an orlicz space approach to superlinear elliptic systems in this paper we study superlinear elliptic systems in hamiltonian form using an orlicz space setting we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense

existence and multiplicity of solutions for an elliptic system with nonlinear boundary conditions abstract in this paper we prove the existence and multiplicity of solutions for the elliptic system u u v v in with nonlinear coupling at the smooth boundary given by u h u v h v on where is a bounded domain of r n with smooth boundary is the outer normal derivative the proofs are done under suitable assumptions on the hamiltonian and based on the local linking theorem and a multiple critical points theorem of the critical point theory

homological approach to problems with jumping non linearity abstract in this paper we use a perturbed version of the rabinowitz floer homology to find solutions to pde s with jumping nonlinearities as applications we find branches for the fucik spectrum for the laplace equation and for systems on manifolds that fiber over s 1

a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

infinitely many solutions for symmetric and non symmetric elliptic systems we study an elliptic system equivalent to a fourth order elliptic equation by using variational and perturbative methods we prove the existence of infinitely many solutions both in the symmetric and in the non symmetric case

waveform inversion of reflection seismic data for kinematic parameters by local optimization a reformulation of the usual least squares waveform inversion problem is proposed to retrieve from seismic data the kinematic parameters the two dimensional 2d background velocity and the source and cable depth by local optimization these last parameters are of paramount importance for a successful inversion of very high resolution vhr seismic data which we are interested in in our inversion the source and cable depth parameters are treated in the same way as the background velocity to avoid the problem of local minima a change of unknowns is performed the depth reflectivity is replaced by its dual variable called the time reflectivity in this way the current value of the reflectivity is stored in the time domain and is strongly decoupled from the current value of the velocity field and cable depth the increase in modeling complexity an additional prestack migration is required for each function evaluation is compensated by the enlargement of the attraction domain of the global minimum which allows the use of a local optimization technique a numerical implementation with born approximation and ray tracing is detailed in which the derivatives of the travel time are computed via an adjoint state technique for more efficiency numerical results illustrate the behavior of the new objective function and inversion of synthetic and real vhr data for kinematic parameters is performed

multiple solutions for asymptotically linear elliptic systems in this article we consider cooperative and noncooperative elliptic systems that are asymptotically linear at infinity we obtain infinitely many solutions with small energy if the potential is even if the noncooperative system is resonant both at zero and at infinity then the number of nontrivial solutions depends on the dimension of the eigenspaces between resonant values

multiscattering inversion for low model wavenumbers abstracta successful full waveform inversion implementation updates the low wavenumber model components first for a proper description of the wavefield propagation and slowly adds the high wavenumber potentially scattering parts of the model the low wavenumber components can be extracted from the transmission parts of the recorded wavefield emanating directly from the source or the transmission parts from the single or double scattered wavefield computed from a predicted scatter field acting as secondary sources we use a combined inversion of data modeled from the source and those corresponding to single and double scattering to update the velocity model and the component of the velocity perturbation responsible for the single and double scattering the combined inversion helps us access most of the potential model wavenumber information that may be embedded in the data a scattering angle filter is used to divide the gradient of the combined inversion so initially the high wavenumber low scatterin

target oriented wave equation least squares migration inversion with phase encoded hessian prestack depth migration produces blurred images resulting from limited acquisition apertures complexities in the velocity model and band limited characteristics of seismic waves this distortion can be partially corrected using the model space least squares migration inversion approach where a target oriented wave equation hessian operator is computed explicitly and then inverse filtering is applied iteratively to deblur or invert for the reflectivity however one difficulty is the cost of computing the explicit hessian operator which requires storing a large number of green s functions making it challenging for large scale applications a new method to compute the hessian operator for the wave equation based least squares migration inversion problem modifies the original explicit hessian formula enabling efficient computation of this operator an advantage is that the method eliminates disk storage of green s functions the modifications however also introduce undesired crosstalk artifacts two different phase encoding schemes plane wave phase encoding and random phase encoding suppress the crosstalk when the randomly phase encoded hessian operator is applied to the sigsbee2a synthetic data set an improved subsalt image with more balanced amplitudes is obtained

existence and multiplicity of solutions for asymptotically hamiltonian elliptic systems in rn abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain solutions for the above system

approximate linearized inversion by optimal scaling of prestack depth migration linearized inversion provides one possible sense of image amplitude correctness an image or bandlimited model perturbation has correct amplitudes if it is an approximate inversion that is if linearized modeling demigration with the image as input reproduces the data approximately the theory of linearized acoustic inverse scattering with slowly varying background or macromodel shows that an approximate inversion may be recovered from the output of prestack depth migration by a combination of scaling and filtering the necessary filter is completely specified by the theory and the scale factor may be estimated via filtering linearized modeling a second migration and the solution of a small auxiliary inverse problem

ground state solutions for superquadratic hamiltonian elliptic systems with gradient terms abstract in this paper we study the following hamiltonian elliptic system with gradient terms u b x u v x u h v x u v v b x v v x v h u x u v for x r n where v x b x and h x u v are 1 periodic in x under weak superquadratic assumptions for the nonlinearities we use much more direct methods to prove that all the cerami sequences for the energy functional are bounded and establish the existence of ground state solutions for this system

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

three dimensional frequency domain full waveform inversion with an iterative solver with the acquisition of wide aperture seismic data sets full waveform inversion is an attractive method for deriving velocity models three dimensional implementations require an efficient solver for the wave equation computing 3d time harmonic responses with a frequency domain solver is complicated because a large linear system with negative and positive eigenvalues must be solved time domain schemes are an alternative nevertheless existing frequency domain iterative solvers with an efficient preconditioner are a viable option when full waveform inversion is formulated in the frequency domain an iterative solver with a multigrid preconditioner is competitive because of a high order spatial discretization numerical examples illustrated the efficiency of the iterative solvers three dimensional full waveform inversion was then studied in the context of deep water ocean bottom seismometer acquisition three dimensional synthetic data inversion results showed the behavior of full waveform inversion with respect to the initial model and the minimum frequency available in the data set results on a 3d real ocean bottom seismometer data set demonstrated the relevance of full waveform inversion especially to image the shallow part of the model

infinitely many solutions for the perturbed bose einstein condensates system abstract we consider the perturbed nonlinear elliptic system as following u j j u j i 1 2 i j u i 2 u j j x u j h 0 1 j 1 2 where is a bounded smooth domain in r n n 3 and j 1 1 is the first eigenvalue of with the dirichlet boundary condition the perturbation terms j l 2 for j 1 2 assume that the matrix i j i j 1 2 is either positive definite when n 1 2 or anti symmetric when n 3 we show the existence of infinitely many solutions to this system the results are also true for the m coupled system i e i j 1 m m 2

a note on superlinear hamiltonian elliptic systems this paper is concerned with the superlinear periodic elliptic systems of hamiltonian type in the whole space the existence of a ground state solution as well as an infinite number of geometrically distinct solutions is obtained

existence and multiplicity of solutions for asymptotically linear noncooperative elliptic systems abstract using the minimax methods in critical point theory and a generalized landesman lazer type condition we establish two existence results of solutions for asymptotically linear noncooperative elliptic systems at resonance besides this we obtain two solutions in the case of near resonance

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

new linking theorem and elliptic systems with nonlinear boundary condition in this paper we establish a new linking theorem without ps type assumption the new theorem produces bounded ps sequences the abstract result will be used to study the existence of nontrivial positive solutions to the elliptic system u u v v in with nonlinear boundary condition u gv u gu on where is a bounded domain of rn with smooth boundary is the outer normal derivative we consider weak superlinear and asymptotically linear cases

on the existence of positive solutions of a perturbed hamiltonian system in rn using the legendre fenchel transformation and the mountain pass theorem due to ambrosetti and rabinowitz we establish an existence result for perturbations of periodic and asymptotically periodic semilinear hamiltonian systems of the type pw u u w2 x v p 1vinrn v v w1 x u q 1uinrn u x v x 0as x u 0 v 0inrn n 2 here the numbers p q 1 are below the critical hyperbola if n 3 that is they satisfy 1 p 1 1 q 1 n 2 n while no additional restrictions on p and q are required if n 2 the functions wi i 1 2 are bounded positive continuous functions

existence of nonnegative solutions for a class of semilinear elliptic systems with indefinite weight abstract we prove the existence of nonnegative solutions to the system u a x f v in v b x g u in u 0 v on where is a bounded domain in r n with a smooth boundary is a positive parameter by the method of monotone iteration and schauder fixed point theorem we prove the existence of a nonnegative solution to the above system for sufficiently small in this study we do not restrict the sign of a and b

accommodating lateral velocity changes in kirchhoff migration by means of fermat s principle when velocity varies laterally as well as with depth an exact kirchhoff depth migration requires that rays be traced from each depth point in the section to each source receiver location because such a procedure is prohibitively expensive kirchhoff migration is usually carried out by using a velocity function that depends only on depth this paper introduces a new method based on fermat s principle which is a compromise between these two extremes the slowness reciprocal velocity function is written as the sum of two functions the first of which is large and depends only on depth while the other is small and varies both with depth and position along the line raypaths are traced for the first slowness function and are used to calculate migration curves for each depth point these same raypaths are used to calculate traveltime perturbations due to the laterally varying part of the slowness the traveltime perturbations are added to the migration curve to obtain an approximation to the exact migrat

multiple positive solutions of strongly indefinite systems with critical sobolev exponents and data that change sign abstract in this paper we study the existence of multiple positive solutions to some hamiltonian elliptic systems v u u p 1 u ef x u v v q 1 v eg x in u 0 v 0 in u v 0 on where is a smooth bounded domain in r n n 3 f g c 1 p q 1 r for the subcritical and critical cases we prove that problem has at least two positive solutions for any e 0 e and has no positive solutions for any e e e in the supercritical case we find that the existence of solutions of problem for 0 is closely related to the existence of nonnegative solutions of some linear elliptic system

2d frequency domain elastic full waveform inversion using the block diagonal pseudo hessian approximation abstractelastic full waveform inversion efwi of multicomponent seismic data is a powerful tool for estimating the subsurface elastic parameters with high accuracy however the trade offs between multiple parameters increase the nonlinearity of efwi although the conventional diagonal approximate hessian matrix describes the illumination and limited bandwidth effects it ignores the trade off effects and decreases the convergence rate of efwi we have developed a block diagonal pseudo hessian operator for 2d frequency domain efwi to take into account the approximate trade offs among the p wave compressional wave velocity s wave shear wave velocity and density without extra computational costs on forward simulations the hessian matrix tends toward a block diagonal matrix as the frequency grows to infinity thus the proposed block diagonal pseudo hessian matrix is more accurate at higher frequencies the inverse of the block diagonal pseudo hessian matrix is used as a preconditioner for the nonlin

nonnegative solutions for nonlinear elliptic systems abstract in this paper we show that the semilinear elliptic systems of the form 0 1 u v g v v u f u x u v 0 x possess at least one nonnegative nontrivial solution pair u v h 0 1 h 0 1 where is a smooth bounded domain in r n

soliton solutions to systems of coupled schrodinger equations of hamiltonian type we study the existence of positive solutions of hamiltonian type systems of second order elliptic pde in the whole space the systems depend on a small parameter and involve a potential having a global well structure we use dual variational methods a mountain pass type approach and fourier analysis to prove positive solutions exist for sufficiently small values of the parameter

inversion of normal incidence seismograms the well known instability of kunetz s 1963 inversion algorithm can be explained by the progressive manner in which the calculations are done descending from the surface and by the fact that completely different impedances can yield indistinguishable synthetic seismograms those difficulties can be overcome by using an iterative algorithm for the inversion of the one dimensional 1 d wave equation together with a stabilizing constraint on the sums of the jumps of the desired impedance for computational efficiency the synthetic seismogram is computed by the method of characteristics and the gradient of the error criterion is computed by optimal control techniques adjoint state equation the numerical results on simulated data confirm the expected stability of the algorithm in the presence of measurement noise tests include noise levels of 50 percent the inversion of two field sections demonstrates the practical feasibility of the method and the importance of taking into account all internal a

elastic least squares reverse time migration abstractwe use elastic least squares reverse time migration lsrtm to invert for the reflectivity images of p and s wave impedances elastic lsrtm solves the linearized elastic wave equations for forward modeling and the adjoint equations for backpropagating the residual wavefield at each iteration numerical tests on synthetic data and field data reveal the advantages of elastic lsrtm over elastic reverse time migration rtm and acoustic lsrtm for our examples the elastic lsrtm images have better resolution and amplitude balancing fewer artifacts and less crosstalk compared with the elastic rtm images the images are also better focused and have better reflector continuity for steeply dipping events compared to the acoustic lsrtm images similar to conventional least squares migration elastic lsrtm also requires an accurate estimation of the p and s wave migration velocity models however the problem remains that when there are moderate errors in the velocity model and strong multiples lsrtm

asymptotic pseudounitary stacking operators stacking operators are widely used in seismic imaging and seismic data processing examples include kirchhoff datuming migration offset continuation dip moveout and velocity transform two primary approaches exist for inverting such operators the first approach is iterative least squares optimization which involves the construction of the adjoint operator the second approach is asymptotic inversion where an approximate inverse operator is constructed in the high frequency asymptotics adjoint and asymptotic inverse operators share the same kinematic properties but their amplitudes weighting functions are defined differently this paper describes a theory for reconciling the two approaches i introduce a pair of asymptotic pseudounitary operators which possess both the property of being adjoint and the property of being asymptotically inverse the weighting function of the asymptotic pseudounitary stacking operators is shown to be completely defined by the derivatives of the operator kinematics i exemplify the general theory by considering several particular examples of stacking operators simple numerical experiments demonstrate a noticeable gain in efficiency when the asymptotic pseudounitary operators are applied for preconditioning iterative least squares optimization

regularization and datuming of seismic data by weighted damped least squares the possibility of improving regularization datuming of seismic data is investigated by treating wavefield extrapolation as an inversion problem weighted damped least squares is then used to produce the regularized datumed wavefield regularization datuming is extremely costly because of computing the hessian so an efficient approximation is introduced approximation is achieved by computing a limited number of diagonals in the operators involved real and synthetic data examples demonstrate the utility of this approach for synthetic data regularization datuming is demonstrated for large extrapolation distances using a highly irregular recording array without approximation regularization datuming returns a regularized wavefield with reduced operator artifacts when compared to a nonregularizing method such as generalized phase shift plus interpolation pspi approximate regularization datuming returns a regularized wavefield for approximately two orders of magnitude less in cost but it is dip limi

on the classification of standing wave solutions to a coupled schrodinger system in this paper we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear schrodinger equation we establish the existence and uniqueness of solutions to the dirichlet problem on the ball and entire space as the parameters within certain regions in addition a complete structure of different types of solutions for the radial case is also provided

existence of solutions to higher order lane emden type systems abstract we prove existence results for the lane emdentype system u v q v u p in b 1 r n r u r 0 r 0 1 on b 1 r v r 0 r 0 1 on b 1 where b 1 is the unitary ball in r n n max 2 2 is the outward pointing normal n 1 and 1 is the polyharmonic operator a continuation method together with a priori estimates will be exploited moreover we prove uniqueness for the particular case 2 1 and p q 1

on the uniqueness and structure of solutions to a coupled elliptic system abstract in this paper we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear schrodinger equation we establish the existence and uniqueness of solutions to the dirichlet problem on the ball in addition the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved

existence of three nontrivial solutions for elliptic systems with critical exponents and weights abstract in this paper an elliptic system with critical sobolev exponents and weights is studied by the linking theorem on product space and the classical minimax theorem by investigating the coefficients of the critical nonlinearities we establish the existence of three nontrivial solutions

multiple solutions for elliptic systems with nonlinearities of arbitrary growth abstract we prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of arbitrary growth moreover if the symmetry of the problem is broken by a small enough perturbation term we find at least three solutions the proofs utilise a variational setting given by de figueiredo and ruf in order to prove an existence s result and the algebraic approach based on the pohozaev s fibering method

asymptotically linear schrodinger equations with sign changing potential abstract in this paper we show that the following system 0 1 u v x u g x v v v x v f x u x r n possesses at least one non trivial solution pair u v where f x t g x t are continuous functions on r n r and asymptotically linear as t and v x is allowed to be sign changing

an algorithm for 3d acoustic time laplace fourier domain hybrid full waveform inversion abstractwhen full waveform inversion fwi is performed in the laplace fourier domain laplace fourier domain wave modeling using finite difference or finite element methods is performed to compute the gradient direction in this case a complex impedance matrix is composed and factored by direct matrix solvers alternatively iterative matrix solvers may be used however solving 3d problems with such methods requires excessive computer memory or computing time which causes problems in the application of the laplace fourier domain fwi to avoid computational overburden in 3d problems we propose a time laplace fourier hybrid fwi in which forward and backward modeling are performed in the time domain and other procedures are conducted in the laplace fourier domain our hybrid fwi is applied to two groups of frequencies inversions for the first and second groups of frequencies correspond to the laplace domain fwi and the laplace fourier domain fwi respectively the graphic processing unit is used to

fast least squares migration with a deblurring filter least squares migration lsm is a linearized waveform inversion for estimating a subsurface reflectivity model that relative to conventional migration improves spatial resolution of migration images the cost however is high because lsm typically requires 10 or more iterations which is at least 20 times more than the cpu cost of conventional migration to alleviate this expense we offer a deblurring filter that can be used in a regularization scheme or a preconditioning scheme to give acceptable lsm images with less than one third the cost of the standard lsm method our results in applying deblurred lsm to synthetic data and field data support this claim in particular a marmousi2 model test shows that the data residual for preconditioned deblurred lsm decreases rapidly in the first iteration which is equivalent to 10 or more iterations of lsm empirical results suggest that regularized deblurred lsm after three iterations is equivalent to about 10 iterations of lsm applying deblurred lsm to 2d marine data gives a higher resolution image compared to those from migration or lsm with three iterations these results suggest that lsm combined with a deblurring filter allows lsm to be a fast practical tool for improved imaging of complicated structures

multiplicity and concentration of solutions for elliptic systems with vanishing potentials abstract in this article we use variational methods to study a strongly coupled elliptic system depending on a positive parameter we suppose that the potentials are nonnegative and the intersection of the sets where they vanish has positive measure a technical condition imposed on the product of the potentials allows us to consider a setting where we do not assume any positive lower bound for the potentials considering the associated functional defined on an appropriated subspace of d 1 2 r n d 1 2 r n we are able to establish results on the existence and multiplicity of solutions for the system when the parameter is sufficiently large we also study the asymptotic behavior of these solutions when

multiple solutions for an elliptic system near resonance abstract we consider an elliptic system with linear part depending on two parameters and a sublinear perturbation we obtain the existence of at least two solutions when the linear part is near resonance the system is associated with a strongly indefinite functional and the solutions are obtained through saddle point theorem and galerkin approximation

an optimal true amplitude least squares prestack depth migration operator in order to define an optimal true amplitude prestack depth migration for multishot and multitrace data we develop a general methodology based on the least squares data misfit function associated with a forward model the amplitude of the migrated events are restored at best for any given geometry and any given preliminary filtering and amplitude correction of the data the migrated section is then the gradient of the cost function multiplied by a weight matrix a study of the hessian associated with this data misfit shows how efficiently to find a good weight matrix via the computation of only few elements of this hessian thanks to this matrix the resulting migration operator is optimal in the sense that it ensures the best possible restoration of the amplitudes among the large class of least squares migrations applied to a forward model based on born ray tracing and diffracting points approximation this optimal migration outperforms or at least equals the classic kirchhoff formula since the latter belongs to the class of least squares migrations and is only optimal for one shot and an infinite aperture numerical results illustrate this construction and confirm the above expectations

asymptotically periodic superquadratic hamiltonian systems abstract we obtain existence of solutions for the system u v x u f v x u v x r n v v x v f u x u v x r n where n 3 v c r n 0 is periodic and f c 1 r n r 2 r is superquadratic at infinity we consider the case that f is periodic and asymptotically periodic in the proofs we apply variational techniques

waveform inversion using a logarithmic wavefield although waveform inversion has been studied extensively since its beginning 20 years ago applications to seismic field data have been limited and most of those applications have been for global seismology or engineering seismology scale problems not for exploration scale data as an alternative to classical waveform inversion we propose the use of a new objective function constructed by taking the logarithm of wavefields allowing consideration of three types of objective function namely amplitude only phase only or both in our waveform inversion we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function we compute the steepest descent directions by using a matrix formalism derived from a frequency domain finite element finite difference modeling technique our numerical algorithms are similar to those of reverse time migration and waveform inversion based on the adjoint state of the wave equation in order to demonstrate the practical applicability of our algorithm we use a synthetic data set from the marmousi model and seismic data collected from the korean continental shelf for noisefree synthetic data the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion when random noise is added the inverted velocity model is also close to the true marmousi model but when frequencies below 5 hz are removed from the data the velocity structure is not as good as those for the noise free and noisy data for field data we compare the time domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short period features of the subsurface although we use wrapped phases in our examples we still obtain reasonable results we expect that if we were to use correctly unwrapped phases in the inversion algorithm we would obtain better results

derivation of forward and adjoint operators for least squares shot profile split step migration the forward and adjoint operators for shot profile least squares migration are derived the forward operator is demigration and the adjoint operator is migration the demigration operator is derived from the born approximation the process begins with a green s function that allows for a laterally varying migration velocity model using the split step approximation next the earth is divided into horizontal layers and within each layer the migration velocity model is made to be constant with respect to depth for a given layer 1 the source side wavefield is propagated down to its top using the background wavefield this gives a background wavefield incident at the layer s upper boundary 2 the layer s contribution to the scattered wavefield is computed using the born approximation to the scattered wavefield and the background wavefield 3 next its scattered wavefield is propagated back up to the measurement surface using again the background wavefield the measured wavefield is approximated by the sum of scattered wavefields from each layer in the derivation of the measured wavefield the shot profile migration geometry is used for each shot the resulting wavefield modeling operator takes the form of a fredholm integral equation of the first kind and this is used to write down its adjoint the shot profile migration operator this forward adjoint pair is used for shot profile least squares migration shot profile least squares migration is illustrated with two synthetic examples the first uses data collected over a four layer acoustic model and the second uses data from the sigsbee 2a model

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

ground state solutions for a diffusion system in this paper we study the following diffusion system t u x u v x u g t x v t v x v v x v f t x u where z u v r r n r 2 v x c r n r is a general periodic function under weaker conditions on nonlinearity we establish the existence of ground state solutions via variational methods

an infinite dimensional linking theorem and applications abstract in this paper we establish a new infinite dimensional linking theorem without ps type assumptions the new theorem needs a weaker linking geometry and produces bounded ps sequences the abstract result will be applied to the study of the existence of solutions of the strongly indefinite partial differential systems for the first application we consider the system u u in v v in u h v x u v on v h u x u v on where is a bounded domain in r n with smooth boundary is the outer normal derivative h r r r is a positive c 1 function one nontrivial solution is obtained the second application we will solve the eigenvalue problem of the system a v f x v w b w g x v w 0 where a b are self adjoint operators on l 2 r n is not necessarily bounded f g are caratheodory functions on r 2 we get infinitely many solutions we deal with asymptotically linear cases for both systems

ground state solutions for superlinear elliptic systems on rn abstract this paper is concerned with the existence of the ground state solutions for the following superlinear elliptic system of hamiltonian type u v x u g x v in r n v v x v f x u in r n u x 0 and v x 0 as x where v c r n r is periodic in x 1 x 2 x n we assume that 0 lies in a gap of the spectrum v and f and g are both superlinear at 0 and infinity but they have different increasing rates at infinity by proving all cerami sequences for the energy functional are bounded existence of a ground state solution is obtained

migration based traveltime waveform inversion of 2 d simple structures a synthetic example migration based traveltime mbtt formulation provides algorithms for automatically determining background velocities from full waveform surface seismic reflection data using local optimization methods in particular it addresses the difficulty of the nonconvexity of the least squares data misfit function the method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity we present an implementation of the mbtt approach for a 2 d finite difference fd full wave acoustic model numerical analysis on a 2 d synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least squares approach even when starting from very poor initial guesses this enlargement of the domain of attraction for the global minima of the least squares misfit has a price each evaluation of the new objective function requires besides the usual fd full wave forward modeling an additional full wave prestack migration hence the fd implementation of the mbtt approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size

an existence result for a linear superlinear elliptic system with neumann boundary conditions abstract in this work we consider an elliptic system of two equations in dimension one with neumann boundary conditions where the nonlinearities are asymptotically linear at and superlinear at we obtain that under suitable hypotheses a solution exists for any couple of forcing terms in l 2 we also present a similar result in which the superlinearity is in only one of the two equations and we discuss the resonant problem too

existence and multiplicity of solutions for asymptotically linear nonperiodic hamiltonian elliptic system abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v v v x v h u x u v x r n u x v x 0 as x where z u v r n r r n 3 assuming the potential v is nonperiodic and sign changing h x z is nonperiodic in x and asymptotically quadratic in z u v existence and multiplicity of solutions are obtained via a variational approach

strongly indefinite systems with critical sobolev exponents we consider an elliptic system of hamiltonian type on a bounded domain in the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms our proof is based on an adaptation of the dual variational method as applied before to the scalar case introduction existence and non existence of solutions of semilinear elliptic srstems has been a subject of active research recently see for example ff l3 ps such systems are called variational if solutions can be viewed as critical points of an associated functional defined on a suitable function space restricting our attention to systems with two unknowns we distinguish two classes of variational elliptic systems a potential systems b hamiltonian systems potential systems are of the form a llu u v ztl2v u v where l1 and l2 are second order selfadjoint elliptic operators formally these are the euler lagrange equations of the functional f u v l1u u i l2v v th u v here is the l2 inner product for functions defined on the underlying domain and tt u v s h u x v x dx the functional f is often called the lagrangian its critical points correspond to weak soltbtions of a the appropr ate choice of the function spaces for u and v follows by requiring that the quadratic part of f be well defined this naturally leads to sobolev spaces with square irltegrable first derivatives since also a must be well defined by virtue of the embedding theorems some growth restrictions on this latter function are required an extra difficulty appears if we consider system a with a plus sign in the second equation for then the corresponding functiorl f has a strongly indefinite quadratic part for systems of superlinear type and with sllbcritical growth the mountain pass theorem ar or in the strongly indefinite case the benci rabinowitz theorem received by the editors june 5 1996 1991 mathematics subject classification primary 35j50 35j55 35j65

solutions of a class of hamiltonian elliptic systems in rn we study the existence of ground state solutions for the following elliptic systems in rn u b xu u hv x u v v b xv v hu x u v where b b1 bn is a constant vector and h c1 rn r2 r is nonperiodic in variables x and super quadratic as z by a recent critical point theorem for strongly indefinite problem we obtain the existence of at least one ground state solution

waveform inversion using a back propagation algorithm and a huber function norm waveforminversionfacesdifficultieswhenappliedtoreal seismic data including the existence of many kinds of noise the 1 norm is more robust to noise with outliers than the least squares method nevertheless the least squares method is preferred as an objective function in many algorithms because the gradient of the 1 norm has a singularity when the residual becomes zero we propose a complex valued huber function for frequency domain waveform inversion that combines the 2 norm for small residuals with the 1 norm for large residuals we also derive a discretized formula for the gradient of the huber function through numericaltestsonsimplesyntheticmodelsandmarmousidata we find the huber function is more robust to outliers and coherent noise we apply our waveform inversion algorithm to field data taken from the continental shelf under the east sea in korea in this setting we obtain a velocity model whose syntheticshotprofilesaresimilartotherealseismicdata

target oriented wave equation inversion a target oriented strategy can be applied to estimate a wave equation least squares inverse lsi image by explicitly computing the wave equation hessian the lsi image is obtained as the solution of a nonstationary least squares inverse filtering problem the rows of the hessian are the nonstationary filters containing information about the acquisition geometry the velocity model and the band limited characteristics of the seismic data by exploiting the sparsity and the structure of the hessian matrix a large number of iterations necessary to achieve convergence can be computed cheaply the results on a structurally complex model show the improvements of the lsi image versus the migrated image

ground state solutions of hamiltonian elliptic systems in dimension two the aim of this paper is to study hamiltonian elliptic system of the form 0 1 where 2 is a bounded domain in the second place we present existence results for the following stationary schrodinger systems defined in the whole plane 0 2 we assume that the nonlinearities f g have critical growth in the sense of trudinger moser by using a suitable variational framework based on the generalized nehari manifold method we obtain the existence of ground state solutions of both systems 0 1 and 0 2

the existence of solutions for superquadratic hamiltonian elliptic systems on rn abstract this paper is concerned with the following hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where v x c r n r and h x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we obtain infinitely many solutions for the above system

accelerating extended least squares migration with weighted conjugate gradient iteration abstractleast squares migration lsm iteratively achieves a mean square best fit to seismic reflection data provided that a kinematically accurate velocity model is available the subsurface offset extension adds extra degrees of freedom to the model thereby allowing lsm to fit the data even in the event of significant velocity error this type of extension also implies additional computational expense per iteration from crosscorrelating source and receiver wavefields over the subsurface offset and therefore places a premium on rapid convergence we have accelerated the convergence of extended least squares migration by combining the conjugate gradient algorithm with weighted norms in range data and domain model spaces that render the extended born modeling operator approximately unitary we have developed numerical examples that demonstrate that the proposed algorithm dramatically reduces the number of iterations required to achieve a given level of fit or gradient reduction compared with conjuga

multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical sobolev exponents abstract in this paper we study the existence of multiple positive solutions to some hamiltonian elliptic systems v u u p ef x u v v q g x in u v 0 in u v 0 on where is a bounded domain in r n n 3 0 f g l 1 p 1 1 q 1 n 2 n p q 1 0 using sub and supersolution method and based on an adaptation of the dual variational approach we prove the existence of at least two nontrivial positive solutions for all 0 1 and e 0 0 where 1 is the first eigenvalue of the laplace operator with zero dirichlet boundary conditions and 0 is a positive number

existence and multiplicity of semiclassical solutions for asymptotically hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system e 2 u v x u h u x u v in r n e 2 v v x v h v x u v in r n u x 0 and v x 0 as x where e 0 is a small parameter and the potential v is bounded below and h is asymptotically linear in z as z with z u v by applying a generalized linking theorem of strongly indefinite functionals we obtain the existence of multiple semiclassical solutions for the above system

least squares gaussian beam migration abstractleast squares migration lsm can be effective to mitigate the limitation of finite seismic acquisition balance the subsurface illumination and improve the spatial resolution of the image but it requires iterations of migration and demigration to obtain the desired subsurface reflectivity model the computational efficiency and accuracy of migration and demigration operators are crucial for applying the algorithm we have developed a test of the feasibility of using the gaussian beam as the wavefield extrapolating operator for the lsm denoted as least squares gaussian beam migration our method combines the advantages of the lsm and the efficiency of the gaussian beam propagator our numerical evaluations including two synthetic data sets and one marine field data set illustrate that the proposed approach could be used to obtain amplitude balanced images and to broaden the bandwidth of the migrated images in particular for the low wavenumber components

existence of solutions for a class of noncooperative elliptic systems we establish the existence of a nontrivial solution for a class of noncooperative elliptic systems with nonlinearities of superlinear growth moreover if there is resonance we also find at least a solution all results are obtained by the minimax methods in critical point theory

on strongly indefinite systems involving the fractional laplacian abstract this paper is devoted to study strongly indefinite systems involving the fractional laplacian on bounded domains explicitly we obtain existence and non existence results a p r i o r i estimates of gidas spruck type and a symmetry result in addition we give a different proof for the a p r i o r i estimate for nonlinear elliptic problems with the fractional laplacian obtained in cabre and tan 2010 and tan 2013

existence and uniqueness of positive radial solutions for the lane emden system in this article we consider component wise positive radial solutions of a weakly coupled system of elliptic equations in a ball with homogeneous nonlinearities the existence is well known in general we give a result for the remaining cases the uniqueness is less studied we complement the known results

liouville type theorems monotonicity results and a priori bounds for positive solutions of elliptic systems this paper is a contribution to the study of boundary value problems for systems of elliptic partial differential equations

existence of solutions for nonperiodic superquadratic hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u r v x u v in r n v v x v r u x u v in r n u x 0 and v x 0 as x where r x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we prove the existence of solution for the above system

multiple solutions of a coupled nonlinear schrodinger system abstract we will consider the relation between the number of positive standing waves solutions for a class of coupled nonlinear schrodinger system in r n and the topology of the set of minimum points of potential v x the main characteristics of the system are that its functional is strongly indefinite at zero and there is a lack of compactness in r n combining the dual variational method with the nehari technique and using the concentration compactness lemma we obtain the existence of multiple solutions associated to the set of global minimum points of the potential v x for sufficiently small in addition our result gives a partial answer to a problem raised by sirakov about existence of solutions of the perturbed system

block row recursive least squares migration abstractrecursive estimates of large systems of equations in the context of least squares fitting is a common practice in different fields of study for example recursive adaptive filtering is extensively used in signal processing and control applications the necessity of solving least squares problems recursively stems from the need for fast real time signal processing strategies the computational cost of least squares solvers can also limit the applicability of this technique in geophysical problems we have considered a recursive least squares solution for least squares wave equation migration with sliding windows involving several fixed rank downdating and updating computations this technique can be applied for dynamic and stationary processes if we use enough data in each windowed setup the spectrum of the preconditioned system is clustered around one and the method will converge superlinearly with probability one numerical experiments were performed to test the effectiveness of the technique

gap metric robustness analysis of linear periodically time varying feedback systems quantitative robust stability results are established in this paper for feedback systems that evolve in continuous time and exhibit linear periodically time varying lptv behavior the results presented are analogous to results known to hold for linear time invariant lti systems although they do not follow directly from these system uncertainty is measured using the gap metric which quantifies the distance between systems in terms of the aperture between their graphs the subspaces corresponding to all input output pairs for each system the main robustness result characterizes the largest gap ball of lptv plants stabilized by a nominal lptv feedback controller known to stabilize a nominal lptv plant at the center of this ball a key step in the proof of this result makes use of a formula derived for the directed gap between lptv systems this formula is essentially a generalization of georgiou s for lti systems importantly all of the results presented apply to a class of sampled data control systems as a special case

flexibly preconditioned extended least squares migration in shot record domain abstractwe have developed a method for accelerating the convergence of iterative least squares migration the algorithm uses a pseudodifferential scaling dip and spatially varying filter preconditioner together with a variant of conjugate gradient cg iteration with iterate dependent flexible preconditioning the migration is formulated without the image stack thus producing a shot dependent image volume that retains offset information useful for velocity updating and amplitude variation with offset analysis numerical experiments indicate that flexible preconditioning with pseudodifferential scaling not only attains considerably smaller data misfit and gradient error for a given computational effort but also produces higher resolution image volumes with more balanced amplitude and fewer artifacts than is achieved with a nonpreconditioned cg method

a unified approach to acoustical reflection imaging ii the inverse problem using the forward matrix model as derived in part i a j berkhout j acoust soc am 93 xxx 1993 it is shown that the first and main part of numerical acoustic imaging consists of a wave field extrapolation process by double matrix inversion physically the wave field extrapolation process means that the downward propagation effects and the upward propagation effects are eliminated from the measurements next the reflection information is extracted from the wave field extrapolation result optionally the reflection information is translated to discipline oriented material parameters by some data fitting process double focusing i e focusing in emission and focusing in detection is closely related to the above numerical imaging process finally it is shown that imaging of zero offset or puls echo data can be formulated by single matrix inversion involving phase shifts only

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

the rabinowitz floer homology for a class of semilinear problems and applications abstract in this paper we construct a rabinowitz floer type homology for a class of non linear problems having a starshaped potential we consider some equivariant cases as well we give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model equations

asymptotically linear elliptic systems in rn abstract in this paper we show that semilinear elliptic systems of the form u u v v g v v v u u f u x r n possess at least one nontrivial solution pair u v h 1 r n h 1 r n where and are positive numbers f t and g t are continuous functions on r and asymptotically linear as t

an iterative inversion of back scattered acoustic waves the application of the born approximation to the scattered wave field followed by a wkbj and far field approximation on the propagation green s function for a slowly varying background medium leads to a simple integral relation between the density and bulk modulus anomalies superimposed on the background medium and the scattered wave field an iterative inversion scheme based on successive back projections of the wave field is used to reconstruct the two acoustic parameters the scheme when applied to data generated using the direct integral relation shows that the variations of the parameters can be reconstructed the procedure is readily applicable to actual data since every iterative step is essentially a prestack kirchhoff migration followed by the application of the direct born approximation and far field operator

strongly indefinite functionals and multiple solutions of elliptic systems we study existence and multiplicity of solutions of the elliptic system left begin array 20 l updelta u h u x u v hfill text in upomega hfill updelta v h v x u v hfill text in upomega quad u x v x 0 quad text on partial upomega hfill end array right where upomega subset mathbb r n n ge 3 is a smooth bounded domain and h in mathcal c 1 overline upomega times mathbb r 2 mathbb r we assume that the nonlinear term h x u v sim left u right p left v right q r x u v text with mathop lim limits left u v right to infty frac r x u v left u right p left v right q 0 where p in 1 2 2 2n n 2 text and q in 1 infty so some supercritical systems are included nontrivial solutions are obtained when h x u v is even in u v we show that the system possesses a sequence of solutions associated with a sequence of positive energies resp negative energies going toward infinity resp zero if p 2 resp p 2 all results are proved using variational methods some new critical point theorems for strongly indefinite functionals are proved