a stream virtual element formulation of the stokes problem on polygonal meshes in this paper we propose and analyze a novel stream formulation of the virtual element method vem for the solution of the stokes problem the new formulation hinges upon the introduction of a suitable stream function space characterizing the divergence free subspace of discrete velocities and it is equivalent to the velocity pressure inf sup stable mimetic scheme presented in l beirao da veiga et al j comput phys 228 2009 pp 7215 7232 up to a suitable reformulation into the vem framework both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

the schur algorithm applied to the one dimensional continuous inverse scattering problem the one dimensional continuous inverse scattering problem can be solved by the schur algorithm in the discrete time domain using sampled scattering data the sampling rate of the scattering data should be increased to reduce the discretization error but the complexity of the schur algorithm is proportional to the square of the sampling rate to improve this tradeoff between the complexity and the accuracy we propose a schur algorithm with the richardson extrapolation sare the asymptotic expansion of the schur algorithm necessary for the richardson extrapolation is derived in powers of the discretization step which shows that the accuracy order with respect to the discretization step of the schur algorithm is 1 the accuracy order of the sare with the n step richardson extrapolation is increased to n 1 with comparable complexity to the schur algorithm therefore the discretization error of the schur algorithm can be decreased in a computationally efficient manner by the sare

convergent adaptive finite elements for the nonlinear laplacian the numerical solution of the homogeneous dirichlet problem for the p laplacian p in 1 infty is considered we propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate solutions converge to the exact one while the algorithm is a rather straight forward generalization of those for the linear case p 2 the proof of its convergence is different in particular it does not rely on a strict error reduction

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

convergence analysis of the mimetic finite difference method for elliptic problems we propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes the a priori error analysis relies on the connection between the mimetic formulation and the lowest order raviart thomas mixed finite element method the theoretical results are confirmed by numerical experiments

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

wave splitting and the reflection operator for the wave equation in r3 the problem of wave splitting in a nonhomogeneous medium in r3 is considered previous results for wave splitting in a planar stratified medium can be generalized to a general nonhomogeneous medium with sufficiently smooth velocity the wave equation is factorized into an up and down going wave system using certain integral and integral differential operators the equation for the reflection operator which relates the up going wave to a down going wave is then obtained and certain properties of the reflection operator are deduced

inverse scattering in multimode structures we consider the inverse scattering problem associated with any number of interacting modes in one dimensional structures the coupling between the modes is contradirectional in addition to codirectional and may be distributed continuously or in discrete points the local coupling as a function of position is obtained from reflection data using a layer stripping type method and the separate identification of the contradirectional and codirectional coupling is obtained using matrix factorization ambiguities are discussed in detail and different a priori information that can resolve the ambiguities is suggested the method is exemplified by applications to multimode optical waveguides with quasi periodical perturbations

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

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

residual based error estimate and quasi interpolation on polygonal meshes for high order bem based fem abstract only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes the bem based finite element method is one of the new discretization strategies which make use of and benefits from the flexibility of these general meshes that incorporate hanging nodes naturally the article in hand addresses quasi interpolation operators for the approximation space over polygonal meshes to prove interpolation estimates the poincare constant is bounded uniformly for patches of star shaped elements these results give rise to the residual based error estimate for high order bem based fem and its reliability as well as its efficiency are proven such a posteriori error estimates can be used to gauge the approximation quality and to implement adaptive fem strategies numerical experiments show optimal rates of convergence for meshes with non convex elements on uniformly as well as on adaptively refined meshes

poisson equation on complete manifolds abstract we develop heat kernel and green s function estimates for manifolds with positive bottom spectrum the results are then used to establish existence and sharp estimates of the solution to the poisson equation on such manifolds with ricci curvature bounded below as an application we show that the curvature of a steady gradient ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above

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

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

factorization of the wave equation in higher dimensions the factorization of the wave equation into a coupled system involving up and down going wave components is obtained for the case where the field quantities are multivariate functions of spatial variables but the velocity c is a function of the z variable only the form of the reflection operator is derived and the quadratic differential integral equation satisfied by its kernel is obtained

estimation of acoustic macro models using a genetic full waveform inversion applications to the marmousi model abstractwe have developed a stochastic full waveform inversion that uses genetic algorithms ga fwi to estimate acoustic macro models of the p wave velocity field stochastic methods such as ga severely suffer the curse of dimensionality meaning that they require unaffordable computer resources for inverse problems with many unknowns and expensive forward modeling to mitigate this issue we have proposed a two grid technique with a coarse grid to represent the subsurface for the ga inversion and a finer grid for the forward modeling we have applied this procedure to invert synthetic acoustic data of the marmousi model and we have developed three different tests the first two tests use a velocity model derived from standard stacking velocity analysis as prior information and differ only for the parameterization of the coarse grid their comparison indicates that a smart parameterization of the coarse grid may significantly improve the final result the third test uses a linearly increasing 1d velocit

on green functions of second order elliptic operators on riemannian manifolds the critical case abstract let p be a second order linear elliptic operator with real coefficients which is defined on a noncompact and connected riemannian manifold m it is well known that the equation p u 0 in m admits a positive supersolution which is not a solution if and only if p admits a unique positive minimal green function on m and in this case p is said to be subcritical in m if p does not admit a positive green function but admits a global positive super solution then such a solution is called a ground state of p in m and p is said to be critical in m we prove for a critical operator p in m the existence of a green function which is dominated above by the ground state of p away from the singularity moreover in a certain class of green functions such a green function is unique up to an addition of a product of the ground states of p and p under some further assumptions we describe the behavior at infinity of such a green function this result extends and sharpens the celebrated result of p li and l f tam concerning the existence of a symmetric green function for the laplace beltrami operator on a smooth and complete riemannian manifold m

levinson like and schur like fast algorithms for solving block slanted toeplitz systems of equations arising in wavelet based solution of integral equations the wiener hopf integral equation of linear least squares estimation of a wide sense stationary random process and the krein integral equation of one dimensional 1 d inverse scattering are fredholm equations with symmetric toeplitz kernels they are transformed using a wavelet based galerkin method into a symmetric block slanted toeplitz bst system of equations levinson like and schur like fast algorithms are developed for solving the symmetric bst system of equations the significance of these algorithms is as follows if the kernel of the integral equation is not a calderon zygmund operator the wavelet transform may not sparsify it the kernel of the krein and wiener hopf integral equations does not in general satisfy the calderon zygmund conditions as a result application of the wavelet transform to the integral equation does not yield a sparse system matrix there is therefore a need for fast algorithms that directly exploit the symmetric block slanted toeplitz structure of the system matrix and do not rely on sparsity the first such o n sup 2 algorithms viz a levinson like algorithm and a schur 1917 like algorithm are presented these algorithms are also applied to the factorization of the bst system matrix the levinson like algorithm also yields a test for positive definiteness of the bst system matrix the results obtained are directly applicable to the problem of constrained deconvolution of a nonstationary signal where the locations of the smooth regions of the signal being deconvolved are known a priori

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

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

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

inverse scattering for a layered acoustic medium using the first order equations of motion the inverse scattering problem for a layered acoustic medium is considered from the first order differential equations of motion resulting in a vector formulation of the problem and using a vector form of the schrodinger inverse scattering methods the result is a vector marchenko equation the differentiability constraints on the acoustic impedance are somewhat relaxed compared to the more standard approach of beginning with the wave equation the solution for plane waves at normal incidence is given along with a good approximate solution which is easily obtainable and takes into account transmission losses not included in the normal wkbj born approximation a new solution for extracting separately the velocity and density of the medium using the reflection response for two different angles of incidence is given which involves a nonlinear integral equation to relate the apparent traveltimes to depth

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

a simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows this paper develops an posteriori error estimate for lowest order locally conservative methods on meshes consisting of very general polytopal elements we focus on the ease of implementation of the methodology based on h1 conforming potential reconstruction and h div conforming flux reconstruction in particular the evaluation of our estimates merely consists in some local matrix vector multiplications where on each mesh element the matrices are either directly inherited from the given numerical method or trivially constructed from the element geometry while the vectors are the degrees of freedom on the given element we then apply this methodology to unsteady nonlinear coupled degenerate problems describing complex multiphase flows in porous media here on each step of the time marching scheme linearization procedure and linear algebraic solver we distinguish the corresponding error components this leads to an easy to implement and fast to run adaptive algorithm with simultaneously guaranteed overall precision and optimal efficiency ensured through the use of adaptive stopping criteria together with adaptive space and time mesh refinements numerous numerical experiments on practical problems in two and three space dimensions illustrate the performance of our methodology

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

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

a wave splitting approach to time dependent inverse scattering for the stratified cylinder this paper examines the inverse problem for the stratified inhomogeneous cylinder in which the unknown velocity function is recovered from reflection data the analysis is based on weston s theoretical wave splitting for stratified three dimensional media j math phys 28 1987 pp 1061 1068 j math phys 29 1988 pp 36 45 the forms of the splitting operators are presented and the inverse problem is solved numerically by means of the r equation an integro differential equation that characterizes the scattering medium independently of the wave field although the r equation is singular at one round trip time corresponding to a caustic when a radially incoming cylindrical wave reaches the z axis several numerical examples indicate that the velocity function can be reconstructed quite accurately

an extension of the discrete variational method to nonuniform grids the discrete variational method is a method used to derive finite difference schemes that inherit the conservation dissipation property of the original equations although this method has mainly been developed for uniform grids we extend this method to multidimensional nonuniform meshes

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

boundary control wave field continuation and inverse problems for the wave equation abstract we consider the problem of the wave field continuation and recovering of coefficients for the wave equation in a bounded domain in r n n 1 the inverse data is a response operator mapping neumann boundary data into dirichlet ones the reconstruction procedure is local this means that observing boundary response for larger times we may recover coefficients deeper in the domain the approach is based upon ideas and results of the boundary control theory yielding some natural multidimensional analogs of the classical gel fand levitan krein equations

a reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks abstract in this paper we propose a reduced basis hybrid method rbhm for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent the rbhm is applied to stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled the rbhm is built upon the reduced basis element method rbem and it takes advantage from both the reduced basis methods rb and the domain decomposition method we move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes on the latter representative solutions corresponding to the same governing partial differential equations are computed for different values of some parameters of interest representing for example the deformation of the blocks a generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration the desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions the geometrical parametrization of the domain by transfinite mapping induces non affine parameter dependence an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline online decomposition of the reduced basis procedure this computational decomposition yields a considerable reduction of the problem complexity results computed on some combinations of 2d and 3d geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses pressure and velocity at subdomain interfaces

the connection between time and frequency domain three dimensional inverse scattering methods we relate three dimensional scattering theory for the time independent schrodinger equation without spherical symmetry to scattering theory for the plasma wave equation pwe we review a recent inverse scattering method for the pwe and find the corresponding method for the schrodinger equation we then review newton s three dimensional marchenko method for the schrodinger equation and transform it to the corresponding pwe method the resulting time domain hyperbolic method clarifies the role of causality in newton s important recent work

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

a formulation for higher dimensional inverse problems for the wave equation abstract two particular inverse scattering problems are of special interest the first concerns the discovery of a perturbation in the speed of sound by analyzing the return signal from a blast wave set off above it the second concerns the determination of a potential from the back scattering of plane waves in one dimensional problems the two cases are very closely related in higher dimensions the situation becomes much more complicated we present here a new approach to four such classical higher dimensional inverse problems for determining the coefficients of a partial differential equation including the two mentioned the idea stems from the work of deift and trubowitz 1 in one space dimension no conclusive theorem is found but the approach provides an algorithm for iteration which might lead to an existence theorem and which will be explored numerically elsewhere

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

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

arbitrary order nodal mimetic discretizations of elliptic problems on polygonal meshes we develop and analyze a new family of mimetic methods on unstructured polygonal meshes for the diffusion problem in primal form these methods are derived from the local consistency condition that is exact for polynomials of any degree m geq1 the degrees of freedom are a solution values at the quadrature nodes of the gauss lobatto formulas on each mesh edge and b solution moments inside polygons the convergence of the method is proven theoretically and an optimal error estimate is derived in a mesh dependent norm that mimics the energy norm numerical experiments confirm the convergence rate that is expected from the theory

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

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

a posteriori error estimates for elliptic problems in two and three space dimensions let u in h be the exact solution of a given selfadjoint elliptic boundary value problem which is approximated by some tilde u in mathcal s mathcal s being a suitable finite element space efficient and reliable a posteriors estimates of the error u tilde u measuring the local quality of tilde u play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh a well known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques in this paper we provide a guideline for the theoretical analysis of such error estimates we further clarify the relation to other concepts our analysis leads to new error estimates which are specially suited to three space dimensions the theoretical results are illustrated by numerical computations

on the feasibility of impulse reflection response data for the two dimensional inverse scattering problem layer stripping algorithms for inverse scattering problems are very fast but have the reputation of being numerically unstable especially when applied to noisy data the goal of this paper is to provide an explicitly discrete framework for layer stripping algorithms for the two dimensional 2 d schrodinger equation inverse scattering problem we determine when 2 d layer stripping algorithms are numerically stable explain why they are stable and specify exactly the discrete problem they solve when they are stable we reformulate the 2 d schrodinger equation as a multichannel two component wave system by fourier transforming the schrodinger equation in the lateral spatial variable discretization results in new 2 d layer stripping algorithms which incorporate multichannel transmission effects this leads to an important new feasibility condition on impulse reflection response data for stability of these algorithms a 2 d discrete schrodinger equation is defined and analogous results are obtained numerical examples illustrate the new results especially how rendering noisy data feasible stabilizes layer stripping algorithms

hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems we present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems under a saturation assumption the global reliability and efficiency of the proposed a posteriori estimator are proved several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies finally we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures

a two level method for mimetic finite difference discretizations of elliptic problems we propose and analyze a two level method for mimetic finite difference approximations of second order elliptic boundary value problems we prove that the two level algorithm is uniformly convergent i e the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underlying partition we also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom numerical results that validate the theory are also presented

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

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

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

hierarchical error estimates for the energy functional in obstacle problems we present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems the main result is that the error in the energy minimum is up to oscillation terms equivalent to an appropriate hierarchical estimator the proof does not invoke any saturation assumption we even show that small oscillation implies a related saturation assumption in addition we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators finally we illustrate our theoretical results by numerical computations

factorization of the wave equation in a nonplanar stratified medium the wave equation is considered for a stratified medium where the stratifications are in the form of a family of nested c2 surfaces along which the velocity c is constant c varying only in a direction normal to the surfaces on each surface c is constant the solution u of the wave equation is decomposed into an outgoing wave component u and an incoming wave component u the associated outgoing and incoming wave conditions are expressed in terms of integral operators kernels being time dependent single and double layer potential type terms operating on u and the normal derivative u n on each surface using the decomposition the scalar wave equation is split into a vector system involving the components u and u the vector system decoupling in a region where c is constant such a splitting is useful for the inverse problem where a reflection operator relating the outgoing wave to an incoming wave can be defined and this in turn can be used to determine c

the calculations of an inverse potential problem a method is developed for solving an inverse problem for the wave equation with potential where the object is to find the potential given cauchy data on a time like surface the computation is carried out with one space variable by an iterative procedure the point of this method is that it can be extended to higher dimensions in principle a coarse mesh finite difference scheme is used which yields fair accuracy

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

mimetic finite difference methods for diffusion equations on non orthogonal non conformal meshes mimetic discretizations based on the support operators methodology are derived for non orthogonal locally refined quadrilateral meshes the second order convergence rate on non smooth meshes is verified with numerical examples

inverse scattering problem for the wave equation with time dependent potential abstract consider the equation u q t x u 0 x r 3 with a time dependent potential q t x compactly supported in x let u s c r s lloc2 r s 2 be the asymptotic wave profiles of a suitably chosen one parameter family of solutions if q1 q2 and the profiles ui related to qi i 1 2 coincide we prove that q1 q2

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

multidimensional inverse scattering an orthogonalization formulation the three dimensional schrodinger equation inverse scattering problem is solved using an orthogonalization approach the plane waves propagating in free space are orthogonalized with respect to an inner product defined in terms of a jost operator the resulting integral equation is identical to the generalized gel fand levitan equation of newton although the present derivation is simpler and more physical than that of newton newton s generalized marchenko equation is derived from the defining integral equation for the jost operator these integral equations are shown to be solved by fast algorithms derived directly from the properties of their solutions this paper thus presents a simple interpretation of newton s two integral equations two fast algorithms for solving these integral equations and relations between the various approaches this is a generalization of previously obtained results which are also reviewed here for the one dimensional inverse scattering problem

efficient and reliable hierarchical error estimates for the discretization error of elliptic obstacle problems we present and analyze novel hierarchical a posteriori error estimates for self adjoint elliptic obstacle problems our approach differs from straightforward but nonreliable estimators by an additional extra term accounting for the deviation of the discrete free boundary in the localization step we prove efficiency and reliability on a saturation assumption and a regularity condition on the underlying grid heuristic arguments suggest that the extra term is of higher order and preserves full locality numerical computations confirm our theoretical findings

direct and inverse scattering in the time domain via invariant imbedding equations this paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain which is applied here to the case of a one dimensional lossless medium the method is based on an integro differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques an example of an exact solution to this equation is given numerical schemes for solving the direct and inverse problems are derived by discretizing this equation the behavior of the resulting algorithms is then tested on several examples the results suggest that the inversion method is stable in the presence of additive noise