high order finite difference simulations of marine csem surveys using a correspondence principle for wave and diffusion fields the computer time required to solve a typical 3d marine controlled source electromagnetic surveying csem simulation can be reduced by more than one order of magnitude by transforming low frequency maxwell equations in the quasi static or diffusive limit to a hyperbolic set of partial differential equations that give a representation of electromagnetic fields in a fictitious wave domain the dispersion and stability analysis can be made equivalent to that of other types of wave simulation problems such as seismic acoustic and elastic modeling second order to eighth order spatial derivative operators are implemented for flexibility fourth order and sixth order methods are the most numerically efficient implementations for this particular scheme an implementation with high order operators requires that both electric and magnetic fields are extrapolated simultaneously into the air layer the stability condition given for high order staggered derivative operators here should be equally valid for seismic wave simulation the bandwidth of recovered fields in the diffusive domain is independent of the bandwidth of the fields in the fictitious wave domain the fields in the fictitious wave domain do not represent observable fields propagation paths and interaction reflection amplitudes are not altered by the transform from the fictitious wave domain to the diffusive frequency domain however the transform contains an exponential decay factor that damps down late arrivals in the fictitious wave domain the propagation paths that contribute most to the diffusive domain fields are airwave shallow water plus typically postcritical events such as refracted and guided waves the transform from the diffusive frequency domain to the fictitious wave domain is an ill posed problem the transform is nonunique this gives a large degree of freedom in postulating temporal waveforms for boundary conditions in the fictitious wave domain that reproduce correct diffusive frequency domain fields

limit cycles of cubic polynomial differential systems with rational first integrals of degree 2 the main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all cubic polynomial differential systems using the averaging theory the computations of this work have been made with mathematica and maple

bifurcation of limit cycles from an n dimensional linear center inside a class of piecewise linear differential systems abstract let n be an even integer we study the bifurcation of limit cycles from the periodic orbits of the n dimensional linear center given by the differential system x 1 x 2 x 2 x 1 x n 1 x n x n x n 1 perturbed inside a class of piecewise linear differential systems our main result shows that at most 4 n 6 n 2 1 limit cycles can bifurcate up to first order expansion of the displacement function with respect to a small parameter for proving this result we use the averaging theory in a form where the differentiability of the system is not needed

full waveform inversion and the truncated newton method full waveform inversion fwi is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield this method consists in minimizing the distance between

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

numerical resolution of an electromagnetic inverse medium problem at fixed frequency the aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency the governing equations are the time harmonic maxwell equations formulated in electric field in a two dimensional bounded domain we express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields our numerical reconstruction algorithm combines the bfgs method and an iterative process called the adaptive eigenspace inversion the unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator both the eigenspace and the mesh are iteratively adapted during the minimization procedure numerical experiments illustrate the performance of the reconstruction for various configurations

chern forms of singular metrics on vector bundles we study singular hermitian metrics on holomorphic vector bundles following berndtsson p un previous work by raufi has shown that for such metrics it is in general not possible to define the curvature as a current with measure coefficients in this paper we show that despite this under appropriate codimension restrictions on the singular set of the metric it is still possible to define chern forms as closed currents of order 0 with locally finite mass

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

efficient calculation of a partial derivative wavefield using reciprocity for seismic imaging and inversion linearized inversion of surface seismic data for a model of the earth s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth s subsurface this sensitivity or jacobian matrix is usually quite expensive to estimate for all but the simplest model parameterizations we exploit the numerical structure of the finite element method modern sparse matrix technology and source receiver reciprocity to develop an algorithm that explicitly calculates the jacobian matrix at only the cost of a forward model solution furthermore we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the hessian matrix as predicted by the classical gauss newton method our method is applicable to the full suite of wave scattering problems amenable to finite element forward modeling we demonstrate our method through some simple 2 d synthetic examples

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

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

fast migration inversion with multivalued ray fields part 1 method validation test and application in 2d to marmousi we address the problem of building an efficient algorithm for ray based migration inversion in complex media characterized by multivalued traveltime since the multivalued ray maps can be computed by wavefront construction the key point lies in the interpolation of these multivalued coarse maps for this goal we propose two new strategies the first one is based on an interpolation of the rayfield after identifying the ray branches on the elementary cells of the coarse grid the second is based on an extrapolation of the arrivals from the coarse grid points both strategies are evaluated in two dimensions with the marmousi data set demonstrating the cpu efficiency and accuracy of both approaches since they both consider multiarrivals locally the associated overhead cost with respect to a single arrival implementation is quite low twice the computing time of the marmousi test case the application also reveals that the new interpolation strategy is more efficient in terms of cpu time compared to the extrapolation strategy but at the expense of a more complex numerical implementation and less accurate results

multiscale adjoint waveform difference tomography using wavelets abstractfull waveform seismic inversions based on minimizing the distance between observed and predicted seismograms are in principle able to yield better resolved earth models than those minimizing misfits derived from traveltimes alone adjoint based methods provide an efficient way of calculating the gradient of the misfit function via a sequence of forward modeling steps which using spectral element codes can be carried out in realistically complex media convergence and stability of full waveform difference adjoint schemes are greatly improved when data and synthetics are progressively presented to the algorithms in a constructive multiscale approximation using a bi orthogonal wavelet transform wavelets provide the nonredundant spectral decomposition that paves the way for the inversion to proceed successively from long wavelength fitting to detailed exploration of the phases in the seismogram the choice of wavelet class and type the initial depth of the multiscale decomposition and the min

finite difference time dispersion transforms for wave propagation the finite difference fd wave equation is widely implemented in seismic imaging for oil exploration but the numerical dispersion due to discretization of time and space derivatives can introduce severe numerical errors we have investigated time dispersion which might distort the phase and introduce severe artifacts to the data and images especially for long time propagation we first studied precisely how the time dispersion was produced and then we developed an efficient approach the time dispersion transforms to cope with time dispersion errors in wave propagation the proposed forward time dispersion transform can predict or simulate the time dispersion errors and it can be applied to remove time dispersion errors in reverse time migration the inverse time dispersion transform can be used to eliminate time dispersion errors from synthetic data after modeling the proposed method is general and works for low and high order time fd schemes by using this method large time steps were allowed in wave propagation thus we can remove the time dispersion errors at a negligible numerical cost and the efficiency is not affected to test the effectiveness and robustness of the proposed time dispersion transforms we used isotropic and anisotropic examples as well as modeling and imaging examples

multiscale seismic waveform inversion iterative inversion methods have been unsuccessful at inverting seismic data obtained from complicated earth models e g the marmousi model the primary difficulty being the presence of numerous local minima in the objective function the presence of local minima at all scales in the seismic inversion problem prevent iterative methods of inversion from attaining a reasonable degree of convergence to the neighborhood of the global minimum the multigrid method is a technique that improves the performance of iterative inversion by decomposing the problem by scale at long scales there are fewer local minima and those that remain are further apart from each other thus at long scales iterative methods can get closer to the neighborhood of the global minimum we apply the multigrid method to a subsampled low frequency version of the marmousi data set although issues of source estimation source bandwidth and noise are not treated results show that iterative inversion methods perform much better when employed with a decomposition by scale furthermore the method greatly reduces the computational burden of the inversion that will be of importance for 3 d extensions to the method

kernel principal component analysis for stochastic input model generation stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used these input models are often constructed from a set of experimental samples of the underlying random field to this end the karhunen loeve k l expansion also known as principal component analysis pca is the most popular model reduction method due to its uniform mean square convergence however it only projects the samples onto an optimal linear subspace which results in an unreasonable representation of the original data if they are non linearly related to each other in other words it only preserves the first order mean and second order statistics covariance of a random field which is insufficient for reproducing complex structures this paper applies kernel principal component analysis kpca to construct a reduced order stochastic input model for the material property variation in heterogeneous media kpca can be considered as a nonlinear version of pca through use of kernel functions kpca further enables the preservation of higher order statistics of the random field instead of just two point statistics as in the standard karhunen loeve k l expansion thus this method can model non gaussian non stationary random fields in this work we also propose a new approach to solve the pre image problem involved in kpca in addition polynomial chaos pc expansion is used to represent the random coefficients in kpca which provides a parametric stochastic input model thus realizations which are statistically consistent with the experimental data can be generated in an efficient way we showcase the methodology by constructing a low dimensional stochastic input model to represent channelized permeability in porous media

waveform inversion by one way wavefield extrapolation one way riemannian wavefield extrapolation rwe on computational meshes conforming to the direction of turning wave propagation is an alternative forward modeling procedure for waveform inversion forward modeling tests demonstrate that the rwe approach may be a sufficiently accurate approximation for calculating the wavefield phases important for early arrival waveform inversion initial results indicate that rwe waveforms are well matched at wide offsets to finite difference data and can be used in a waveform inversion scheme to invert synthetic data for moderate 1d velocity perturbations

constrained full waveform inversion by model reparameterization the waveform inversion problem is inherently ill posed traditionally regularization schemes are used to address this issue for waveform inversion where the model is expected to have many details reflecting the physical properties of the earth regularization and data fitting can work in opposite directions the former smoothing and the latter adding details to the model we propose constraining estimated velocity fields by reparameterizing the model this technique also called model space preconditioning is based on directional laplacian filters it preserves most of the details of the velocity model while smoothing the solution along known geological dips preconditioning also yields faster convergence at early iterations the laplacian filters have the property to smooth or kill local planar events according to a local dip field by construction these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models we illustrate with 2d syn

a parallel sweeping preconditioner for heterogeneous 3d helmholtz equations a parallelization of a sweeping preconditioner for three dimensional helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models the setup and application costs of the sequential preconditioner are shown to be o gamma 2 n 4 3 and o gamma n log n where gamma omega denotes the modestly frequency dependent number of grid points per perfectly matched layer several computational and memory improvements are introduced relative to using black box sparse direct solvers for the auxiliary problems and competitive runtimes and iteration counts are reported for high frequency problems distributed over thousands of cores two open source packages are released along with this paper parallel sweeping preconditioner psp and the underlying distributed multifrontal solver clique

cascaded wavefield tomography and inversion using extended common image point gathers a case study abstractimage domain wavefield tomography wt exploits focusing characteristics of extended images for updating the velocity field to make good use of this information one must understand how such images behave if the migration velocity is accurate this is not trivial because focusing depends on not only the model error but also on the acquisition setup the data bandwidth and illumination variation caused by the overburden we address this problem by constructing penalty functions based on the point spread functions of the imaging operator that characterize focusing in extended images moreover instead of sampling the extended images at preset distances along the surface we sample the image by constructing common image point gathers which are more economical from a computational point of view and also measure the spatial and temporal focusing of the wavefield coupled with image residuals exploiting illumination based penalty functions we construct robust wavefield tomographic updates from spar

correlation based reflection full waveform inversion because modeling for full waveform inversion fwi cannot produce reflections unless the velocity model has the scattering potential high wavenumbers using a migration demigration process to generate modeling data which is a key step in what is now known as reflection fwi rfwi is a credible alternative to tackle the reflection nonlinearity associated with fwi however because rfwi depends on a conventional data residual or zero lag correlation objective function high nonlinearity can still exist when the true amplitude migration is not used as well as at far offsets due to cycle skipping to avoid the cycle skipping and the need for a true amplitude migration we have developed a correlation based reflection full waveform inversion method to update the low wavenumber components of the velocity model the success of this method relies on a sensitivity kernel decomposition and a correlation based objective function the sensitivity kernel decomposition makes it possible to separate out the contributions of different subkernels and to smear the reflected wave residuals along the rabbit ear wavepath to obtain middle and deep background model estimates the correlation based objective function measures differences in kinematic information and behaves in a more linear way than the traditional waveform residual misfit moreover our approach is less sensitive to the frequency content and amplitude information of the seismic data enabling reliable background velocity estimates to be obtained without the need for low frequencies and full physics modeling because the kinematic features of reflected waves are described correctly the inversion result of the proposed method can be used as a migration model or an initial model for conventional fwi to achieve a correct high wavenumber model update

seismic envelope inversion and modulation signal model abstractwe recognized that the envelope fluctuation and decay of seismic records carries ultra low frequency ulf i e the frequency below the lowest frequency in the source spectrum signals that can be used to estimate the long wavelength velocity structure we then developed envelope inversion for the recovery of low wavenumber components of media smooth background so that the initial model dependence of waveform inversion can be reduced we derived the misfit function and the corresponding gradient operator for envelope inversion to understand the long wavelength recovery by the envelope inversion we developed a nonlinear seismic signal model the modulation signal model as the basis for retrieving the ulf data and studied the nonlinear scale separation by the envelope operator to separate the envelope data from the wavefield data envelope extraction a demodulation operator envelope operator was applied to the waveform data numerical tests using synthetic data for the marmousi model pro

bergman kernels and the pseudoeffectivity of relative canonical bundles a criterion for the pseudoeffectivity of twisted relative canonical bundles of surjective projective maps

inversion gradients for acoustic vti wavefield tomography abstractwavefield tomography can handle complex subsurface geology better than ray based techniques and ultimately provide a higher resolution here we implement forward and adjoint wavefield extrapolation for vti transversely isotropic with a vertical symmetry axis media using a generalized pseudospectral operator based on a separable approximation for the p wave dispersion relation this operator is employed to derive the gradients of the differential semblance optimization dso and modified image power objective functions we also obtain the gradient expressions for a data domain objective function that can more easily incorporate borehole information necessary for stable vti velocity analysis these gradients are similar to the ones obtained with a space time finite difference fd scheme for a system of coupled wave equations but the pseudospectral method is not hampered by the imprint of the shear wave artifact numerical examples also show the potential advantages of the modified image power

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

imaging of local surface displacement on an infinite ground plane the multiple frequency case this paper is concerned with the inverse electromagnetic scattering from a perfectly conducting local surface displacement on an infinite ground plane by using multiple frequency data to image the profile of the surface displacement a stable and accurate reconstruction method is presented and investigated with numerical simulations an integral equation approach is also introduced for the solution of the forward scattering problem by recovering the main features at low frequencies and capturing the fine details at high frequencies the proposed method yields a stable and convergent reconstruction of the general profile in particular for a multiple scale profile the method resolves the fine scales with sufficiently high frequency information numerical examples are provided to illustrate the effectiveness of our approach

wave field reciprocity and optimization in remote sensing a unified approach to local optimization techniques and wave field reciprocity as applied to constructing solutions to remote sensing imaging and inversion problems in acoustic elastic and electromagnetic wave theory is presented the starting point is a system of linear first order partial differential equations in space time of the class to which the indicated wave phenomena gives rise for this system three types of remote sensing problems the inverse source the inverse scattering and the inverse transducted wave field problems are formulated and the construction of their solutions via local optimization techniques is discussed emphasis is placed on iterative algorithms that are based on a guaranteed decrease in the mismatch between modelled and observed data at each update of the medium subsequently the wave field reciprocity theorems of the time convolution and the time correlation types are derived and their occurrence in the optimization procedures is discussed also attention is paid to approximate methods in particular to the rayleigh gans born approximation approximations of this sort provide the means to invoke the method of preconditioning in the process of inverting the operator equations exotic media for example chiral media in electromagnetics are included in the analysis

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

velocity model building by 3d frequency domain full waveform inversion of wide aperture seismic data we assessed 3d frequency domain fd acoustic fullwaveforminversionfwidataasatooltodevelophigh resolution velocity models from low frequency global offset data the inverse problem was posed as a classic leastsquares optimization problem solved with a steepest descent method inversion was applied to a few discrete frequencies allowing management of a limited subset of the 3d data volume the forward problem was solved with a finite difference frequency domain method based on a massively paralleldirectsolver allowingefficientmultiple shotsimulations the inversion code was fully parallelized for distributedmemory platforms taking advantage of a domain decomposition of the modeled wavefields performed by the direct solver after validation on simple synthetic tests fwi was applied to two targets channel and thrust system of the 3d seg eageoverthrustmodel correspondingto3ddomains of 78 752 25 km and 13 513 54 65 km respectively the maximum inverted frequencies are 15 and 7 hz for the two applications amaximum of 30 dual core biprocessor nodes with 8 gb of shared memory per node were used for the second target the main structures were imaged successfully at a resolution scale consistent with the inverted frequencies our study confirms the feasibility of 3d frequency domain fwi of global offset data on large distributed memory platforms to develop high resolution velocity models these high velocity models may provide accurate macromodelsforwave equationprestackdepthmigration

inexact interior point method for pde constrained nonlinear optimization starting from the inexact interior point framework from curtis schenk and wachter siam j sci comput 32 2012 pp 3447 3475 we propose an effective schur complement slack control preconditioner for the full lagrangian hessian matrix needed at each newton iteration together they yield a scalable robust and highly parallel method for the numerical solution of large scale nonconvex pde constrained optimization problems with inequality constraints because it uses the full hessian matrix modifying it whenever needed the method not only is globally convergent but also converges fast locally our preconditioner is not tailored to any particular class of pdes or constraints but instead judiciously exploits the sparsity structure of the hessian numerical examples from pde constrained optimal control parameter estimation and full waveform inversion demonstrate the robustness and efficiency of the method even in the presence of active inequality constraints

linearized wave equation migration velocity analysis by image warping seismic imaging produces images of contrasts in physical parameters in the subsurface e g velocity or impedance to build such images a background model describing the wave kinematics in the earth is necessary in practice the structural image and background velocity model are unknown and have to be estimated from the acquired data migration velocity analysis deals with estimation of the background model in the framework of seismic migration and relies on two main elements data redundancy and invariance of the structures with respect to different seismic experiments because all the experiments probe the same model the reflectors must be invariant in suitable domains e g shots or reflection angle the semblance principle is the tool used to measure the invariance of a set of multiple images we measure the similarity of the structural features between pairs of single shot migrated images obtained from adjacent experiments by using the estimated warping vector field between two migrated images we construct an image perturbation which describes the difference in reflectivity observed by two shots we derive an expression for the image perturbation that drives a migration velocity analysis procedure based on a linearization of the wave equation with respect to the model parameters synthetic 2d examples show promising results in retrieving errors in the velocity model this methodology can be directly applied to 3d

segre forms and kobayashi lubke inequality starting from the description of segre forms as direct images of powers of the first chern form of the anti tautological line bundle on the projectivized bundle of a holomorphic hermitian vector bundle we derive a version of the pointwise kobayashi lubke inequality

fast interior point solution of quadratic programming problems arising from pde constrained optimization interior point methods provide an attractive class of approaches for solving linear quadratic and nonlinear programming problems due to their excellent efficiency and wide applicability in this paper we consider pde constrained optimization problems with bound constraints on the state and control variables and their representation on the discrete level as quadratic programming problems to tackle complex problems and achieve high accuracy in the solution one is required to solve matrix systems of huge scale resulting from newton iteration and hence fast and robust methods for these systems are required we present preconditioned iterative techniques for solving a number of these problems using krylov subspace methods considering in what circumstances one may predict rapid convergence of the solvers in theory as well as the solutions observed from practical computations

full waveform inversion with extrapolated low frequency data abstractthe availability of low frequency data is an important factor in the success of full waveform inversion fwi in the acoustic regime the low frequencies help determine the kinematically relevant low wavenumber components of the velocity model which are in turn needed to avoid convergence of fwi to spurious local minima however acquiring data less than 2 or 3 hz from the field is a challenging and expensive task we have explored the possibility of synthesizing the low frequencies computationally from high frequency data and used the resulting prediction of the missing data to seed the frequency sweep of fwi as a signal processing problem bandwidth extension is a very nonlinear and delicate operation in all but the simplest of scenarios it can only be expected to lead to plausible recovery of the low frequencies rather than their accurate reconstruction even so it still requires a high level interpretation of band limited seismic records into individual events each of which can be extr

adaptive eigenspace for multi parameter inverse scattering problems abstract a nonlinear optimization method is proposed for inverse scattering problems in the frequency domain when the unknown medium is characterized by one or several spatially varying parameters the time harmonic inverse medium problem is formulated as a pde constrained optimization problem and solved by an inexact truncated newton type method combined with frequency stepping instead of a grid based discrete representation each parameter is projected to a separate finite dimensional subspace which is iteratively adapted during the optimization each subspace is spanned by the first few eigenfunctions of a linearized regularization penalty functional chosen a priori the small and slowly increasing finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard tikhonov type regularization numerical results illustrate the accuracy and efficiency of the resulting adaptive eigenspace regularization for single and multi parameter problems including the well known marmousi model from geosciences

regularized seismic full waveform inversion with prior model information full waveform inversion fwi delivers high resolution quantitative images and is a promising technique to obtain macroscale physical property model of the subsurface in most geophysical applications prior information such as that collected in wells is available and should be used to increase the image reliability for this we propose to introduce three terms in the definition of the fwi misfit function the data misfit itself the first order tikhonov regularization term acting as a smoothing operator and a prior model norm term this last term is the way to smoothly introduce prior information into the fwi workflow on a selected target of the marmousi synthetic example significant improvement was obtained when using the prior model term for noise free and noisy synthetic data the prior model term may significantly reduce the inversion sensi tivity to incorrect initial conditions the limited range of spatial wavenumber sampling by the acquisition may be compensated with the prior model information for multiple free and multiplecontaminated data prior and initial models play different roles in the inversion scheme the starting model is used for wave propagation and therefore drives the data misfit gradient whereas the prior model is never explicitly used for solving the wave equation and only drives the optimization step as an additional constraint to minimize the total objective function thus the prior model is not required to follow kinematic properties as precisely as the initial model except in zones of poor illumination in addition we investigate the influence of a simple dynamic decreasing weighting of the prior model term once the cycle skipping problem has been solved the impact of the prior model term is gradually reduced within the misfit function to be driven by seismic data only

computations of bott chern classes on ℙ e we compute the bott chern classes of the metric eulersequence describing the relative tangent bundle of the variety p e of hyperplans of a holomorphic hermitian vector bundle e h on a complex manifold we give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle and to the computation of the height of p e with respect to the tautological quotient bundle o e 1

limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n the subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter this upper bound is reached for proving these results we use the averaging theory in a form where the differentiability of the system is not needed

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

optimal solvers for pde constrained optimization optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering particularly in problems of design the solution of such pde constrained optimization problems is usually a major computational task here we consider simple problems of this type distributed control problems in which the 2 and 3 dimensional poisson problem is the pde the large dimensional linear systems which result from discretization and which need to be solved are of saddle point type we introduce two optimal preconditioners for these systems which lead to convergence of symmetric krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem these preconditioners are block structured and involve standard multigrid cycles the optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases the theoretical proof indicates that these approaches may have much broader applicability for other pdes

an inexact newton method for nonconvex equality constrained optimization we present a matrix free line search algorithm for large scale equality constrained optimization that allows for inexact step computations for strictly convex problems the method reduces to the inexact sequential quadratic programming approach proposed by byrd et al siam j optim 19 1 351 369 2008 for nonconvex problems the methodology developed in this paper allows for the presence of negative curvature without requiring information about the inertia of the primal dual iteration matrix negative curvature may arise from second order information of the problem functions but in fact exact second derivatives are not required in the approach the complete algorithm is characterized by its emphasis on sufficient reductions in a model of an exact penalty function we analyze the global behavior of the algorithm and present numerical results on a collection of test problems

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