conservative modified cranknicolson and time splitting wavelet methods for modeling boseeinstein condensates in delta potentials this paper explores two wavelet based energy conserving algorithms for the grosspitaevskii equation with delta potentials in boseeinstein condensates named modified cranknicolson wavelet method and time splitting wavelet method respectively both proposed methods can preserve the intrinsic properties of original problems as much as possible meanwhile the rigorous error estimates and some conservative properties are investigated they are proved to preserve the charge conservation exactly the global energy conservation laws can be preserved under several conditions in practical computations to avoid a large drift in energy values caused by discontinuous potential well an improved discrete delta function is implemented numerical experiments for attractive and repulsive cases are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

symplectic and multisymplectic lobatto methods for the good boussinesq equation in this paper we construct second order symplectic and multisymplectic integrators for the good boussineq equation using the two stage lobatto iiia iiib partitioned runge kutta method which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative numerical dispersion properties and the stability of both integrators are investigated numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preserving local and global energy and momentum

a fast collocation method for an inverse boundary value problem in this paper we present an implementation of a fast multiscale collocation method for boundary integral equations of the second kind and its application to solving an inverse boundary value problem of recovering a coefficient function from a boundary measurement we illustrate by numerical examples the insensitive nature of the map from the coefficient to measurement and design and test a gauss newton iteration algorithm for obtaining the best estimate of the unknown coefficient from the given measurement based on a least squares formulation

numerical approaches to simulation of multi core fibers abstract we propose generalizations of two numerical algorithms to solve the system of linearly coupled nonlinear schrodinger equations nlses describing the propagation of light pulses in multi core optical fibers an iterative compact dissipative second order accurate in space and fourth order accurate in time scheme is the first numerical method this compact scheme has strong stability due to inclusion of the additional dissipative term the second algorithm is a generalization of the split step fourier method based on pade approximation of the matrix exponential we compare a computational efficiency of both algorithms and show that the compact scheme is more efficient in terms of performance for solving a large system of coupled nlses we also present the parallel implementation of the numerical algorithms for shared memory systems using openmp

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

direct multichannel predictive deconvolution the levinson principle generally can be used to compute recursively the solution of linear equations it can also be used to update the error terms directly this is used to do single channel deconvolution directly on seismic data without computing or applying a digital filter multichannel predictive deconvolution is used for seismic multiple attenuation in a standard procedure the prediction error filter matrices are computed with a levinson recursive algorithm using a covariance matrix of the input data the filtered output is the prediction errors or the nonpredictable part of the data starting with the classical levinson recursion wehave derived new algorithms for direct recursive calculationof the prediction errors without computing the data covariance matrix or computing the prediction error filters one algorithm generates recursively the one step forward and backward predic tion errors and the l step forward prediction error computing only the filter matrices with the highest index a numeri

solvability of concatenated runge kutta equations for second order nonlinear pdes in this paper we consider the solvability of concatenated runge kutta crk equations which are the nonlinear algebraic equations arising from crk methods applied to some kind of second order nonlinear partial differential equations to guarantee the existence and uniqueness of the solution of crk equations temporal and spatial step sizes of crk methods should be chosen appropriately under different assumptions with the homotopy continuation method we choose different step sizes to ensure the solvability of crk equations

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

conjugate gradient method for the robin inverse problem associated with the laplace equation this paper studies a non linear inverse problem associated with the laplace equation of identifying the robin coefficient from boundary measurements a variational formulation of the problem is suggested thereby transforming it into an optimization problem mathematical properties relevant to its numerical computation are established the optimization problem is solved using the conjugate gradient method in conjunction with the discrepancy principle and the algorithm is implemented using the boundary element method numerical results are presented for several benchmark problems with both exact and noisy data and the convergence of the algorithm with respect to mesh refinement and decreasing the amount of noise in the data is investigated copyright 2006 john wiley sons ltd

numerical simulation of interaction between schrodinger field and klein gordon field by multisymplectic method abstract we propose a multisymplectic scheme to solve the coupled klein gordon schrodinger system the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law and can simulate the original waves well in a long time this scheme also has discrete quasi norm conservation law numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results

detection of mines by electric measurements in a homogeneous region omega of earth with conductivity 1 a mine is possibly buried occupying a region d and having conductivity ne 1 d is unknown but meas d geqq c 0 we let electric current enter omega at some point of partial omega and exit it at another point of partial omega and we measure the resulting potential at some open portion of partial omega it is proved that if the error in the measurements is small enough then one can assert whether or not a mine d is in fact buried in omega some results on the location of d are also given

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

symplectic and multi symplectic methods for the nonlinear schrodinger equation the hamiltonian and the multi symplectic formulations of the nonlinear schrodinger equation are considered for the multi symplectic formulation a new six point difference scheme which is equivalent to the multi symplectic preissman integrator is derived numerical experiments are also reported c 2002 elsevier science ltd all rights reserved

discrete convolution by means of forward and backward modeling an approach to discrete convolution is presented which obviates the zero assumption the method is structurally similar to the method of j p burg stanford univ may 1975 which estimates the autocorrelation coefficients of a series in a manner which does not require a predefinition of the behavior of the signal outside of the known interval the basic principle of the present approach is that each term of the convolution is recursively determined from previous terms in a manner consistent with the optimal modeling of one signal in terms of the other the recursion uses forward and backward modeling together with the algorithm of m morf et al ibid vol assp 25 p 429 43 oct 1977 for computation of the prediction error filter the method is illustrated by application to the computation of the analytic signal and its derivative

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

accuracy of classical conservation laws for hamiltonian pdes under runge kutta discretizations we investigate conservative properties of runge kutta methods for hamiltonian partial differential equations it is shown that multi symplecitic runge kutta methods preserve precisely the norm square conservation law based on the study of accuracy of runge kutta methods applied to ordinary and partial differential equations we present some results on the numerical accuracy of conservation laws of energy and momentum for hamiltonian pdes under runge kutta discretizations

the numerical study of a nonlinear one dimensional dirac equation we study the convergence of a difference scheme for a nonlinear one dimensional dirac equation also we show some numerical experiments by using this scheme

a levinson type algorithm for a class of non toeplitz systems with applications to multichannel iir filtering a very flexible levinson type recursion for a class of non toeplitz systems of linear equations is demonstrated a complete solution is expressed as a linear combination of a partial solution and three auxiliary solutions the class of systems possesses a special structure in that the coefficient matrices can be partitioned into four block toeplitz submatrices the number of multiplications and additions required to compute an n dimensional solution is o n sup 2 the recursion is then applied to multichannel iir filtering specifically a lattice structure is established for linear minimum mean square error predictors having independently and arbitrarily specified numbers of poles and zeros next the recursion is used to develop a fast time and order recursive least squares algorithm for arx system identification the novelty of the algorithm is that it can be used to efficiently determine parameter estimates of a family of arx models

on uniqueness of recovery of a discontinuous conductivity coefficient on considere le probleme de retrouver le coefficient a de l equation elliptique div a u 0 dans un domaine avec la condition aux limites u sur quand u n est donnee pour tout regulier on demontre l unicite pour a a 0 b discontinu ou est la fonction indicateur d un ensemble ouvert a o est une fonction c 2 donnee et b est une fonction c 2 inconnue

energy and quadratic invariants preserving equip multi symplectic methods for hamiltonian wave equations abstract it is well known that a numerical method which is at the same time geometric structure preserving and physical property preserving cannot exist in general for hamiltonian partial differential equations motivated by equip methods proposed in brugnano et al 2012 13 in this paper we present a novel class of parametric multi symplectic runge kutta methods called equip multi symplectic methods for hamiltonian wave equations which can also conserve energy simultaneously in a weaker sense with a suitable parameter the existence of such a parameter which enforces the energy preserving property is proved under certain assumptions on the fixed step sizes and the fixed initial condition we compare the proposed method with the classical multi symplectic runge kutta method in numerical experiments which shows the remarkable energy preserving property of the proposed method and illustrates the validity of theoretical results these theoretical and numerical results show that equip methods can be well adapted to handle hamiltonian partial differential equations

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

identification of contact regions in semiconductor transistors by level set methods in this paper we present the formulation of level set methods for the inverse problem of identifying an interface in the coefficient of an elliptic equation from a boundary measurement this problem arises from the modeling of the identification of contact regions by boundary measurements for semiconductor transistors we propose the gauss newton direction as the interface velocity and implement the scheme for a parameterized class of interfaces

on the preservation of phase space structure under multisymplectic discretization in this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and fourier spectral approximations multisymplectic ms schemes are developed for two benchmark nonlinear wave equations the sine gordon and nonlinear schrodinger equations we examine the implications of preserving the ms structure under discretization on the numerical scheme s ability to preserve phase space structure as measured by the nonlinear spectrum of the governing equation we find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws dynamical invariants and complicated phase space structures

multi symplectic runge kutta nystrom methods for nonlinear schrodinger equations with variable coefficients in this paper we consider runge kutta nystrom rkn methods applied to nonlinear schrodinger equations with variable coefficients nlsevc concatenating symplectic nystrom methods in spatial direction and symplectic runge kutta methods in temporal direction for nlsevc leads to multi symplectic integrators i e to numerical methods which preserve the multi symplectic conservation law mscl we present the corresponding discrete version of mscl it is shown that the multi symplectic rkn methods preserve not only the global symplectic structure in time but also local and global discrete charge conservation laws under periodic boundary conditions we present a 4 order multi symplectic rkn method and use it in numerical simulation of quasi periodically solitary waves for nlsevc and we compare the multi symplectic rkn method with a non multi symplectic rkn method on the errors of numerical solutions the numerical errors of discrete energy discrete momentum and discrete charge the precise conservation of discrete charge under the multi symplectic rkn discretizations is attested numerically some numerical superiorities of the multi symplectic rkn methods are revealed

fast bayesian approach for parameter estimation this paper presents two techniques i e the proper orthogonal decomposition pod and the stochastic collocation method scm for constructing surrogate models to accelerate the bayesian inference approach for parameter estimation problems associated with partial differential equations pod is a model reduction technique that derives reduced order models using an optimal problem adapted basis to effect significant reduction of the problem size and hence computational cost scm is an uncertainty propagation technique that approximates the parameterized solution and reduces further forward solves to function evaluations the utility of the techniques is assessed on the non linear inverse problem of probabilistically calibrating scalar robin coefficients from boundary measurements arising in the quenching process and non destructive evaluation a hierarchical bayesian model that handles flexibly the regularization parameter and the noise level is employed and the posterior state space is explored by the markov chain monte carlo the numerical results indicate that significant computational gains can be realized without sacrificing the accuracy

energy dissipation preserving birkhoffian multi symplectic methods for maxwell s equations with dissipation terms in this paper we propose two new energy dissipation preserving birkhoffian multi symplectic methods birkhoffian and birkhoffian box for maxwell s equations with dissipation terms after investigating the non autonomous and autonomous birkhoffian formalism for maxwell s equations with dissipation terms we first apply a novel generating functional theory to the non autonomous birkhoffian formalism to propose our birkhoffian scheme and then implement a central box method to the autonomous birkhoffian formalism to derive the birkhoffian box scheme we have obtained four formal local conservation laws and three formal energy global conservation laws we have also proved that both of our derived schemes preserve the discrete version of the global local conservation laws furthermore the stability dissipation and dispersion relations are also investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable dissipation preserving for maxwell s equations in a perfectly matched layer pml medium and have second order accuracy in both time and space numerical experiments for problems with exact theoretical results are given to demonstrate that the birkhoffian multi symplectic schemes are much more accurate in preserving energy than both the exponential finite difference time domain fdtd method and traditional hamiltonian scheme we also solve the electromagnetic pulse emp propagation problem and the numerical results show that the birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation

partitioned least squares operator for large scale geophysical inversion least squares ls problems are encountered in many geophysical estimation and data analysis problems where a large number of observations data are combined to determine a model some aspect of the earth structure examples of least squares in seismic exploration include several data processing algorithms theoretically accurate ls migration inversion for reservoir parameters and background velocity estimation a frequently encountered problem is that the volume of data in 3d is so large that the matrices required for the ls solution cannot be stored within the memory of a single computer a new technique is described for parallel computation of the ls operator that is based on a partitioned matrix algorithm the classical ls method for solution of block toeplitz systems of normal equation ne to the general case of block hermitian and non toeplitz systems of ne is generalized specifically a solution of a block hermitian system of ne is shown that may be obtained recursively by linearly combining the solutions of lesser order that are related to the forward and backward subsystems of equations this results in an efficient parallel algorithm in which each partitioned system can be evaluated independently the application of the algorithm to the problem of 3d plane wave transformation is demonstrated

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

fast algorithms for discrete time wiener filters with optimum lag in many applications including geophysical signal processing and system identification the computation of a fir wiener filter corresponding to the optimum lag between the input signal and the desired response or of an optimum prediction distance predictor are often required these problems lead to the solution of a family of toeplitz systems of equations having the same associated matrix but right hand side vectors which are shifted versions of each other a very efficient method for the solution of the optimum lag problem is the well known simpson s sideways recursions the purpose of this paper is to introduce two new algorithms for the efficient computation of the optimum lag wiener filter as well as the optimum prediction distance predictor their main characteristic is a coupled step down step up recursion which takes full advantage of both the toeplitz nature of the matrix and the shifted structure of the right hand side vectors the proposed algorithms are featured by increased speed they are about twice as fast as simpson s scheme and computational simplicity in addition they provide recursive error computation capability and give further insight into the problem

numerical studies on the split step finite difference method for nonlinear schrodinger equations in this study we use the split step finite difference method to solve various nonlinear schrodinger equations including coupled ones we study the numerical accuracy of the method detailed numerical results including higher dimensions show that the method provides accurate and stable solutions for nonlinear schrodinger equations

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

multisymplectic geometry and multisymplectic preissmann scheme for the kdv equation the multisymplectic structure of the kdv equation is presented directly from the variational principle from the numerical view point we give a multisymplectic twelve points scheme which is equivalent to the multisymplectic preissmann scheme finally we test the twelve points scheme on solitary waves over long time intervals

the multisymplectic diamond scheme we introduce a class of general purpose linear multisymplectic integrators for hamiltonian wave equations based on a diamond shaped mesh on each diamond the pde is discretized by a symplectic runge kutta method the scheme advances in time by filling in each diamond locally leading to greater efficiency and parallelization and easier treatment of boundary conditions compared to methods based on rectangular meshes

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

on multisymplecticity of sinc gauss legendre collocation discretizations for some hamiltonian pdes abstract in this paper a new multisymplectic scheme based on sinc collocation discretization in space and gauss legendre collocation discretization in time for some hamiltonian pdes is developed the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law moreover the discrete energy and momentum conservative properties of the multisymplectic integrator are also discussed in order to testify the superiority of the multisymplectic method it is applied to nonlinear dirac equation numerical experiments are given to illustrate the accuracy of the approximation and the conserved quantities

numerical analysis of a multi symplectic scheme for a strongly coupled schrodinger system abstract in this paper we analyze a multi symplectic scheme for a strongly coupled nonlinear schrodinger equations we derive a general box scheme which is equivalent to the multi symplectic scheme by reduction method based on the general box scheme we prove that the multi symplectic scheme preserves not only the multi symplectic structure of the equation but also conservation law of mass in general the multi symplectic schemes are not conservative to energy in the nonlinear case so it is difficult to obtain the estimates of numerical solutions in norm hence proofs of convergence and stability are difficult for multi symplectic schemes of nonlinear equations a deduction argument and the energy analysis method are used to prove that the numerical solution is stable for initial values and second order convergent to the exact solutions in 2 norm a fixed point theorem is introduced and used to prove the unique solvability of the numerical solutions

high order compact splitting multisymplectic method for the coupled nonlinear schrodinger equations in this paper we develop a new kind of multisymplectic integrator for the coupled nonlinear schrodinger cnls equations the cnls equations are cast into multisymplectic formulation then it is split into a linear multisymplectic formulation and a nonlinear hamiltonian system the space of the linear subproblem is approximated by a high order compact hoc method which is new in multisymplectic context the nonlinear subproblem is integrated exactly for splitting and approximation we utilize an hoc sms integrator its stability and conservation laws are investigated in theory numerical results are presented to demonstrate the accuracy conservation laws and to simulate various solitons as well for the hoc sms integrator they are consistent with our theoretical analysis

inverse problems for metal oxide semiconductor field effect transistor contact resistivity inverse problems arising from the identification of transistor contact resistivity and contact window location are studied a one point boundary measurement of the potential is shown to be sufficient to identify each from a one parameter monotone family and such identification is both stable and continuously dependent on the parameter for a nonmonotone family of locations with resistivity given similar results are obtained for measurement made on almost all boundary points

inversion of robin coefficient by a spectral stochastic finite element approach this paper investigates a variational approach to the nonlinear stochastic inverse problem of probabilistically calibrating the robin coefficient from boundary measurements for the steady state heat conduction the problem is formulated into an optimization problem and mathematical properties relevant to its numerical computations are investigated the spectral stochastic finite element method using polynomial chaos is utilized for the discretization of the optimization problem and its convergence is analyzed the nonlinear conjugate gradient method is derived for the optimization system numerical results for several two dimensional problems are presented to illustrate the accuracy and efficiency of the stochastic finite element method

multi symplectic runge kutta methods for nonlinear dirac equations in this paper we consider the multi symplectic runge kutta msrk methods applied to the nonlinear dirac equation in relativistic quantum physics based on a discovery of the multi symplecticity of the equation in particular the conservation of energy momentum and charge under msrk discretizations is investigated by means of numerical experiments and numerical comparisons with non msrk methods numerical experiments presented reveal that msrk methods applied to the nonlinear dirac equation preserve exactly conservation laws of charge and momentum and conserve the energy conservation in the corresponding numerical accuracy to the method utilized it is verified numerically that msrk methods are stable and convergent with respect to the conservation laws of energy momentum and charge and msrk methods preserve not only the inner geometric structure of the equation but also some crucial conservative properties in quantum physics a remarkable advantage of msrk methods applied to the nonlinear dirac equation is the precise preservation of charge conservation law

numerical methods for the simulation of trapped nonlinear schrodinger systems we propose analyze and compare the efficiency and accuracy of different numerical schemes for the solution of the nonlinear schrodinger equation with a trapping potential in particular we study schemes of finite difference pseudospectral and spectral types for the space discretization together with explicit symplectic multistep split step and standard variable step integrators to solve the time evolution all of these schemes are evaluated comparatively and some recommendations based on their accuracy and computational efficiency are made

numerical methods for nonlinear dirac equation abstract this paper presents a review of the current state of the art of numerical methods for nonlinear dirac nld equation several methods are extendedly proposed for the 1 1 dimensional nld equation with the scalar and vector self interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge energy and linear momentum those methods are the crank nicolson cn schemes the linearized cn schemes the odd even hopscotch scheme the leapfrog scheme a semi implicit finite difference scheme and the exponential operator splitting os schemes the nonlinear subproblems resulted from the os schemes are analytically solved by fully exploiting the local conservation laws of the nld equation the effectiveness of the various numerical methods with special focus on the error growth and the computational cost is illustrated on two numerical experiments compared to two high order accurate runge kutta discontinuous galerkin methods theoretical and numerical comparisons show that the high order accurate os schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency a fourth order accurate os scheme is further applied to investigating the interaction dynamics of the nld solitary waves under the scalar and vector self interaction the results show that the interaction dynamics of two nld solitary waves depend on the exponent power of the self interaction in the nld equation collapse happens after collision of two equal one humped nld solitary waves under the cubic vector self interaction in contrast to no collapse scattering for corresponding quadric case

numerical estimation of piecewise constant robin coefficient this paper considers a numerical method for reconstructing piecewise constant robin coefficients from boundary measurements a novel variational formulation of the inverse problem based on the celebrated modica mortola functional is proposed and its gamma convergence to the formulation with total variation regularization is shown the differentiability of the functional is investigated based on which the concave convex procedure is justified for its numerical solution the finite element method is utilized for discretizing the functional and the convergence of the finite element approximation is established three sets of numerical experiments are presented to illustrate its distinct features

stable and efficient numerical schemes for two dimensional maxwell equations in lossy medium abstract in this paper some stable and efficient numerical methods are developed for two dimensional maxwell equations to avoid solving large scale algebraic equations we use local one dimensional lod technique and split the original equations into several lod equations then the resulted lod equations are discretized by wendroff scheme which is usually applied to simulate hyperbolic type equations the lie trotter composition and strang composition are employed to improve the computational efficiency and the convergent accurate the space direction is approximated by high order compact method some benchmark symbols such as stability conservation laws of the schemes are analyzed some numerical examples confirm the theoretical analysis numerical results show that the new schemes can not only accurately simulate the electromagnetic waves including profiles amplitude but also capture the energy structures exactly

levinson type extensions for non toeplitz systems it is shown levinson s basic principle for the solution of normal equations which are of toeplitz form may be extended to the case where these equations do not possess this specific symmetry the method is illustrated by application to various examples which are chosen so that the coefficient matrix possesses various symmetries specifically the solution of the normal equations when the associated matrix is the doubly symmetric non toeplitz covariance matrix is considered next the solution of extended yule walker equations where the coefficient matrix is toeplitz but nonsymmetric is obtained finally the approach is illustrated by considering the determination of the prediction error operator when the normal equations are of symmetric toeplitz form

a survey on symplectic and multi symplectic algorithms we simply review symplectic and multi symplectic algorithms in the paper these algorithms have many superiorities to general non symplectic algorithms such as they have long time behavior they can preserve the symplectic geometry structure of the hamiltonian system exactly and they can simulate original physical phenomena well some commonly used methods to construct symplectic and multi symplectic schemes are simply surveyed numerical examples are also enumerated to illustrate their performances

inverse boundary value problems at the boundary continuous dependence we use the methods of microlocal analysis to give a new proof of a theorem of kohn and vogelius showing that the boundary values of a continuous isotropic conductivity can be recovered from voltage and current measurements at the boundary moreover we prove sharp estimates to establish the continuous dependence of the boundary values of the conductivity on the voltage to current maps

a note on multisymplectic fourier pseudospectral discretization for the nonlinear schrodinger equation based on the multisymplectic fourier pseudospectral scheme for the nonlinear schrodinger nls equation we investigate some discrete properties corresponding to local conservation laws of the original equation the discrete normal conservation law is proved and the error estimation of local and global energy conservation laws are also obtained numerical experiments for cubic nls equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results

multi symplectic preissmann scheme for a high order wave equation of kdv type the high order wave equation of kdv type which describes many important physical phenomena has been investigated widely in last several decades here the new multi symplectic formulations for the high order wave equation of kdv type are presented and the associated local conservation laws are shown the multi symplectic discretization of each formulation is exemplified by the multi symplectic preissmann scheme the numerical experiments are given and the results verify the efficiency of the multi symplectic preissmann scheme

explicit multi symplectic methods for klein gordon schrodinger equations in this paper we propose explicit multi symplectic schemes for klein gordon schrodinger equation by concatenating suitable symplectic runge kutta type methods and symplectic runge kutta nystrom type methods for discretizing every partial derivative in each sub equation it is further shown that methods constructed in this way are multi symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions in the aim of the commonly practical applications a novel 2 order one parameter family of explicit multi symplectic schemes through such concatenation is constructed and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods furthermore some high order explicit multi symplectic schemes of such category are given as well good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

decoupled local global energy preserving schemes for the n coupled nonlinear schrodinger equations abstract we develop two local energy preserving integrators and a global energy preserving integrator for the general multisymplectic hamiltonian system when applied to the 1d and multi dimensional n coupled nonlinear schrodinger equations the given schemes have the exact preservation of the local global conservation law and are decoupled in the components n n 1 2 n i e each of the components can be solved independently the decoupled feature is significant and helpful for overcoming the computational difficulty of the n coupled n 3 nonlinear schrodinger equations especially of the multi dimensional case the composition method is employed to improve the accuracy of the schemes in time and the discrete fast fourier transform is used to reduce the computational complexity several numerical experiments are carried out to exhibit the behaviors of the wave solutions numerical results confirm the theoretical results

analysis of an adaptive finite element method for recovering the robin coefficient based on a new a posteriori error estimator an adaptive finite element method is proposed for recovering the robin coefficient involved in a diffusion system from some boundary measurement the a posteriori error estimator cannot be derived for this ill posed nonlinear inverse problem as was done for the existing a posteriori error estimators for direct problems instead we shall derive the a posteriori error estimator from our convergence analysis of the adaptive algorithm we prove that the adaptive algorithm guarantees a convergent subsequence of discrete solutions in an energy norm to some exact triplet the robin coefficient state and costate variables determined by the optimality system of the least squares formulation with tikhonov regularization for the concerned inverse problem some numerical results are also reported to illustrate the performance of the algorithm

a fourth order time splitting laguerre hermite pseudospectral method for bose einstein condensates a fourth order time splitting laguerre hermite pseudospectral method is introduced for bose einstein condensates becs in three dimensions with cylindrical symmetry the method is explicit time reversible and time transverse invariant it conserves the position density and is spectral accurate in space and fourth order accurate in time moreover the new method has two other important advantages i it reduces a three dimensional 3 d problem with cylindrical symmetry to an effective two dimensional 2 d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain the method is applied to vector gross pitaevskii equations vgpes for multicomponent becs extensive numerical tests are presented for the one dimensional 1 d gpe the 2 d gpe with radial symmetry the 3 d gpe with cylindrical symmetry as well as 3 d vgpes for two component becs to show the efficiency and accuracy of the new numerical method

the multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this article we consider partitioned runge kutta prk methods for hamiltonian partial differential equations pdes and present some sufficient conditions for multi symplecticity of prk methods of hamiltonian pdes

split step alternating direction implicit difference scheme for the fractional schrodinger equation in two dimensions in this paper we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space fractional schrodinger equation with the riesz fractional derivative the scheme is constructed by combining the split step method for handling the nonlinearity with the alternating direction implicit adi method for resolving the multi dimensions difficulty the riesz space fractional derivative is approximated by the second order accurate fractional centered difference based on matrix analysis we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems the unconditional convergence is proved rigorously in the linear case numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme

schur type algorithms for spatial ls estimation with highly pipelined architectures a family of schur type spatial least squares algorithms is presented for solving the spatial ls estimation problem in which the correlation matrix is neither toeplitz nor near toeplitz by order recursion normalized spatial levinson and schur type algorithms are also derived highly pipelined architectures are designed to realize these recursions the reflection coefficients are first computed using the spatial schur type recursions then the forward and backward filter parameters are calculated by the spatial levinson type recursions a pyramid systolic array is demonstrated to calculate not only the filter parameters but also the ldu decomposition of the inverse cross correlation matrix at every clock phase this pyramid array can be mapped onto a two dimensional systolic array which has a simpler structure a square systolic array is developed to implement the levinson and schur type temporal recursive ls rls algorithms a highly concurrent architecture which exploits the parallelism of the spatial schur type recursions is illustrated to perform the ldu decomposition of the cross correlation matrix

semi implicit operator splitting pade method for higher order nonlinear schrodinger equations abstract we propose a semi implicit finite difference operator splitting pade ospd method for solving the higher order nonlinear schrodinger equation which describes the optical soliton wave propagation in fibers the method achieves fourth order of accuracy in space and has been proven to be stable by linear stability analysis numerical experiments and comparisons are investigated to show the advantages and effectivity of the ospd method and some interesting collision behaviors are also observed

a uniqueness theorem for an inverse boundary value problem in two dimensions abstract we consider the inverse problem to determine the potential q entering the schrodinger equation u qu 0 in a bounded smooth domain in r 2 in terms of the boundary measurements dirichlet to neumann map when n 3 the uniqueness for the inverse problem was well established in this paper we prove the uniqueness in two dimensions under the assumption that the lp p 2 norm of q is small this result improves a uniqueness result of sylvester uhlmann under the assumption that the w1 norm of q is small

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

quadratic invariants and multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this paper we study the preservation of quadratic conservation laws of runge kutta methods and partitioned runge kutta methods for hamiltonian pdes and establish the relation between multi symplecticity of runge kutta method and its quadratic conservation laws for schrodinger equations and dirac equations it reveals that multi symplectic runge kutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation respectively

multi symplectic runge kutta nystrom methods for nonsmooth nonlinear schrodinger equations abstract in this paper we discuss multi symplectic ms runge kutta nystrom rkn methods applied to a kind of nonsmooth nonlinear schrodinger equations nlses i e the nlses with delta potentials based on some theoretical suggestions proposed in 1 e g the weak reformulation of multi symplectic hamiltonian system mshs for the nlses and some newly addressed local and global conservation laws concatenating rkn algorithms for this weak mshs are novelly constructed and the multi symplecticity of them is revealed under the msrkn discretizations we show that some intrinsic characters of this weak mshs including the global symplectic structure in time and the normalization conservation laws are preserved analytically just like that for the smooth nlses 11 numerical experiments validate our methods and the errors of numerical solutions discrete energies and discrete parity symmetry are also investigated numerically comparisons with non ms schemes our ones exhibit some advantages especially for the precise preservation of normalization conservation laws

levinson type algorithms for polynomial fitting and for cholesky and q factors of hankel and vandermonde matrices this paper presents levinson 1947 type algorithms for i polynomial fitting ii obtaining a q decomposition of vandermonde matrices and a cholesky factorization of hankel matrices iii obtaining the inverse of hankel matrices the algorithm for the least squares solution of hankel systems of equations requires 3n sup 2 9n 3 multiply and divide operation mdo the algorithm for obtaining an orthogonal representation of an m spl times n vandermonde matrix x and computing the cholesky factors f of hankel matrices requires 5mn n sup 2 2n 3m mdo and the algorithm for generating the inverse of hankel matrices requires 3 n sup 2 n 2 2 mdo our algorithms have been tested by means of fitting of polynomials of various orders and fortran versions of all subroutines are provided in the appendix