the variable background born approximation a constant density acoustic medium is probed by a wide band point source and the scattered field is observed along a curved receiver array located outside the region where the medium velocity is different from the assumed background velocity function the solution proposed relies on the introduction of a backpropagated field this field is obtained by using a finite difference scheme backward in time to backpropagate into the medium the scattered field observed along the receiver array the backpropagated field is imaged at the source travel times giving an image of the same type as obtained by reverse time finite difference migration techniques the gradient of this image is then taken along rays linking the source to points in the medium and after scaling this gives the reconstructed potential to relate the reconstructed potential to the true scattering potential high frequency

a multi frequency inverse source problem this paper is concerned with an inverse source problem that determines the source from measurements of the radiated fields away at multiple frequencies rigorous stability estimates are established when the background medium is homogeneous it is shown that the ill posedness of the inverse problem decreases as the frequency increases under some regularity assumptions on the source function it is further proven that by increasing the frequency the logarithmic stability converts to a linear one for the inverse source problem

robustness of onionlike correlated networks against targeted attacks recently it was found by schneider et al proc natl acad sci usa 108 3838 2011 using simulations that scale free networks with onion structure are very robust against targeted high degree attacks the onion structure is a network where nodes with almost the same degree are connected motivated by this work we propose and analyze based on analytical considerations an onionlike candidate for a nearly optimal structure against simultaneous random and targeted high degree node attacks the nearly optimal structure can be viewed as a set of hierarchically interconnected random regular graphs the degrees and populations of whose nodes are specified by the degree distribution this network structure exhibits an extremely assortative degree degree correlation and has a close relationship to the onion structure after deriving a set of exact expressions that enable us to calculate the critical percolation threshold and the giant component of a correlated network for an arbitrary type of node removal we apply the theory to the cases of random scale free networks that are highly vulnerable against targeted high degree node removal our results show that this vulnerability can be significantly reduced by implementing this onionlike type of degree degree correlation without much undermining the almost complete robustness against random node removal we also investigate in detail the robustness enhancement due to assortative degree degree correlation by introducing a joint degree degree probability matrix that interpolates between an uncorrelated network structure and the onionlike structure proposed here by tuning a single control parameter the optimal values of the control parameter that maximize the robustness against simultaneous random and targeted attacks are also determined our analytical calculations are supported by numerical simulations

solving the multi frequency electromagnetic inverse source problem by the fourier method abstract this work is concerned with an inverse problem of identifying the current source distribution of the time harmonic maxwell s equations from multi frequency measurements motivated by the fourier method for the scalar helmholtz equation and the polarization vector decomposition we propose a novel method for determining the source function in the full vector maxwell s system rigorous mathematical justifications of the method are given and numerical examples are provided to demonstrate the feasibility and effectiveness of the method

stable determination of a scattered wave from its far field pattern the high frequency asymptotics we deal with the stability issue for the determination of outgoing time harmonic acoustic waves from their far field patterns we are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding helmholtz equation and in understanding the high wavenumber that is frequency asymptotics applications include stability results for the determination from far field data of solutions of direct scattering problems with sound soft obstacles and an instability analysis for the corresponding inverse obstacle problem the key tool consists of establishing precise estimates on the behavior of hankel functions with large argument or order

increasing stability in the inverse source problem with many frequencies abstract we study increasing stability in the interior inverse source problem for the helmholtz equation from boundary cauchy data for multiple wave numbers by using the fourier transform with respect to the wave number explicit bounds for the analytic continuation uniqueness of the continuation results and exact observability bounds for the wave equation a sharp uniqueness result and an increasing with larger wave numbers intervals stability estimate are obtained numerical examples in 3 spatial dimension support the theoretical prediction

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

invertible point transformations painleve analysis and anharmonic oscillators the techniques of an invertible point transformation and the painleve analysis can be used to construct integrable ordinary differential equations the authors compare both techniques for anharmonic oscillators

efficient calculation of the robustness measure r for complex networks abstract in a recent work schneider et al 2011 proposed a new measure r for network robustness where the value of r is calculated within the entire process of malicious node attacks in this paper we present an approach to improve the calculation efficiency of r in which a computationally efficient robustness measure r is introduced when the fraction of failed nodes reaches to a critical threshold q c simulation results on three different types of network models and three real networks show that these networks all exhibit a computationally efficient robustness measure r the relationships between r and the network size n and the network average degree k are also explored it is found that the value of r decreases with n while increases with k our results would be useful for improving the calculation efficiency of network robustness measure r for complex networks

on an inverse boundary problem arising in brain imaging abstract we consider the inverse problem of recovering both an unknown electric current and the surrounding electromagnetic parameters of a medium from boundary measurements this inverse problem arises in brain imaging we show that under generic conditions one can recover both the source and the electromagnetic parameters if these are piecewise constant and the source current is invariant in a fixed direction or a harmonic function respectively

inverse acoustic scattering problem in half space with anisotropic random impedance abstract we study an inverse acoustic scattering problem in half space with a probabilistic impedance boundary value condition the robin coefficient surface impedance is assumed to be a gaussian random function with a pseudodifferential operator describing the covariance we measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of our main result is to show that under certain conditions the principal symbol of the covariance operator of is uniquely determined most importantly no approximations are needed and we can solve the full non linear inverse problem we concentrate on anisotropic models for the principal symbol which leads to the analysis of a novel anisotropic spherical radon transform and its invertibility

identifying influential nodes in weighted networks based on evidence theory the design of an effective ranking method to identify influential nodes is an important problem in the study of complex networks in this paper a new centrality measure is proposed based on the dempster shafer evidence theory the proposed measure trades off between the degree and strength of every node in a weighted network the influences of both the degree and the strength of each node are represented by basic probability assignment bpa the proposed centrality measure is determined by the combination of these bpas numerical examples are used to illustrate the effectiveness of the proposed method

inverse random source scattering for elastic waves this paper is concerned with the direct and inverse random source scattering problems for elastic waves where the source is assumed to be driven by an additive white noise given the source the direct problem is to determine the displacement of the random wave field the inverse problem is to reconstruct the mean and variance of the random source from the boundary measurement of the wave field at multiple frequencies the direct problem is shown to have a unique mild solution by using a constructive proof based on the explicit mild solution fredholm integral equations of the first kind are deduced for the inverse problem the regularized block kaczmarz method is developed to solve the ill posed integral equations the convergence is proved for the method numerical experiments are shown to demonstrate the effectiveness of the proposed method

stability on the inverse random source scattering problem for the one dimensional helmholtz equation abstract consider the one dimensional stochastic helmholtz equation where the source is assumed to be driven by the white noise this paper concerns the stability analysis of the inverse random source problem which is to reconstruct the statistical properties of the source such as the mean and variance our results show that increasing stability can be obtained for the inverse problem by using suitable boundary data with multi frequencies

boundary value problems for stochastic differential equations in this paper we study stochastic differential equations with boundary conditions at the endpoints of a time interval instead of the customary initial condition we present existence and uniqueness results and study the markov property of the solution in the one dimensional case we prove that the solution is a markov field operatorname iff the drift is affine

inverse source problem in nonhomogeneous background media the scalar wave inverse source problem isp of determining an unknown radiating source from knowledge of the field it generates outside its region of localization is investigated for the case in which the source is embedded in a nonhomogeneous medium with known index of refraction profile n mathbf r it is shown that the solution to the isp having minimum energy the so called minimum energy source can be obtained via a simple method of constrained optimization this method is applied to the special case when the nonhomogeneous background is spherically symmetric n mathbf r n r and it yields the minimum energy source in terms of a series of spherical harmonics and radial wave functions that are solutions to a sturm liouville problem the special case of a source embedded in a spherical region of constant index is treated in detail and results from computer simulations are presented for this case

integrability and symmetries for the helmholtz oscillator with friction this paper deals with the helmholtz oscillator which is a simple nonlinear oscillator whose equation presents a quadratic nonlinearity and the possibility of escape when a periodic external force is introduced the width of the stochastic layer which is a region around the separatrix where orbits may exhibit transient chaos is calculated in the absence of friction and external force it is well known that analytical solutions exist since it is completely integrable when only friction is included there is no analytical solution for all parameter values however by means of the lie theory for differential equations we find a relation between parameters for which the oscillator is integrable this is related to the fact that the system possesses a symmetry group and the corresponding symmetries are computed finally the analytical explicit solutions are shown and related to the basins of attraction