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

reversing the drift of the ehrenfest urn model and three conditionings we consider a drift reversed version of the celebrated ehrenfest urn process with n balls for this dual process the boundaries are assumed to be absorbing and so the killing times at the boundaries play a central role three natural conditionings on the fixation extinction events pertaining to this model are investigated some spectral information on the conditioned markoff chains is obtained allowing us to draw precise new conclusions on their limiting behaviors

on the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes let x t be a diffusion process on r d with generator l 1 2 nabla cdot a nabla b nabla and let p x x in r d be the corresponding measures on paths pick 0 s which are always satisfied if a 1 b is a gradient function we show that y t s is an inhomogeneous diffusion process and that as t rightarrow infty y t s 0 leq s leq t converges to a limiting homogeneous positive recurrent diffusion y s 0 leq s leq t with state space g since t is arbitrary we actually obtain a limiting process y s 0 leq s infty the generator of the limiting process may be written in the form l g 1 2 nabla cdot a nabla b nabla a nabla g 0 g 0 nabla a nabla h g 0 nabla where g 0 is the square root of the density of a measure mu 0 which minimizes the i function for the process over all gamma in mathscr p bar g the set of probability measures on bar g the function h g 0 appears in the explicit calculation of i mu 0 and solves a certain variational equation the invariant measure for the process is mu 0

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

quasi stationary distributions for structured birth and death processes with mutations we study the probabilistic evolution of a birth and death continuous time measure valued process with mutations and ecological interactions the individuals are characterized by phenotypic traits that take values in a compact metric space each individual can die or generate a new individual the birth and death rates may depend on the environment through the action of the whole population the offspring can have the same trait or can mutate to a randomly distributed trait we assume that the population will be extinct almost surely our goal is the study in this infinite dimensional framework of quasi stationary distributions when the process is conditioned on non extinction we firstly show in this general setting the existence of quasi stationary distributions this result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators we then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi stationary distribution with maximal exponential decay rate the proof of uniqueness is based on an absolute continuity property with respect to a reference measure

on the imaging of reflectors in the earth in this paper i present a modification of the beylkin inversion operator this modification accounts for the band limited nature of the data and makes the role of discontinuities in the sound speed more precise the inversion presented here partially dispenses with the small parameter constraint of the born approximation this is shown by applying the proposed inversion operator to upward scattered data represented by the kirchhoff approximation using the angularly dependent geometrical optics reflection coefficient a fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these kirchhoff approximate data for the forward problem the inversion of these data involves integration over the source receiver surface the reflecting surface and frequency the spatial integrals are computed by the method of stationary phase the output is asymptotically a scaled singular function of the reflecting surface the singular function of a

velocity inversion using a stratified reference we present an inversion algorithm for backscattered stacked seismic data which will reconstruct the velocity profile in realistic earth conditions the basic approach follows that of the original constant reference speed cohen and bleistein paper 1979a in that hig hfrequency asymptotics and perturbation methods are used however we use a reference speed which may vary with depth and this greatly enhances the validity of the perturbation assumption and hence the inversion results the new algorithm enjoys the same economies and stability properties of the original algorithm making it very competitive with current migration schemes four major assumptions are made 1 the acoustic wave equation is an adequate model 2 stacked data have amplitude information worth preserving fairly accurately 3 the actual reflectivity coefficients can be adequately modeled as perturbations from a continuous reference velocity which depends only on the depth variable and 4 the subsurface can be adequately

exponential convergence to quasi stationary distribution and q process for general almost surely absorbed markov processes we obtain necessary and sufficient conditions for exponential convergence to a unique quasi stationary distribution in the total variation norm these conditions also ensure the existence and exponential ergodicity of the q process the process conditioned to never be absorbed we apply these results to one dimensional birth and death processes with catastrophes multi dimensional birth and death processes infinite dimensional population models with brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain

quasistationary distributions for one dimensional diffusions with killing we extend some results on the convergence of one dimensional diffusions killed at the boundary conditioned on extended survival to the case of general killing on the interior we show under fairly general conditions that a diffusion conditioned on long survival either runs off to infinity almost surely or almost surely converges to a quasistationary distribution given by the lowest eigenfunction of the generator in the absence of internal killing only a sufficiently strong inward drift can keep the process close to the origin to allow convergence in distribution an alternative that arises when general killing is allowed is that the conditioned process is held near the origin by a high rate of killing near oo we also extend to the case of general killing the standard result on convergence to a quasistationary distribution of a diffusion on a compact interval

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

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

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

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

reverse time wave field extrapolation imaging and inversion the scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background which are expressed as a scattering potential in general there is no closed form expression for the kernel of this integral representation although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array when the receiver array completely surrounds the scatterer the kernel reduces to the imaginary part of the green s function for the background medium this integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential two such relations are given 1 for the migrated image obtained by deconvolving the extrapolated field with the incident field and 2 for the reconstructed image obtained by applying a one way wave operator to the extrapolated field and then deconvolving by the incident field the migrated image highlig

existence of quasi stationary distributions a renewal dynamical approach we consider markov processes on the positive integers for which the origin is an absorbing state quasi stationary distributions qsd s are described as fixed points of a transformation phi in the space of probability measures under the assumption that the absorption time at the origin r of the process starting from state x goes to infinity in probability as x rightarrow infty we show that the existence of a operatorname qsd is equivalent to e xe lambda r infty for some positive lambda and x we also prove that a subsequence of phi n delta x converges to a minimal operatorname qsd for a birth and death process we prove that phi n delta x converges along the full sequence to the minimal operatorname qsd the method is based on the study of the renewal process with interarrival times distributed as the absorption time of the markov process with a given initial measure mu the key tool is the fact that the residual time in that renewal process has as stationary distribution the distribution of the absorption time of phi mu

a brief comparison of some kirchhoff integral formulas for migration and inversion kirchhoff migration has traditionally been viewed as an imaging procedure usually few claims are made regarding the amplitudes in the imaged section in recent years a number of inversion formulas similar in form to those of kirchhoff migration have been proposed a kirchhoff type inversion produces not only an image but also an estimate of velocity variations or perhaps reflection coefficients the estimate is obtained from the peak amplitudes in the image in this paper prestack kirchhoff migration and inversion formulas for the one parameter acoustic wave equation are compared following a heuristic approach based on the imaging principle a migration formula is derived which turns out to be identical to one proposed by bleistein for inversion prestack kirchhoff migration and inversion are thus seen to be the same both in terms of the image produced and the peak amplitudes of the output

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

fluctuation theory for moran s genetics model abstract a genetics model of moran which includes the bernoulli laplace urn as a special case is examined to give a picture of the model s fluctuation theory and asymptotics as the number of states grows large with particular attention to the time of passage between extreme states these passage times are shown to have means which grow exponentially with the number of states and when normalised converge weakly to an exponential distribution

fast 2 d ray born migration inversion in complex media in this paper we evaluate the capacity of a fast 2 d ray born migration inversion algorithm to recover the true amplitude of the model parameters in 2 d complex media the method is based on a quasi newtonian linearized inversion of the scattered wavefield asymptotic green s functions are computed in a smooth reference model with a dynamic ray tracing based on the wavefront construction method the model is described by velocity perturbations associated with diffractor points both the first traveltime and the strongest arrivals can be inverted the algorithm is implemented with several numerical approximations such as interpolations and aperture limitation around common midpoints to speed the algorithm both theoritical and numerical aspects of the algorithm are assessed with three synthetic and real data examples including the 2 d marmousi example comparison between logs extracted from the exact marmousi perturbation model and the computed images shows that the amplitude of the velocity perturbations

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

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

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

wave field extrapolation techniques in seismic migration a tutorial the objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration i e the kirchhoff summation approach the plane wave method k f method and the finite difference technique particular emphasis is given to the relationship between the different methods by formulating the problem in the space frequency domain x y omega domain a systems approach can be adopted which results in simple and concise expressions these expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure in the situation of lateral velocity variations the de convolution procedure becomes space variant the space frequency domain is most suitable for recursive depth migration in addition frequency dependent properties such as absorption dispersion and spatial bandwidth can be handled easily it is shown that all extrapolation methods are based on two equations taylor series and wave equation in the kirchhoff summation approach all terms of the taylor series are summed to an exact analytical expression the kirchhoff integral for plane surfaces it formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications the fourier transformed version of the kirchhoff integral is used in the plane wave method k f method this actually means that spatial de convolution in the x y omega domain is translated into multiplication in the k x k y omega domain of course this is not allowed if the extrapolation operators are space variant in explicit finite difference techniques a truncated version of the taylor series is used with some optimum adjustments of the coefficients for only one or two terms in the taylor series a spatial low pass filter must be applied to compensate for the amplitude errors at high tilt angles explicit methods are simple and most suitable for three dimensional 3 d applications in implicit finite difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator properly designed implicit schemes do not show amplitude errors and therefore amplitude correction filters need not be applied in comparison with explicit schemes implicit schemes are more sensitive to improper boundary conditions at both ends of the data file it is shown that the forward seismic model can be elegantly described by a matrix equation using separate operators for downward and upward traveling waves using this model inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to compensate for the upward propagation effects

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 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

an explicit iteration for zeros of accretive operators abstract in this paper for lipschitz accretive operator a an iteration scheme is defined as follows x n 1 1 n x n n u n ax n its strong convergence is established for finding some zero of a whenever n n 0 1 satisfying conditions lim n n 0 n 1 n lim n n 0 furthermore some applications for equilibrium problems are given also in particular the iteration coefficient is simpler and more general

on mann implicit iterations for strongly accretive and strongly pseudo contractive mappings abstract for a lipschitz strongly accretive map considered by chidume in c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 it is known that a classical picard type iteration process converges strongly to a zero of the operator he also proved that the rate of convergence in this case is at least as fast as a geometric progression in this paper we study the mann implicit iteration sequence for strongly accretive and strongly pseudo contractive mappings we showed that this implicit scheme gives better convergence rate estimate presented results improve the corresponding results of c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 l liu approximation of fixed points of a strictly pseudo contractive mapping proc am math soc 125 2 1997 1363 1366 w r sastry g v r babu approximation of fixed points of strictly pseudo contractive mappings on arbitrary closed convex sets in a banach space proc amer math soc 128 2000 2907 2909 y song r chen viscosity approximative methods to cesaro means for non expansive mappings appl math comput 186 2 2007 1120 1128

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

a computer implementation of 2 5 d common shot inversion the computer implementation of a two and one half dimensional 2 5 d constant density prestack inversion formalism with laterally and depth dependent background propagation speed is a kirchhoff type inversion summing data from a line of receivers over traveltime curves in the depth dependent background medium with weights determined from born kirchhoff inversion theory this theory predicts that the output will be a reflector map with peak amplitudes on each reflector being in known proportion to the angularly dependent geometrical optics reflection coefficient the 2 5 d feature provides for out of plane spreading correction consistent with the prescribed background medium the method is applied to a synthetic data set and to a physically modeled data set generated at the seismic acoustic laboratory the graphical output demonstrates the validity of the formalism as a kirchhoff migration parameter estimation for the synthetic data confirmed the theory parameter estimation for the experimental data wa

siegmund duality with applications to the neutral moran model conditioned on never being absorbed we first consider the classical neutral moran model with two alleles whose fate is either to become extinct or to reach fixation we study an ergodic version of the moran model obtained by conditioning it to never hit the boundaries making use of a doob transform we call it the recurrent moran model we show that the siegmund dual of the recurrent moran process exists and is a substochastic birth and death chain conditioning this process to exit in its natural absorbing state we construct a process with a unique absorbing state which is intertwined to the original recurrent moran process the time needed for the intertwined process to first hit its absorbing state is related to the time needed to reach stationarity for the recurrent moran process using spectral information on the intertwined chain we extract limiting information on this first hitting time that shows that there is no abrupt relaxation to equilibrium for the recurrent moran chain this makes use of the relation between duality and intertwining and strong stationary times other related transition times of the recurrent moran chain are also briefly investigated namely the first return time to the ground state and the expected time needed to move from one end to the other end of the state space

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

classification of killed one dimensional diffusions we show necessary and sufficient conditions for r recurrence and mbox r positivity of one dimensional diffusions killed at the origin these conditions are stated in terms of the bottom eigenvalue function

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

ray kirchhoff migration in inhomogeneous media a kirchhoff method that avoids possible singularities on the surface of integration and is more accurate than previous kirchhoff methods has been developed for seismic migration in laterally inhomogeneous media it is based on a newly derived integral solution to the acoustic wave equation this solution indicates that wave fields in an inhomogeneous medium can be expressed as a summation of ray solutions determined by the transport and extended eikonal equations the extended eikonal equation is in turn solved by an asymptotic series for implementation a perturbation scheme is developed for solving the ray and transport equations in addition to computing higher order terms of the asymptotic series this scheme can be used to avoid most of the ray tracing required for wave extrapolation in a medium where vertical variations may be large but lateral variations are small compared to velocity itself where an analytic ray solution for the reference velocity used in the perturbation scheme exists ray kirchhoff migration in such a medium can be carried out without numerical ray tracing

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

ehrenfest urn revisited playing the game on a realistic fluid model the ehrenfest urn process also known as the dogs and fleas model is realistically simulated by molecular dynamics of the lennard jones fluid the key variable is delta z i e the absolute value of the difference between the number of particles in one half of the simulation box and in the other half this is a pure jump stochastic process induced under coarse graining by the deterministic time evolution of the atomic coordinates we discuss the markov hypothesis by analyzing the statistical properties of the jumps and of the waiting times between jumps in the limit of a vanishing integration time step the distribution of waiting times becomes closer to an exponential and therefore the continuous time jump stochastic process is markovian the random variable delta z behaves as a markov chain and in the gas phase the observed transition probabilities follow the predictions of the ehrenfest theory

a homotopy method for well log constraint waveform inversion the primary difficulty posed by the method of waveform inversion for the acoustic wave equation is the presence of local minima of the objective function waveform inversion fails to converge to the global minimum unless the wavefield contains very low frequencies or the starting model is very close to the true model constraints can improve the convergence of the method however if the starting model is far from the correct model normal iterative methods will still get trapped in local minima we designed a homotopy method to improve the robustness of waveform inversion it makes natural use of constraints such as well logs in addition to further condition the inverse problem we incorporate standard tikhonov regularization we demonstrate through a synthetic example that our method is more likely to find a global minimum than normal iterative methods

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

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

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

imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized radon transform this paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers the linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids this integral equation is connected with the causal generalized radon transform grt and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the grt the first term of this asymptotic expansion is interpreted as a migration algorithm as a result this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium also a partial reconstruction operator is explicitly derived for a limited aperture when specialized to a constant background velocity and specific source receiver geometries our results are directly related to some known migration algorithms

extremality of invariant measures and ergodicity of stochastic systems in this paper we first present some conditions on the extremality of invariant measures of a markov process these conditions involve average convergence and invariant functions of the process we then combine these with coupling and duality to study some aspects of convergence to invariant measures of markov processes in particular we suggest a new way of studying complete convergence applications to interacting particle systems are given

power law behavior and condensation phenomena in disordered urn models we investigate the equilibrium statistical properties of urn models with disorder two urn models are proposed one belongs to the ehrenfest class and the other corresponds to the monkey class these models are introduced from the view point of the power law behavior and randomness it is clarified that quenched random parameters play an important role in generating the power law behavior we evaluate the occupation probability p k with which an urn has k balls by using the concept of statistical physics of disordered systems in the disordered urn model belonging to the monkey class we find that above critical density c for a given temperature condensation phenomenon occurs and the occupation probability changes its scaling behavior from an exponential law to a heavy tailed power law in large k regime we also discuss an interpretation of our results for explaining macro economy in particular the emergence of wealth differentials

quasi stationary distributions and diffusion models in population dynamics in this paper we study quasi stationarity for a large class of kolmogorov diffusions the main novelty here is that we allow the drift to go to infty at the origin and the diffusion to have an entrance boundary at infty these diffusions arise as images by a deterministic map of generalized feller diffusions which themselves are obtained as limits of rescaled birth death processes generalized feller diffusions take nonnegative values and are absorbed at zero in finite time with probability 1 an important example is the logistic feller diffusion we give sufficient conditions on the drift near 0 and near infty for the existence of quasi stationary distributions as well as rate of convergence in the yaglom limit and existence of the q process we also show that under these conditions there is exactly one quasi stationary distribution and that this distribution attracts all initial distributions under the conditional evolution if and only if infty is an entrance boundary in particular this gives a sufficient condition for the uniqueness of quasi stationary distributions in the proofs spectral theory plays an important role on l 2 of the reference measure for the killed process

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

quantum operators in classical probability theory ii the concept of duality in interacting particle systems duality has proved to be a powerful tool in the theory of interacting particle systems the approach in this paper is algebraic rather than via harris diagrams a form of duality is found which includes coalescing and annihilating duality as special cases this enables new results for the branching annihilating random walk and the biased annihilating branching process to be derived

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

imaging method utilizing attenuation and speed parameters in inverse scattering techniques methods for imaging the internal structures of an object using an acoustic wave field are provided in one aspect for example a method of imaging internals of a physical object using acoustic waves may include transmitting an acoustic wave field toward the object receiving a resultant acoustic wave field with a receiver where the resultant acoustic wave field is in response to the transmitted acoustic wave field reflected from or transmitted through the object and determining a predicted resultant acoustic wave field derived from a model of the object the method may also include determining a residual between the predicted resultant acoustic wave field and the resultant acoustic wave field and back propagating the residual to determine corrections to the model of the object in another aspect the above recited steps may be further iterated to successively refine the model of the object over a number of iterations until a predefined condition is reached

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

inversion of n dimensional spherical averages in this paper we give a simple inversion formula for determining the even part of a function from its averages over the surfaces of spheres in mathbb r n centered on a hyperplane y 0 we emphasize the relation between the backprojection approximation to the function and the full inversion result a simple necessary and sufficient condition for data to correspond to the spherical averages of a function is given

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

asymptotic laws for one dimensional diffusions conditioned to nonabsorption if x t is a one dimensional diffusion corresponding to the operator l 1 2 xx x starting from x 0 and t is the hitting time of we prove that under suitable conditions on the drift coefficient the following limit exists s 0 a f s lim t p x x a t 0 t we characterize this limit as the distribution of an h like process h satisfying lh h h 0 0 h 0 1 where lim t 1 t log p x t 0 t moreover we show that this parameter can only take two values 0 or where is the smallest point of increase of the spectral distribution of the operator l 1 2 xx x

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

a born wkbj inversion method for acoustic reflection data density and bulk modulus variations in an acoustic earth are separately recoverable from standard reflection surveys by utilizing the amplitude versus offset information present in the observed wave fields both earth structure and a variable background velocity can be accounted for by combining the born and wkbj approximations in a before stack migration with two output sections one for density variations and the other for bulk modulus variations for the inversion the medium is considered to be composed of a known low spatial frequency variation the background plus an unknown high spatial frequency variation in bulk modulus and density the reflectivity the division between the background and the reflectivity depends upon the frequency content of the source for constant background parameters computations are done in the fourier domain where the first part of the algorithm includes a frequency shift identical to that in an f k migration the modulus and density variations are then determined by observing in a least squares sense amplitude versus offset wavenumber for a spatially variable background wkbj green s operators that model the direct wave in a medium with a smoothly varying background are used a downward continuation with these operators removes the effects of variable velocity from the problem and consequently the remainder of the inversion essentially proceeds as if the background were constant if the background is strictly depth dependent the inversion can be expressed in closed form the method neglects multiples and surface waves and it is restricted to precritical reflections density is distinguishable from bulk modulus only if a sufficient range of precritical incident angles is present in the data

acoustic inverse scattering via helmholtz operator factorization and optimization we present a joint acoustic seismic inverse scattering and finite frequency reflection tomography program formulated as a coupled set of optimization problems in terms of inhomogeneous helmholtz equations we use a higher order finite difference scheme for these helmholtz equations to guarantee sufficient accuracy we adapt a structured approximate direct solver for the relevant systems of algebraic equations which addresses storage requirements through compression to yield a complexity for computing the gradients or images in the optimization problems that consists of two parts viz the cost for all the matrix factorizations which is roughly o rn for example o rnlogn when d 2 times the number of frequencies and the cost for all solutions by substitution which is roughly o n for example o nlog rlogn when d 2 times the number of frequencies times the number of sources events where n n d if n is the number of grid samples in any direction and r is a parameter depending on the preset accuracy and the problem at hand with this complexity the multi frequency approach to inverse scattering and finite frequency tomography becomes computationally feasible with large data sets in dimensions d 2 and 3

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

a note on the paper a new iteration process for generalized lipschitz pseudo contractive and generalized lipschitz accretive mappings abstract in this note we will modify several gaps in chidume and ofoedu c e chidume e u ofoedu a new iteration process for generalized lipschitz pseudo contractive and generalized lipschitz accretive mappings nonlinear anal 2006 in press doi 10 1016 j na 2006 05 012

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

two and one half dimensional born inversion with an arbitrary reference multidimensional inversion algorithms are presented for both prestack and poststack data gathered on a single line these algorithms both image the subsurface i e give a migrated section and given relative true amplitude data estimate reflection strength or impedance on each reflector the algorithms are two and one half dimensional 2 5 d in that they incorporate three dimensional 3 d wave propagation in a medium which varies in only two dimensions the use of 3 d sources does not entail any computational penalty and it avoids the serious degradation of amplitude incurred by using the 2 d wave equation our methods are based on the linearized inversion theory associated with the born inversion thus we assume that the sound speed profile is well approximated by a given background velocity plus a perturbation it is this perturbation that we seek to reconstruct we are able to treat the case of an arbitrary continuous background profile however the cost of implementation increases as one

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

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