chebyshev hankel matrices and the splitting approach for centrosymmetric toeplitz plus hankel matrices abstract centrosymmetric toeplitz plus hankel matrices are investigated on the basis of their splitting property which is their similarity to the direct sum of two special toeplitz plus hankel matrices these matrices can be considered as hankel matrices moment matrices in bases of chebyshev polynomials and are called chebyshev hankel matrices chebyshev hankel matrices have similar properties like hankel matrices this concerns inversion formulas and fast algorithms a superfast algorithm for solving chebyshev hankel and centrosymmetric toeplitz plus hankel systems is presented that is based on real trigonometric transforms the main tool of investigation is the interpretation of chebyshev hankel matrices as matrices of restricted multiplication operators with respect to chebyshev bases

centrosymmetric and centro skewsymmetric toeplitz plus hankel matrices and bezoutians abstract in the paper algebraic properties of centrosymmetric and centro skewsymmetric toeplitz plus hankel t h matrices are investigated it is shown that in the nonsingular case the inverse can be constructed with the help of the solutions of four related toeplitz systems with symmetric or skewsymmetric right hand sides in the singular case the related toeplitz matrices determine the kernel of the t h matrices finally centrosymmetric and centro skewsymmetric t h bezoutians are characterized

axisymmetric elastic waves in weakly coupled layered media of infinite radial extent abstract the propagation of axisymmetric waves in layered periodic elastic media of infinite radial extent is investigated hankel and laplace transforms are employed to convert radial and time dependence of displacement and stress within a layer to frequency and radial wavenumber continuity of stress and displacement is imposed at the interface between layers yielding transfer matrices the structure of propagation and attenuation zones pz s and az s of the system with infinite number of layers is studied when the ratios of shear and longitudinal mechanical impedances t l of the two layers are large the pz s of the layered system become narrow approaching certain limiting curves in the frequency wavenumber plane for large t l an asymptotic analysis is employed analytically to approximate the width of the pz s numerical computations of pz s are given which are in good agreement with the analytical predictions the spacing of the resonance points of a layered system with finite number of layers is found to depend mainly on the structure of the pz s of the corresponding system with an infinite number of layers

estimation of acoustic source strength by inverse methods part i conditioning of the inverse problem this paper deals with the discrete inverse problem in acoustics it is assumed that a number of acoustic sources are located at known spatial positions and that the acoustic pressure is measured at a number of spatial positions in the radiated field the transfer functions relating the strengths of the acoustic sources to the radiated pressures are also assumed known in principle the strengths of the acoustic sources can be deduced from the measured acoustic pressures by inversion of this matrix of transfer functions the accuracy of source strength reconstruction in the presence of noise which contaminates the measured pressures is crucially dependent on the conditioning of the matrix to be inverted this paper examines the conditioning of this inverse problem particularly with regard to the geometry and number of sources and measurement positions and the non dimensional frequency a preliminary investigation is also presented of methods such as tikhonov regularization and singular value discarding which can improve the accuracy of source strength reconstruction in poorly conditioned cases results are also presented which enable the solution of the inverse problem when the time histories of the acoustic sources are time stationary random processes and the spectra and cross spectra are measured at a number of positions in the radiated field the paper illustrates the possibilities and limitations of the use of inverse methods in the deduction of acoustic source strength from radiated field measurements

differential and integral methods for multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this procedure operates by recursively reconstructing the radon transform of the potential from the jump in the radon transform of the scattered field at the wave front this reconstructed potential is then used to propagate the wave front and scattered field differentially further into the support of the potential the connections between this differential procedure and integral equation procedures are then illustrated by the derivations of two well known exact integral equation procedures using the radon transform and a generalized radon transform these procedures as well as the layer stripping procedure are then reduced to the familiar born approximation result for this problem by neglecting multiple scattering events this illustrates the central role of the radon transform in both exact and approximate inversion procedures

some matrix factorization identities for discrete inverse scattering this note analyzes the relationships between several inverse scattering methods and points out that all classical approaches implicitly construct a lower upper triangular factorization of a given positive definite toeplitz matrix it is shown that the various inversion methods implicitly obtain the factorization by solving different nested sets of linear equations and expressing the factors in terms of the solutions obtained the fact that the triangular factorization of a matrix with nonzero leading minors is unique immediately yields various identities e g involving the so called central mass sequence that were derived in the literature with considerably more manipulation in fact our basic factorization results are somewhat more general since they do not require the toeplitz assumption

scattering by a potential using hyperbolic methods etude du probleme de diffusion inverse developpement d un algorithme correspondant pour deux variables

reconstructing the transition rate function of a broadwell random walk from exit times in this paper we utilize the layer stripping method a method used in seismology by geophysicists to study a stochastic inverse problem arising from a one dimensional broadwell process the broadwell process can be described as a random walk of a particle that transitions with a spatially dependent flip rate between two states associated with the same speed but with different signs of velocities our goal is to reconstruct the spatially dependent flip rate function of a one dimensional broadwell process from exit time distributions out of a finite interval in principle we are able to reconstruct flip rate functions with error within o 10 2 from o 10 5 exit times when the particle speed is unity for smaller particle speeds the noise increases for a fixed number of exit times this method is less time consuming compared with traditional projection methods which involve optimization

maximum likelihood estimation with side information of a 1 d discrete layered medium from its noisy impulse reflection response we consider the problem of computing the maximum likelihood estimates of the reflection coefficients of a discrete 1 d layered medium from noisy observations of its impulse reflection response we have side information in that a known subset of the reflection coefficients are known to be zero this knowledge could come from either a priori knowledge of a homogeneous subregion inside the scattering medium or from a thresholding operation in which noisy reconstructed reflection coefficients with absolute values below a threshold are known to be zero our procedure converges in one or two iterations each of which requires only setting up and solving a small system of linear equations and running the levinson algorithm numerical examples are provided that demonstrate not only the operation of the algorithm but also that the side information improves the reconstruction of unconstrained reflection coefficients as well as constrained ones due to the nonlinearity of the inverse scattering problem

adjoint optimization of one dimensional hyperbolic equations with constrained periodic boundary conditions abstract this paper presents a continuous adjoint based optimization theory for a general closed loop transport hyperbolic model controlled via a periodic boundary control to minimize a cost functional the periodic boundary control is subject to a nonlinear differential equation constraint thus resulting in a coupling between the hyperbolic equation and the ordinary differential equation variational principles are used to derive the pontryagin s minimum principle for optimality that results in a dual adjoint system a numerical optimization method is implemented using the adjoint based second order gradient method to solve for the optimal trajectory of the control numerical methods for solving the hyperbolic equation using an explicit scheme wave splitting method and for solving the adjoint equation using an implicit scheme and a quasi steady state method are described

a layer stripping solution of the inverse problem for a one dimensional elastic medium a fast algorithm for recovering profiles of density and lame parameters as functions of depth for the inverse seismic problem in an elastic medium is obtained the medium is probed with planar impulsive p and sv waves at oblique incidence and the medium velocity components are measured at the surface the interconversion of p and sv waves defines reflection coefficients from which the medium parameter profiles are obtained recursively the algorithm works on a layer stripping principle and it is specified in both differential and recursive forms a physical interpretation of this procedure is given in terms of a lattice filter where the first reflections of the downgoing waves in each layer yield the various reflection coefficients for that layer a computer run of the algorithm on the synthetic impulsive plane wave responses of a twenty layer medium shows that the algorithm works satisfactorily

acoustic pulse spreading in a random fractal fractal medium models are used to model for instance the heterogeneous earth and the turbulent atmosphere a wave pulse propagating through such a medium will be affected by multiscale medium fluctuations for a class of one dimensional fractal random media defined in terms of fractional brownian motion we show how the wave interacts with the medium fluctuations the modification in the pulse shape depends on the roughness of the medium and can be described in a deterministic way when the pulse is observed at its random arrival time for very rough media the coherent wave is confined to a surface layer

a kalman filter solution of the inverse scattering problem with a rational reflection coefficient abstract this paper presents a new inverse scattering method for reconstructing the reflectivity function of symmetric two component wave equations or the potential of a schrodinger equation when the reflection coefficient is rational this method relies on the so called chandrasekhar equations which implement the kalman filter associated to a stationary state space model these equations are derived by using first a general layer stripping principle to obtain some differential equations for reconstructing a general scattering medium and by specializing these recursions to the case when the probing waves have a state space model

neumann boundary control of hyperbolic equations with pointwise state constraints we consider optimal control problems for hyperbolic equations with controls in neumann boundary conditions with pointwise constraints on the control and state functions focusing on the multidimensional wave equation with a nonlinear term we derive new necessary optimality conditions in the form of a pointwise pontryagin maximum principle for the state constrained problem under consideration our approach is based on modern methods of variational analysis that allow us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional

goupillaud inverse problem with arbitrary input a new impulse inverse formula and it s generalization including arbitrary input are proposed the new generalized inverse formula has the advantage of not being necessary to specify the first digital value of the input numerical instability of the inverse formula for arbitrary input is investigated carefully an analysis of the perturbed stability and lyapunov exponent are carried out as the result of this analysis it is shown that the estimated error is sensitive to the experimental error and previously estimated error as a sequential algorithm simulation using experimental data is also shown in the paper

differential methods in inverse scattering this paper discusses a new set of differential methods for solving the inverse scattering problem associated to the propagation of waves in an inhomogeneous medium by writing the medium equations in the form of a two component system describing the interaction of rightward and leftward propagating waves the causality of the propagation phenomena is exploited in order to identify the medium layer by layer the recursive procedure that we obtain constitutes a continuous version of an algorithm first derived by schur in order to test for the boundedness of functions analytic inside the unit circle it recovers the local reflectivity function of the medium using similar ideas some other differential methods can also be derived to reconstruct alternative parametrizations of the layered medium in terms of the local impedance or of the potential function the differential inverse scattering methods turn out to be very efficient since in some sense they let the medium perform the inversion by itself and thus

application of the schur algorithm to the inverse problem for a layered acoustic medium the schur algorithm is a signal processing algorithm which works on a layer stripping principle its time domain version the fast cholesky recursion is a fast and efficient algorithm well suited for high speed data processing in this paper these algorithms are applied to the inverse problem for a continuous layered acoustic medium three different excitations of the medium are considered impulsive plane waves at normal incidence impulsive plane waves at oblique incidence and spherical waves emanating from an impulsive point source the fast algorithms obtained for each of these problems seem to be computationally superior to past work done on these problems that employed gelfand levitan theory to reconstruct the potential of a schrodinger equation

inversion of reflection traveltimes and amplitudes a bayesian nonlinear inversion scheme is used to invert traveltime and amplitude data from supercritical reflections reflection amplitudes are determined using a spherical wave model where the amplitude of a reflected spherical wave is given by a sommerfeld integral this integral is evaluated numerically by integrating over a contour in the complex plane the spherical wave model differs from plane wave models in the regions around the critical points for p and s refraction in these regions the spherical wave amplitudes are sensitive to frequency with a dependance on the ratio h between the reflector depth h and the seismic wave length the accuracies of the inversion scheme and the spherical reflection model are tested using control data from the university of houston physical model tank the solution obtained by inversion agrees with the known solution to within 3 percent for all parameters and the amplitudes predicted by the spherical wave model agree remarkably well with the observed data

dispersion of stress waves in one dimensional semi infinite weakly coupled layared systems an analytical study of the dispersion of transient stress waves in the first layer of a weakly coupled semi infinite bi layered system is carried out the analysis is based on approximate fourier transform inversions and makes use of the fact that the weakly coupled system possesses small propagation zones pzs in the frequency domain low and high frequency asymptotic approximations to the transient waves are computed taking into account frequency components of the transfer function in the first and second pzs respectively the derived analytic expressions are superpositions of non oscillating terms and convolution integrals with decaying oscillatory kernels it is shown that depending on the frequency and the amplitude of the convolution kernels the dispersed waves overshoot or undershoot the applied impulsive excitation this result is of significant practical importance in the design of layered systems as stress attenuators

sensitivity analysis of an auto correlation function based damage index and its application in structural damage detection abstract structural damage detection using time domain vibration responses has advantages such as simplicity in calculation and no requirement of a finite element model which attracts more and more researchers in recent years in present paper a new approach to detect the damage based on the auto correlation function is proposed the maximum values of the auto correlation function of the vibration response signals from different measurement points are formulated as a vector called auto correlation function at maximum point value vector amv for short the relative change of the normalized amv before and after damage is used as the damage index to locate the damage sensitivity analysis of the normalized amv with respect to the local stiffness shows that the normalized amv has a sharp change around the local stiffness change location which means the normalized amv is a good indicator to detect the damage even when the damage is very small stiffness reduction detection of a 12 story frame structure is provided to illustrate the validity effectiveness and the anti noise ability of the proposed method comparison of the normalized amv and the other correlation function based damage detection method shows the normalized amv has a better detectability

fast matrix factorizations via discrete transmission lines abstract fast algorithms associated with the names of schur and levinson are known for the triangular factorization of symmetric positive definite toeplitz matrices and their inverses in this paper we show that these algorithms can be derived from simple arguments involving causality symmetry and energy conservation in discrete lossless transmission lines the results not only provide a nice interpretation of the classical schur and levinson algorithms and a certain toeplitz inversion formula of gohberg and semencul but they also show immediately that the same fast algorithms apply not only to toeplitz matrices but to all matrices with so called displacement inertia 1 1 the results have been helpful in suggesting new digital filter structures and in the study of nonstationary second order processes

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

a new iterative method for solving the coefficient inverse problem of the wave equation on donne la definition de l equation operateur non lineaire du probleme inverse coefficient de l equation d onde dans le domaine des variables d espace et de temps sans les transformees de laplace et de fourier

self inversive hilbert space operator polynomials with spectrum on the unit circle abstract hilbert space operator polynomials with self inversive structure are studied if the inner numerical radius of an associated polynomial is greater than or equal to one then the spectrum lies on the unit circle and consists of normal approximate characteristic values

reconstruction of a potential on the line that is a priori known on the half line in this paper the authors study the inverse schrodinger scattering on the real line a method is given that allows unique reconstruction of a potential that is a priori known on the half line from the knowledge of the reflection coefficient and the bound state energies in particular no information on the forming constants is required the method is based on an appropriate trace formula and on the solution of the nonlinear ordinary differential equation that is obtained when the potential is replaced by its trace formula in the schrodinger equation the deift trubowitz approach to inverse scattering is followed the main new point is the way in which bound states are treated in addition to its mathematical interest the case when the potential is a priori known on the half line is particularly interesting in many applications one can consider for example a potential that has compact support or that it is zero on a half line

the round trip theory for reconstruction of green s functions at passive locations an expression for the green s function at an arbitrary set of passive locations no applied force is derived and validated by experiment three sets of points are involved the passive reconstruction points c which lie on a virtual boundary and two sets of auxiliary points denoted a and b located either side the reconstruction is achieved using green s functions forming a round trip from and to the reconstruction points via a and b a two stage measurement procedure is described involving excitation at b and a but with no excitation required at the reconstruction points a known round trip relationship is first introduced which is theoretically exact for points on a multi point interface between two linear time invariant subsystems experimental results for frequency response functions of a beam plate structure show that this relationship gives good results in practice it is then shown that the theory provides an nth order approximation for the green s function at arbitrary points where n is the number of points at b the expression is validated by reconstructing point and transfer frequency response functions at two passive points on an aluminum plate

frequency content of randomly scattered signals the statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it the asymptotic analy

source wavelet and its angular spectrum from plane wave seismograms in a typical seismic experiment land or marine the generated source wavelet is either unknown or known only approximately in order to solve many geophysical problems an accurate wavelet estimation is crucial this paper demonstrates a fast numerical algorithm which allows not only estimating the source wavelet but also its angular spectrum radiation pattern from plane wave decomposed seismograms surprisingly literature on this subject is virtually missing although a nonuniform angular spectrum of the generated wavelet can substantially affect the recorded data the technique presented here is based on the downward continuation dc and reverse time extrapolation rte of the recorded data both pressure and its vertical derivative the proposed method is iterative the rate of convergence does not depend on phase characteristics of the generated wavelet we demonstrate that both the wavelet signature and its angular spectrum can be estimated accurately from plane wave seismograms we also illu

the poles of symmetric linear prediction models lie on the unit circle symmetric linear prediction models have their poles on the unit circle a simple proof of this result is presented

high frequency perturbational analysis of the surface point source response of a layered fluid abstract we linearize the relation between the parameters density and velocity profiles of a layered fluid subject to a specified surface point traction and the precitically reflected part of its surface motion we decompose the resulting linear map into a high frequency leading term and a lower order smoother remainder and show that the spectral analysis of the leading term may be described in terms of ray geometry the leading term is generally well conditioned but may become poorly conditioned in low velocity zones these results imply stability estimates for the linearized acoustic reflection inversion problem as well as for the nonlinear problem and yield insight into the behavior of numerical algorithms for the determination of the density and velocity of a layered fluid from its surface response

sufficient optimality conditions for dirichlet boundary control of wave equations we study optimal control problems for wave equations focusing on the multidimensional wave equation with control functions in the dirichlet boundary conditions under pointwise control and we admit state by assuming weak hypotheses constraints

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

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

the use of the local truncation error to improve arbitrary order finite elements for the linear wave and heat equations abstract the local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high order finite elements in the 1 d 2 d and 3 d cases on uniform and non uniform meshes several applications of the local truncation error are considered in the paper it is proven that for the 1 d wave equation with a piece wise constant wave velocity the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity the lumped mass matrix and the central difference method with the time increments equal to the stability limit are used it is shown in the 1 d and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy the minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high order elements with improved accuracy i e accuracy is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations new high order boundary conditions are developed in order to keep a high order accuracy of the developed technique the new elements can be equally applied to linear wave propagation and heat transfer problems it is also shown that non uniform meshes may lead to inaccurate results due to the increase in the local truncation error the difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach the presented numerical examples are in good agreement with the theoretical results the approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches

immittance domain levinson algorithms several computationally efficient versions of the levinson algorithm for solving linear equations with toeplitz and quasi toeplitz matrices are presented motivated by a new stability test the new versions require half the number of multiplications and the same number of additions as the conventional form of the levinson algorithm the saving is achieved by using three term rather than two term recursions and propagating them in an impedance admittance or immittance domain rather than the conventional scattering domain one of the recursions coincides with recent results of p delsarte and y genin ieee trans acoust speech signal proc vol assp 34 p 470 8 june 1986 on split levinson algorithms for symmetric toeplitz matrices where the efficiency is gained by using the symmetric and skew symmetric versions of the usual polynomials this special structure is lost in the quasi toeplitz case but one still can obtain similar computational reductions by suitably using three term recursions in the immittance domain

acoustic impulse responses for nonuniform media abstract the acoustic impulse responses of a family of nonuniform lossless one dimensional media are determined the nonuniform portions of the media are confined to layers of either finite or infinite thicknesses the acoustic impedances of the nonuniform layers as functions of travel time are assumed known the impulse responses are determined by evaluating the fourier transforms of their reflection coefficients using infinite series expansions and contour integrations

fast algorithms for estimation and signal processing an inverse scattering framework the inverse scattering problem for a lossless transmission line has been used as a framework for deriving the levinson and schur fast algorithms for toeplitz matrices here the same framework is used to derive and attach novel simple scattering interpretations to the lattice algorithm and the split algorithms this demonstrates that the scattering and interpretation is more fundamental to fast signal processing algorithms than previously noted

reflectance of acoustic horns and solution of the inverse problem a method is described for solving the inverse problem of determining the profile of an acoustic horn when time domain reflectance tdr is known only at the entrance the method involves recasting webster s horn equation in terms of forward and backward propagating wave variables an essential feature of this method is a requirement that the backward propagating wave be continuous at the wave front at all locations beyond the entrance derivation of the inverse solution raises questions about the meaning of causality in the context of wave propagation in non uniform tubes exact reflectance expressions are presented for infinite exponential conical and parabolic horns based on exact solutions of the horn equation diameter functions obtained with the inverse solution are a good match to all three horn profiles

exact analytical solutions for elastodynamic impact abstract we consider the one dimensional impact problem in which a semi infinite flyer collides with and adheres to the front face of a stationary target plate of finite thickness with the back face of the target bonded to another semi infinite medium all three bodies are assumed to be linear elastic and homogeneous our interest is in explicit expressions for the stress and velocity in the target at all times after impact the analysis of this problem is simplified by reducing it to an initial boundary value problem for the target only which is solved by combining the d alembert solution of the wave equation with the laplace transform method an appropriate impact boundary condition is required on the front face of the target in the literature this is usually taken as a prescribed step in stress or velocity but the correct boundary condition involves a linear combination of the unknown stress and velocity at the impact face our solutions are expressed in an apparently new compact form involving the floor or greatest integer function the results are amenable to asymptotic analysis in particular solutions for stress free or rigid back faces follow easily as limiting cases of the backing impedance and the long time asymptotes of stress and velocity in the target are seen to be independent of the target s elastic properties all of our results are corroborated by derivation of exact discrete solutions from recursive equations for the impact problems

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

an iterative method for the inversion of the two dimensional wave equation with a potential a numerical method for a nonlinear inversion problem for the 2d wave equation with a potential is discussed in order to avoid the ill posedness we substitute a coupled system of one way wave equations for the original wave equation an iterative algorithm is constructed to improve the accuracy of the inversion numerical experiments are performed on several examples to examine the effectiveness of this method

layer stripping solutions of multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this method operates by recursively reconstructing the potential from the jump in the scattered field at the wave front and then using the reconstructed potential to propagate the wave front and the scattered field further into the inhomogeneous region it is thus a generalization of algorithms that have been developed for one dimensional inverse scattering problems although the procedure has not yet been numerically tested the corresponding one dimensional algorithms have performed well on synthetic data the procedure is applied to a two dimensional inverse seismic problem connections between simplifications of this method and born approximation inverse scattering methods are also noted

some nniqueness results related to soil and building structural identification the need for accurate predictions of the dynamic responses of soil and structural systems has led to the widespread use of forced vibration testing for obtaining knowledge of the dynamic properties of the systems under consideration this paper is concerned with the uniqueness of the results in the identification of such properties more specifically the damping and stiffness distributions which are of importance in the linear range of response have been investigated an n storied structure or an n layered soil system is modeled as an n degree of freedom lumped mass system consisting of masses springs and dampers assuming that the mass distribution is known the conditions under which unique identification of the stiffness and damping distributions can be obtained are established several useful results on the ability to identify uniquely some but not all of the properties of the system have also been obtained the paper extends some of the known results of one dimensional reflection seismology to

a new method to extract modal parameters using output only responses this work proposes a new output only modal analysis method to extract mode shapes and natural frequencies of a structure the proposed method is based on an approach with a single degree of freedom in the time domain for a set of given mode isolated signals the un damped mode shapes are extracted utilizing the singular value decomposition of the output energy correlation matrix with respect to sensor locations the natural frequencies are extracted from a noise free signal that is projected on the estimated modal basis the proposed method is particularly efficient when a high resolution of mode shape is essential the accuracy of the method is numerically verified using a set of time histories that are simulated using a finite element method the feasibility and practicality of the method are verified using experimental data collected at the newly constructed king storm water bridge in california united states

reconstruction of a spherically symmetric speed of sound consider the inverse acoustic scattering problem for a spherically symmetric inhomogeneity of compact support that arises among other places in nondestructive testing define the corresponding homogeneous and inhomogeneous interior transmission problems see e g d colton and p monk quart j mech math 41 1988 pp 97 125 here the authors study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem it is shown in mclaughlin and polyakov j differential equations to appear that these eigenvalues are the zeros of an average of the scattering amplitude and a uniqueness theorem for the inverse acoustic scattering problem is presented where these eigenvalues are the given data in the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented the method given here completely determines the sound

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

parametric estimation of the cross power spectral density a new cross spectral analysis procedure is proposed for the parametric estimation of the relationship between two time sequences in the frequency domain in this method the two observable outputs are modeled as a pair of autoregressive moving average and moving average armama models under the assumption that the two outputs are driven by a common input and independent ones simultaneously cross and auto power spectral densities psds of a pair of armama models can be derived as forms of rational polynomial functions the coefficients of these functions can be estimated from the cross correlation function or the auto correlation functions of the two observed sequences by using the method presented in this paper the main advantage of the present procedure is that the physical parameters of an unknown system can be easily estimated from the coefficients of the cross and auto psd functions to illustrate the effectiveness of the proposed procedure numerical and practical examples of a mechanical vibration problem are analyzed the results show that the proposed procedure gives accurate cross and auto psd estimates moreover the physical properties of the unknown system can be well estimated from the obtained cross and auto psds

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

inverse scattering and minimal partial realizations we present an inverse scattering interpretation of the classical minimal partial realization problem posed in a slightly generalized context our approach starts by considering a canonical cascade form structure for the realization of arbitrary transfer functions where the cascade structure can be interpreted as the description of a layered wave scattering medium in this context the partial realization problem calls for a recursive process of layer identification from a given input response pair the scattering data the realization algorithm uses a causality principle to progressively determine the parameters of cascaded linear 2 ports that model the successive wave interaction layers this method for approaching the realization problem turns out to fit nicely into a framework that was also used to obtain fast structured linear estimation algorithms and cascade realizations for digital filters

inverse problem in nonstationary multidimensional medium the problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered the boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method the integral differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method the order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary it was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium the cause and character of the multidimensional inverse problem overdefiniteness are considered the application of the proposed procedure for a statistical problem is discussed

matrix representations of split bezoutians abstract inverses of symmetric or skewsymmetric toeplitz matrices as well as of centrosymmetric or centro skewsymmetric toeplitz plus hankel matrices can be represented as sums of two split bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors thus it is desirable to find matrix representations for split bezoutians b this is the main aim of the present paper recursion formulas for the entries of b are presented bases of very simple split bezoutians or of sparse matrices are constructed and b is represented as a corresponding linear combination moreover matrix representations of gohberg semencul type are established

one channel time reversal in chaotic cavities theoretical limits one channel time reversal experiments in closed chaotic cavities produce excellent but not perfect time reversed focusing this paper investigates such experiments by a simple eigenmode analysis of the system it shows that the process is even for long reversed signals subject to an information loss during recording and re emission which prevents perfect time reversal this fact can be expressed by a simple equation called the cavity equation which states that the signal of a one channel time reversal is equal to the signal of a perfect time reversal after convolution with the backscattering impulse response of the reversal point i e from this point to this point the latter convolution describes the introduction of the loss of information furthermore arguments are presented suggesting that one third of the total energy of the time reversed wave field is actually contained in the refocusing wavefront the predictions are verified by numerical finite difference time domain simulations

study on an auto correlation function based damage index sensitivity analysis and structural damage detection abstract the damage index based on the auto correlation function to detect the damage of the structure under white noise excitation is studied in detail in this paper the maximum values of the auto correlation function of the vibration response signals displacement velocity and acceleration from different measurement points of the structure are collected and formulated as a vector called auto correlation function at maximum point value vector amv which is expressed as a weighted combination of the hadamard product of two mode shapes amv is normalized by its root mean square value so that the influence of the excitation can be eliminated sensitivity analysis for the different parts of the normalized amv shows that the sensitivity of the normalized amv to the local stiffness is dependent most on the sensitivity of the hadamard product of the two lower order mode shapes to the local stiffness which has a sudden change of the value around the local stiffness change position the sensitivity of the normalized amv has the similar shape and same trend that shows it is a very good damage indicator even for the very small damage the relative change of the normalized amv before and after damage occurs in the structure is adopted as the damage index to show the damage location several examples of the stiffness reduction detection of a 12 story shear frame structure are utilized to validate the results in sensitivity analysis illustrate the effectiveness and anti noise ability of the amv based damage detection method and compare the effect of the response type on the detectability of the normalized amv

study of some practical issues in inversion with the born iterative method using time domain data the performance of the born iterative method of nonlinear two dimensional profile inversion is examined for the reconstruction of large objects and in the presence of measurement noise time domain data are used it is shown that objects at least as large as about nine wavelengths can be inverted without any convergence problems the algorithm is shown to perform well in the presence of 10 noise or 20 db signal to noise ratio the simultaneous inversion of permittivity and conductivity profiles is formulated and solved using the born iterative method objects with various loss tangents are reconstructed and the limits of applicability of the algorithm are investigated

numerical methods for reflection inverse problems convergence and nonimpulsive sources we study numerical methods for a one dimensional reflection inverse problem a normally incident impulsive plane wave is sent into a stratified elastic half space from an adjoining homogeneous elastic half space and the reflected wave the reflectance is measured the characteristic impedance of the medium is to be recovered as a function of travel time this inverse problem occurs in several applications including reflection seismology and determining vocal tract shape for speech synthesis we show that if the step reflectance or ramp reflectance the first or second time integral of the reflectance is sampled appropriately and the problem is discretized accordingly then the solution of the discrete inverse problem converges uniformly to the impedance profile and the convergence is second order in the mesh width the proofs involve analyzing a discretization of an integral equation obtained by burridge which is a variant of the marchenko equation the discrete inverse problem can be solved by integ

the phase space of interactions in neural networks with definite symmetry we calculate the typical fraction of the phase space of interactions which solve the problem of storing a given set of p patterns represented as n spin configurations as a function of the storage ratio a p n of the stability parameter k and of the symmetry 7 of the interaction matrices the calculation is performed for strongly diluted networks where the connectivity of each spin c is of the order of in n for each value of k and 17 there is a maximal value of a above which the volume of solutions vanishes for each value of k and a there is a typical value of 7 at which this volume is maximal the analytical studies are supplemented by numerical simulations on fully connected and diluted networks using specific learning algorithms

estimation of coefficients and boundary parameters in hyperbolic systems we consider semi discrete galerkin approximation schemes in connection with inverse problems for the estimation of spatially varying coefficients and boundary condition parameters in second order hyperbolic systems typical of those arising in 1 d surface seismic problems spline based algorithms are proposed for which theoretical convergence results along with a representative sample of numerical findings are glven

inversion of p sv seismic data in a layered elastic material density shear velocity and compressional velocity can be found at any depth from broadband surface measurements at two distinct nonzero precritical values of plane wave incidence angle layer stripping inversion uses three component surface velocity measurements generated by a polarized surface source to determine subsurface properties incrementally the surface velocity measurements initialize a first order nonlinear matrix riccati equation derived from the elastic wave equation which takes advantage of an attractive fixed point condition in the complex frequency plane to extract subsurface mechanical impedances subsurface density and velocities are recovered from the inverted impedances at two or more plane wave incidence angles general properties of the matrix riccati equation in the complex frequency plane aid in incorporating bandwidth constraints inversion of synthetic plane wave data from a piece wise continuous model illustrates inversion effects when only

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

iterative trace deconvolution and noncausal transform for processing band limited data in 1983 carrion and patton showed that if recorded seismic data do not have low frequencies in their spectrum inversion for acoustic impedance is unstable this concept is commonly accepted here we infer acoustic impedance from band limited data without low frequencies using iterative trace deconvolution with the so called noncausal projection which moves initial data samples corresponding to small eigenvalues to negative times we show that this procedure can solve ill posed problems by migrating large residuals uncertainties in the trend of the recovered impedance to the bottom of the model an important property of the proposed method is that it converges to the true solution independently of the chosen initial model another advantage of the proposed algorithm is that unlike conventional dynamic deconvolution characteristic integration schemes it increases neither errors nor the condition number with depth the method is illustrated on synthetic and real data

statistical theory of the vector random decrement technique the vector random decrement technique has previously been introduced as an efficient method to transform ambient responses of linear structures into vector random decrement functions which are equivalent to free decays of the current structure the modal parameters can be extracted from the free decays due to the speed and or accuracy of the vector random decrement technique it was introduced as an attractive alternative to the random decrement technique in this paper the theory of the vector random decrement technique is extended by applying a statistical description of the stochastic processes describing the ambient measurements the vector random decrement functions are linked to the correlation functions of the stochastic processes provided they are stationary and gaussian distributed furthermore a new approach for quality assessment of the vector random decrement functions is given on the basis of the derived results the work presented in this paper makes the theory of the vector random decrement technique equivalent to the theory of the random decrement technique the theoretical derivations are illustrated by the analysis of the response of a 3dof system loaded by white noise

on the stability of periodic orbits of high dimensional autonomous hamiltonian systems we study the stability of periodic orbits of autonomous hamiltonian systems with n 1 degrees of freedom or equivalently of 2n dimensional symplectic maps with n 1 we classify the different stability types introducing a new terminology which is perfectly suited for systems with many degrees of freedom since it clearly reflects the configuration of the eigenvalues of the corresponding monodromy matrix on the complex plane the different stability types correspond to different regions of the n dimensional parameter space s defined by the coefficients of the characteristic polynomial of the monodromy matrix all the possible direct transitions between different stability types are classified and the corresponding transition hypersurface in s is determined the dimension of the transition hypersurface is an indicator of how probable to happen is the corresponding transition as an application of the general results we consider the well known cases of hamiltonian systems with two and three degrees of freedom we also describe in detail the different stability regions in the three dimensional parameter space s of a hamiltonian system with four degrees of freedom or equivalently of a six dimensional symplectic map 2001 elsevier science b v all rights reserved

perceptron like learning in time summating neural networks this paper investigates the ability of a single layer time summating neural network to associate and store temporal sequences in particular the associative learning of temporal sequences is reformulated as an equivalent classification task involving static patterns this leads to a generalization of the perceptron learning rule and convergence theorem to the case of temporal sequences using geometrical arguments based on linear separability it is shown how a time summating network can handle temporal features such as ordering and coarticulation effects such an ability is a consequence of the fact that the time summating network develops an activity trace consisting of a decaying sum of all previous inputs to the network on the other hand such an activity trace may also lead to an accumulation of errors in the presence of noisy inputs this motivates a modification of the perceptron learning rule involving the introduction of a stability parameter that guarantees a certain level of robustness to noise the performance of the network in the presence of random input sequences is then analysed using statistical mechanical techniques finally it is shown how with small modifications the time summating network can be trained to store and recall complex sequences

convergence of numerical inversion methods for discontinuous impedance profiles we study numerical methods for the one dimensional inverse problem of reflection seismology an impulsive plane wave in pressure is applied to the surface of a piecewise smooth stratified elastic half space and the resulting particle velocity is measured at the surface the characteristic impedance of the medium is to be recovered as a function of travel time we consider two methods for the numerical solution of this inverse problem a discretization of the gopinath sondhi integral equation and a downward continuation method both methods are exact for so called goupillaud layered media both use second order difference schemes to reconstruct a solution of the system of partial differential equations governing motion in the medium a formally second order accurate equation based on the theory of geometrical acoustics is used to recover the impedance by analyzing the discretization of the gopinath sondhi integral equation we prove that if the impulse response is measured by sampling the step response or

system identification from ambient vibration measurements on a bridge abstract the cross correlation function between two response measurements made on an ambiently excited structure is shown to have the same form as the system s impulse response function therefore standard time domain curve fitting procedures which are typically applied to impulse response functions can now be applied to the cross correlation functions to estimate the resonant frequencies and modal damping of the structure this derivation is based on the assumption that the ambient vibration source is a white noise random process curve fitting cross correlation functions to obtain modal properties offers advantages over standard procedures that identify resonant frequencies from peaks in the power spectrum and damping from the width of the power spectrum the primary advantage is the ability to identify closely spaced modes and their associated damping the resonant frequencies of a highway bridge that were identified by curve fitting the cross correlation functions using traffic excitation as the ambient vibration source are compared to modal properties identified by standard forced vibration testing methods results of this comparison showed a maximum discrepancy of 3 63 percent similar comparisons for the average modal damping values identified by the two methods showed a 9 82 percent difference this experimental verification implies that the proposed method of analyzing ambient vibration data can be used to accurately assess the dynamic properties of structures in a non intrusive manner

temporal sequence storage capacity of time summating neural networks the maximum capacity for storage of temporal sequences in a time summating neural network is analysed within a statistical mechanical framework each neuron of such a network maintains an activity trace consisting of a decaying sum of all previous inputs to that neuron the maximum storage capacity is expressed as a function of the temporal correlations of the local fields of the neurons and evaluated using a perturbation expansion in powers of the decay rate

on some properties of positive definite toeplitz matrices and their possible applications abstract various properties of a real symmetric toeplitz matrix m with elements jk a j k 1 j k m are reviewed here matrices of this kind often arise in applications in statistics econometrics psychometrics structural engineering multichannel filtering reflection seismology etc and it is desirable to have techniques which exploit their special structure possible applications of the results related to their inverse determinant and eigenvalue problem are suggested

z transform methods for the optimal design of one dimensional layered elastic media in this work we develop a finite trigonometric series representation for the stress in a multilayered goupillaud type elastic strip subjected to transient heaviside loading on one end while the other end is held fixed this representation is achieved by means of the z transform method and involves the so called base angles generally different layered designs could share the same set of base angles and the more layers the design has the more base angles are expected necessary conditions for the base angles and design parameters for any given design are described as a result of the stress representation we are able to identify optimal layered designs which provide the smallest stress amplitude for two and three layered designs for which the coefficients of the stress representation are easy to find the optimization results are achieved using a custom made discrete optimization technique applied in a p velo and g a gazonas int j solids structures 40 2003 pp 6417 6428 for other laye

wave focusing on the line focusing of waves in one dimension is analyzed for the plasma wave equation and the wave equation with variable speed the existence of focusing causal solutions to these equations is established and such wave solutions are constructed explicitly by deriving an orthogonality relation for the time independent schrodinger equation the connection between wave focusing and inverse scattering is studied the potential at any point is recovered from the incident wave that leads to focusing to that point it is shown that focusing waves satisfy certain temporal antisymmetry and support properties discontinuities in the spatial and temporal derivatives of the focusing waves are examined and related to the discontinuities in the potential of the schrodinger equation the theory is illustrated with some explicit examples

half solitons as solutions to the zakharov shabat eigenvalue problem for rational reflection coefficient ii potentials on infinite support it is shown how to invert the zakharov shabat eigenvalue problem for rational scattering coefficients in order to produce potentials defined over the whole real line the method reduces the problem to finding two semi infinite potentials which can be efficiently calculated using the soliton lattice algorithm described in a previous paper the inversion is usually unique for given rational scattering coefficients since they usually specify the scattering data of the system uniquely in the case of nonunique inversion it is possible to obtain a family of potentials from the inversion each member of the family has the same scattering data except for differing residues examples of the inversion include an adiabatic inversion pulse for use in nuclear magnetic resonance and a demonstration of how the cubic nonlinear schrodinger equation may be integrated

asymptotic impact behavior of goupillaud type layered elastic media abstract in this paper we use the method of riemann invariants to develop a system of recurrence relations for the one dimensional elastic impact problem in which a semi infinite flyer collides with and adheres to a stationary goupillaud type layered elastic target plate of finite thickness that is bonded to a semi infinite half space we derive explicit expressions for the long time asymptotes of stress and particle velocity that are independent of the layered target s elastic properties this result is verified by numerical simulations of impact into layered targets with periodic or randomly distributed elastic properties and holds when the half space backing the target is replaced by either free or fixed boundary conditions our discrete recurrence relation solutions for impact into layered media generalize our recent analytical results concerning impact into homogeneous single layer targets as well as prior discrete recurrence relation solutions for stress wave propagation in goupillaud type layered elastic media