kernel based hard clustering methods in the feature space with automatic variable weighting abstract this paper presents variable wise kernel hard clustering algorithms in the feature space in which dissimilarity measures are obtained as sums of squared distances between patterns and centroids computed individually for each variable by means of kernels the methods proposed in this paper are supported by the fact that a kernel function can be written as a sum of kernel functions evaluated on each variable separately the main advantage of this approach is that it allows the use of adaptive distances which are suitable to learn the weights of the variables on each cluster providing a better performance moreover various partition and cluster interpretation tools are introduced experiments with synthetic and benchmark datasets show the usefulness of the proposed algorithms and the merit of the partition and cluster interpretation tools

quadratic operator relations and bethe equations for spin 1 2 richardson gaudin models in this work we demonstrate how one can in a generic approach derive a set of n simple quadratic bethe equations for integrable richardson gaudin rg models built out of n spins 1 2 these equations depend only on the n eigenvalues of the various conserved charges so that any solution of these equations defines indirectly through the corresponding set of eigenvalues one particular eigenstate the proposed construction covers the full class of integrable rg models of the xyz including the subclasses of xxz and xxx models type realised in terms of spins 1 2 coupled with one another through terms including as well magnetic field like terms linear in the pauli matrices the approach exclusively requires integrability defined here only by the requirement that n conserved charges r i with such that exist the result is therefore valid and equally simple for models with or without symmetry with or without a properly defined pseudo vacuum as well as for models with non skew symmetric couplings

clustering via kernel decomposition spectral clustering methods were proposed recently which rely on the eigenvalue decomposition of an affinity matrix in this letter the affinity matrix is created from the elements of a nonparametric density estimator and then decomposed to obtain posterior probabilities of class membership hyperparameters are selected using standard cross validation methods

a high precision integration scheme for the spectral element time domain method in electromagnetic simulation a high precision integration hpi scheme combined with the spectral element time domain setd method is presented to solve the time dependent maxwell s equations spatial discretization by spectral elements will lead to block diagonal mass matrices thus greatly alleviating the computational burden of inverting the mass matrices a high precision time integration method based on the matrix exponential is then employed to solve the discretized setd system numerical examples demonstrate that this algorithm is unconditionally stable and very accurate

numerical dispersion in dg fetd method using brick and tetrahedral edge elements a numerical dispersion analysis nda of the discontinuous galerkin finite element time domain dg fetd method is presented vector basis functions under the nda are with brick and tetrahedral elements this nda shows that there exist both normal and spurious modes in the dg fetd method a dg fetd modeling of transverse electromagnetic tem wave propagation in a parallel waveguide is applied to verify the nda for zeroth order vector bases the effect of wave propagation direction and electrical size of elements on numerical dispersion is investigated it is shown from the nda for higher order interpolatory vector bases that the phase error of normal modes can be significantly reduced by using higher order bases

sympletic runge kutta shemes i order conditions much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating hamiltonian systems of differential equations runge kutta schemes are symplectic subject to a simple algebraic condition starting with butcher s formalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods involving only s s 1 2 free parameters for a symplectic s stage scheme a graph theoretical process for determining the new order conditions is outlined furthermore it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the parametrized coefficients when coupled with the standard simplifying assumptions for implicit schemes the number of order conditions may be further reduced in the new framework a simple symmetry of the parameter matrix yields not necessarily symplectic self adjoint methods in this case the order conditions associated with even trees become redundant

high order symplectic integration methods for finite element solutions to time dependent maxwell equations in this paper we motivate the use of high order integration methods for finite element solutions of the time dependent maxwell equations in particular we present a symplectic algorithm for the integration of the coupled first order maxwell equations for computing the time dependent electric and magnetic fields symplectic methods have the benefit a conserving total electromagnetic field energy and are therefore preferred over dissipative methods such as traditional runge kutta in applications that require high accuracy and energy conservation over long periods of time integration we show that in the context or symplectic methods several popular schemes can be elegantly cast in a single algorithm we conclude with some numerical examples which demonstrate the superior performance of high order time integration methods

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

the force gradient symplectic finite difference time domain scheme for maxwell s equations in this paper a novel force gradient explicit symplectic finite difference time domain sfdtd algorithm based on the rm t rm v type hamiltonian decomposition is derived to investigate the propagation characteristics of electromagnetic waves in which maxwell s equations are numerically integrated by third order symplectic map combined with force gradient in the time direction the numerical stability and dispersion analyses are presented the numerical results effectively confirm that the force gradient sfdtd f sfdtd method is reliable for electromagnetic wave computation its accuracy is superior to the non force gradient sfdtd method and the traditional fdtd method the implementation of f sfdtd method is easily applied to compute the back scattered radar cross sections rcs using 2 d tm mode for a perfectly conducting cylinder

creation operators and bethe vectors of the osp 1 2 gaudin model a gaudin model based on the orthosymplectic lie superalgebra osp 1 2 is studied the eigenvectors of the osp 1 2 invariant gaudin hamiltonians are constructed by algebraic bethe ansatz corresponding creation operators are defined by a recurrence relation furthermore explicit solution to this recurrence relation is found the action of the creation operators on the lowest spin vector yields bethe vectors of the model the relation between the bethe vectors and solutions to the knizhnik zamolodchikov equation of the corresponding super conformal field theory is established

radon penrose transform for d modules abstract let formula be a correspondence of complex analytic manifolds fbe a sheaf onx and m be a coherent d x module consider the associated sheaf theoretical and d module integral transforms given by sf rg f 1f d and s m g f 1 m whererg andf 1 resp g and f 1 denote the direct and inverse image functors for sheaves resp for d modules andd ds dyis the difference of dimension betweensandy in this paper assuming thatfis smooth gis proper and f g is a closed embedding we prove some general adjunction formulas for the functors sand s moreover under an additional geometrical hypothesis we show that the transformation sestablishes an equivalence of categories between coherent d x modules modulo flat connections and coherent d y modules with regular singularities along an involutive manifoldv modulo flat connections herevis determined by the geometry of the correspondence applications are given to the case of penrose s twistor correspondence

a symplectic integration algorithm for separable hamiltonian functions abstract we derive an algorithm to numerically integrate differential equations derivable from a separable hamiltonian function this symplectic algorithm is accurate to fourth order in the time step and preserves exactly the poincare cartan integral invariants associated with the topology of the phase flow we compare the efficiency and accuracy of this method to that of existing integrators both symplectic and non symplectic by integrating the equations of motion corresponding to a nonlinear pendulum a particle in the field of a standing wave and a harmonic oscillator perturbed by a plane wave

improved conditioning of finite element matrices using new high order interpolatory bases the condition number of finite element matrices constructed from interpolatory bases will grow as the polynomial degree of the basis functions is increased the worst case scenario for this growth rate is exponential and in this paper we demonstrate through computational example that the traditional set of uniformly distributed interpolation points yields this behavior we propose a set of nonuniform interpolation points which yield a much improved polynomial growth rate of condition number these points can be used to construct several types of popular hexahedral basis functions including the 0 form standard lagrangian 1 form curl conforming and 2 form divergence conforming varieties we demonstrate through computational example the benefits of using these new interpolatory bases in finite element solutions to maxwell s equations in both the frequency and time domain

on clusterings good bad and spectral we motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures a simple recursive heuristic is shown to have poly logarithmic worst case guarantees under the new measure the main result of the article is the analysis of a popular spectral algorithm one variant of spectral clustering turns out to have effective worst case guarantees another finds a good clustering if one exists

a vector finite element time domain method for solving maxwell s equations on unstructured hexahedral grids in this paper the vector finite element time domain vfetd method is derived analyzed and validated the vfetd method uses edge vector finite elements as a basis for the electric field and face vector finite elements as a basis for the magnetic flux density the galerkin method is used to convert maxwell s equations to a coupled system of ordinary differential equations the leapfrog method is used to advance the fields in time the method is shown to be stable and to conserve energy and charge for arbitrary hexahedral grids a numerical dispersion analysis shows the method to be second order accurate on distorted hexahedral grids several computational experiments are performed to determine the accuracy and efficiency of the method

symplectic partitioned runge kutta methods for constrained hamiltonian systems this article deals with the numerical treatment of hamiltonian systems with holonomic constraints a class of partitioned runge kutta methods consisting of the couples of s stage lobatto iiia and lobatto iiib methods has been discovered to solve these problems efficiently these methods are symplectic preserve all un derlying constraints and are superconvergent with order 2s 2 for separable hamiltonians of the form h q p frac 1 2 p t m 1 p u q the rattle algorithm based on the verlet method was up to now the only known symplectic method preserving the constraints in fact this method turns out to be equivalent to the 2 stage lobatto iiia iiib method of order 2 numerical examples have been performed which illustrate the theoretical results

kernel fuzzy c means with automatic variable weighting this paper presents variable wise kernel fuzzy c means clustering methods in which dissimilarity measures are obtained as sums of euclidean distances between patterns and centroids computed individually for each variable by means of kernel functions the advantage of the proposed approach over the conventional kernel clustering methods is that it allows us to use adaptive distances which change at each algorithm iteration and can either be the same for all clusters or different from one cluster to another this kind of dissimilarity measure is suitable to learn the weights of the variables during the clustering process improving the performance of the algorithms another advantage of this approach is that it allows the introduction of various fuzzy partition and cluster interpretation tools experiments with synthetic and benchmark datasets show the usefulness of the proposed algorithms and the merit of the fuzzy partition and cluster interpretation tools

a nondiffusive finite volume scheme for the three dimensional maxwell s equations on unstructured meshes we prove a sufficient cfl like condition for the l2 stability of the second order accurate finite volume scheme proposed by remaki for the time domain solution of maxwell s equations in heterogeneous media with metallic and absorbing boundary conditions we yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions numerical tests show the potential of this original finite volume scheme in one two and three space dimensions for the numerical solution of maxwell s equations in the time domain

multidomain pseudospectral time domain algorithm using a symplectic integrator the novel multidomain pseudospectral time domain method based on a symplectic integrator scheme is presented in this paper a symplectic integrator scheme is a decomposition of the exponential operator or a general propagation technique the scheme is nondissipative and does not require additional memory for temporary storage of data for the internal stages the good performance of the scheme is demonstrated by numerical examples

compatible lie brackets related to elliptic curve for the direct sum of several copies of sln a family of lie brackets compatible with the initial one is constructed the structure constants of these brackets are expressed in terms of functions associated with an elliptic curve the structure of casimir elements for these brackets is investigated a generalization of this construction to the case of vector valued functions is presented a different procedure for constructing compatible lie brackets based on the argument shift method for quadratic poisson brackets is discussed

large scale 3d geoelectromagnetic modeling using parallel adaptive high order finite element method abstractwe have investigated the use of the adaptive high order finite element method fem for geoelectromagnetic modeling because high order fem is challenging from the numerical and computational points of view most published finite element studies in geoelectromagnetics use the lowest order formulation solution of the resulting large system of linear equations poses the main practical challenge we have developed a fully parallel and distributed robust and scalable linear solver based on the optimal block diagonal and auxiliary space preconditioners the solver was found to be efficient for high finite element orders unstructured and nonconforming locally refined meshes a wide range of frequencies large conductivity contrasts and number of degrees of freedom dofs furthermore the presented linear solver is in essence algebraic i e it acts on the matrix vector level and thus requires no information about the discretization boundary conditions or physical source used making it readily ef

toward the construction of a fourth order difference scheme for transient em wave simulation staggered grid approach a compact central difference approximation in conjunction with the yee 1966 grid is used to compute the spatial derivatives in maxwell s equations to advance the semi discrete equations the four stage runge kutta rk integrator is invoked this combination of spatial and temporal differencing leads to a scheme that is fourth order accurate conditionally stable and highly efficient moreover the use of compact differencing allows one to apply the compact operator in the vicinity of a perfect conductor an attribute not found in other higher order methods results are provided that quantify the spectral properties of the method simulations are conducted on problem spaces that span one and three dimensions and whose domains are of the open and closed type results from these simulations are compared with exact closed form solutions the agreement between these results is consistent with numerical analysis

splitting and composition methods for explicit time dependence in separable dynamical systems we consider splitting methods for the numerical integration of separable non autonomous differential equations in recent years splitting methods have been extensively used as geometric numerical integrators showing excellent performances both qualitatively and quantitatively when applied on many problems they are designed for autonomous separable systems and a substantial number of methods tailored for different structures of the equations have recently appeared splitting methods have also been used for separable non autonomous problems either by solving each non autonomous part separately or after each vector field is frozen properly we show that both procedures correspond to introducing the time as two new coordinates we generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques we show that the performance as well as the order of the final method can strongly depend on the particular choice we present a simple analysis which in many relevant cases allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems this technique is applied to different problems and its performance is illustrated for several numerical examples

trigonometric osp 1 2 gaudin model the problems connected with gaudin models are reviewed by analyzing model related to the trigonometric osp 1 2 classical r matrix the eigenvectors of the trigonometric osp 1 2 gaudin hamiltonians are found using explicitly constructed creation operators the commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated the coordinate representation of the bethe states is presented the relation between the bethe vectors and solutions to the knizhnik zamolodchikov equation yields the norm of the eigenvectors the generalized knizhnik zamolodchikov system is discussed both in the rational and in the trigonometric case

orthogonal series density estimation and the kernel eigenvalue problem kernel principal component analysis has been introduced as a method of extracting a set of orthonormal nonlinear features from multivariate data and many impressive applications are being reported within the literature this article presents the view that the eigenvalue decomposition of a kernel matrix can also provide the discrete expansion coefficients required for a nonparametric orthogonal series density estimator in addition to providing novel insights into nonparametric density estimation this article provides an intuitively appealing interpretation for the nonlinear features extracted from data using kernel principal component analysis

a high order mixed vector finite element method for solving the time dependent maxwell equations on unstructured grids we present a mixed vector finite element method for solving the time dependent coupled ampere and faraday laws of maxwell s equations on unstructured hexahedral grids that employs high order discretization in both space and time the method is of arbitrary order accuracy in space and up to 4th order accurate in time making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods in addition the method correctly models both the jump discontinuities and the divergence free properties of the electric and magnetic fields is charge and energy conserving conditionally stable and free of spurious modes several computational experiments are performed to demonstrate the accuracy efficiency and benefits of the method

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

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

higher order hybrid implicit explicit fdtd time stepping both partially implicit fdtd methods and symplectic fdtd methods of high temporal accuracy 3rd or 4th order are well documented in the literature in this paper we combine them we construct a conservative fdtd method which is fourth order accurate in time and is partially implicit we show that the stability condition for this method depends exclusively on the explicit part which makes it suitable for use in e g modelling wave propagation in plasmas

conservative and provably stable fdtd subgridding the finite difference time domain fdtd method is a common robust simulation technique for transient electromagnetic interactions with complicated structures however the standard fdtd method is limited to cartesian grids everywhere in the computational grid many practitioners have extended fdtd to handle multiresolution problems by using finer grids near structures with small geometrical features abutted to coarse grids in regions of empty space unfortunately subgridding implementations based on interpolation or extrapolation of neighboring field values can exhibit late time instabilities herein a subgridding method based on multigrid finite element principles will be developed and its stability proven numerical results will assess its performance in 2 d and 3 d

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

integrable systems on so 4 related to xxx spin chains with boundaries we consider two site xxx heisenberg magnets with different boundary conditions which are integrable systems on so 4 possessing additional cubic and quartic integrals of motion the separated variables for these models are constructed using the sklyanin method

prediction of evanescent coupling efficiency in two parallel silica nanowires abstract in this study the efficiency of evanescent wave coupling between two air clad silica nanowires in single mode operation is numerically predicted in time domain using the finite difference method to this end a three dimensional scheme developed for solving the maxwell s equations on staggered grids will be employed the electric and magnetic field solutions are sought subjected to the zero divergence condition or gauss s law in discrete level in addition we aim to conserve hamiltonians existing in ideal maxwell s equations at all times by adopting the explicit second order accurate symplectic partitioned runge kutta temporal scheme moreover all the spatial derivative terms in the faraday s and ampere s equations are approximated by the proposed scheme which can render not only fourth order spatial accuracy but also minimize the discrepancy between the exact and the derived numerical phase velocities

finite difference solution of a nonlinear klein gordon equation with an external source in this paper we consider the darboux problem for a 1 1 dimensional cubic nonlinear klein gordon equation with an external source stable finite difference scheme is constructed on a four point stencil which does not require additional iterations for passing from one level to another it is proved that the finite difference scheme converges with the rate o h 2 when the exact solution belongs to the sobolev space w 2 2

a survey of kernel and spectral methods for clustering clustering algorithms are a useful tool to explore data structures and have been employed in many disciplines the focus of this paper is the partitioning clustering problem with a special interest in two recent approaches kernel and spectral methods the aim of this paper is to present a survey of kernel and spectral clustering methods two approaches able to produce nonlinear separating hypersurfaces between clusters the presented kernel clustering methods are the kernel version of many classical clustering algorithms e g k means som and neural gas spectral clustering arise from concepts in spectral graph theory and the clustering problem is configured as a graph cut problem where an appropriate objective function has to be optimized an explicit proof of the fact that these two paradigms have the same objective is reported since it has been proven that these two seemingly different approaches have the same mathematical foundation besides fuzzy kernel clustering methods are presented as extensions of kernel k means clustering algorithm

a generalized mass lumping technique for vector finite element solutions of the time dependent maxwell equations time domain finite element solutions of maxwell s equations require the solution of a sparse linear system involving the mass matrix at every time step this process represents the bulk of the computational effort in time dependent simulations as such mass lumping techniques in which the mass matrix is reduced to a diagonal or block diagonal matrix are very desirable in this paper we present a special set of high order 1 form also known as curl conforming basis functions and reduced order integration rules that together allow for a dramatic reduction in the number of nonzero entries in a vector finite element mass matrix the method is derived from the nedelec curl conforming polynomial spaces and is valid for arbitrary order hexahedral basis functions for finite element solutions to the second order wave equation for the electric or magnetic field intensity we present a numerical eigenvalue convergence analysis of the method and quantify its accuracy and performance via a series of computational experiments

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

approximate kernel orthogonalization for antenna array processing we present a method for kernel antenna array processing using gaussian kernels as basis functions the method first identifies the data clusters by using a modified sparse greedy matrix approximation then the algorithm performs model reduction in order to try to reduce the final size of the beamformer the method is tested with simulations that include two arrays made of two and seven printed half wavelength thick dipoles in scenarios with 4 and 5 users coming from different angles of arrival the antenna parameters are simulated for all doas and include the dipole radiation pattern and the mutual coupling effects of the array the method is compared with other state of the art nonlinear processing methods to show that the presented algorithm has near optimal capabilities together with a low computational burden

island coalescence using parallel first order system least squares on incompressible resistive magnetohydrodynamics this paper investigates the performance of a parallel newton first order system least squares fosls finite element method with local adaptive refinement and algebraic multigrid amg applied to incompressible resistive magnetohydrodynamics in particular an island coalescence test problem is studied that models magnetic reconnection using a reduced two dimensional 2d model of a tokamak fusion reactor the results show that using an appropriate temporal and spatial resolution these methods are capable of resolving the physical instabilities accurately at small computational cost the time dependent nonlinear system of pdes is solved using work equivalent to about 50 60 simple relaxation sweeps gauss seidel iterations per time step experiments show that unless the time step is sufficiently small nonphysical numerical instabilities may occur further decreasing the time step size does not proportionally increase the cost of the computation because amg convergence is improved in addition

nonnegative matrix factorization in polynomial feature space plenty of methods have been proposed in order to discover latent variables features in data sets such approaches include the principal component analysis pca independent component analysis ica factor analysis fa etc to mention only a few a recently investigated approach to decompose a data set with a given dimensionality into a lower dimensional space is the so called nonnegative matrix factorization nmf its only requirement is that both decomposition factors are nonnegative to approximate the original data the minimization of the nmf objective function is performed in the euclidean space where the difference between the original data and the factors can be minimized by employing l 2 norm in this paper we propose a generalization of the nmf algorithm by translating the objective function into a hilbert space also called feature space under nonnegativity constraints with the help of kernel functions we developed an approach that allows high order dependencies between the basis images while keeping the nonnegativity constraints on both basis images and coefficients two practical applications namely facial expression and face recognition show the potential of the proposed approach

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

on multi symplectic partitioned runge kutta methods for hamiltonian wave equations many conservative pdes such as various wave equations schrodinger equations kdv equations and so on allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes in this note we show the discretization to hamiltonian wave equations in space and time using two symplectic partitioned runge kutta methods respectively leads to multi symplectic integrators which preserve a symplectic conservation law under some conditions we discuss the energy and momentum conservative property of partitioned runge kutta methods for the wave equations with a quadratic potential

numerical stability of nonorthogonal fdtd methods in this paper a sufficient test for the numerical stability of generalized grid finite difference time domain fdtd schemes is presented it is shown that the projection operators of such schemes must be symmetric positive definite without this property such schemes can exhibit late time instabilities the origin and the characteristics of these late time instabilities are also uncovered based on this study nonorthogonal grid fdtd schemes nfdtd and the generalized yee gy methods are proposed that are numerically stable in the late time for quadrilateral prism elements allowing these methods to be extended to problems requiring very long time simulations the study of numerical stability that is presented is very general and can be applied to most solutions of maxwell s equations based on explicit time domain schemes

symplectic discretization for spectral element solution of maxwell s equations applying the spectral element method sem based on the gauss lobatto legendre gll polynomial to discretize maxwell s equations we obtain a poisson system or a poisson system with at most a perturbation for the system we prove that any symplectic partitioned runge kutta prk method preserves the poisson structure and its implied symplectic structure numerical examples show the high accuracy of sem and the benefit of conserving energy due to the use of symplectic methods

separation of variables in the elliptic gaudin model for the elliptic gaudin model a degenerate case of the xyz integrable spin chain a separation of variables is constructed in the classical case the corresponding separated coordinates are obtained as the poles of a suitably normalized baker akhiezer function the classical results are generalized to the quantum case where the kernel of the separating integral operator is constructed the simplest one degree of freedom case is studied in detail

an efficient vector finite element method for nonlinear electromagnetic modeling we have developed a mixed vector finite element method vfem for maxwell s equations with a nonlinear polarization term the method allows for discretization of complicated geometries with arbitrary order representations of the b and e fields in this paper we will describe the method and a series of optimizations that significantly reduce the computational cost additionally a series of test simulations will be presented to validate the method finally a nonlinear waveguide mode mixing example is presented and discussed

a time discontinuous implicit variational integrator for stress wave propagation analysis in solids this paper proposes a time discontinuous and space continuous variational integration tdsc vi method for accurate elastic stress wave propagation computations in one dimensional bar the algorithm employs a limiter akin to classical artificial viscosity treatment to mitigate the deleterious gibbs jumps across the stress discontinuities and a parametrized consistent mass to alleviate dispersion error when the stepsizes are different from the courant stability limit which becomes necessary for elastic unloading and internal reflections in plastic deformation problems stability and accuracy analyses of the proposed tdsc vi method are carried out and compared with several well known traditional integration algorithms numerical experiments are carried out with the proposed implicit and explicit methods which show that the proposed methods perform favorably compared to the trapezoidal rule and the central difference method

lattice electromagnetic theory from a topological viewpoint the language of differential forms and topological concepts are applied to study classical electromagnetic theory on a lattice it is shown that differential forms and their discrete counterparts cochains provide a natural bridge between the continuum and the lattice versions of the theory allowing for a natural factorization of the field equations into topological field equations i e invariant under homeomorphisms and metric field equations the various potential sources of inconsistency in the discretization process are identified distinguished and discussed a rationale for a consistent extension of the lattice theory to more general situations such as to irregular lattices is considered

symplectic mixed finite element approximation of linear acoustic wave equations we apply mixed finite element approximations to the first order form of the acoustic wave equation the semidiscrete method exactly conserves the system energy a fully discrete method employing the symplectic euler time method in time exactly conserves a positive definite pertubed energy functional that is equivalent to the actual energy under a cfl condition in addition to proving optimal order l infty l 2 l l2 estimates we also develop a bootstrap technique that allows us to derive stability and error bounds for the time derivatives and divergence of the vector variable beyond the standard under some additional regularity assumptions

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

time domain finite element methods various time domain finite element methods for the simulation of transient electromagnetic wave phenomena are discussed detailed descriptions of test trial spaces explicit and implicit formulations nodal and edge facet element basis functions are given along with the numerical stability properties of the different methods the advantages and disadvantages of mass lumping are examined finally the various formulations are compared on the basis of their numerical dispersion performance

painleve vi rigid tops and reflection equation we show that the painleve vi equation has an equivalent form of the non autonomous zhukovsky volterra gyrostat this system is a generalization of the euler top in mathbb c 3 and includes the additional constant gyrostat momentum the quantization of its autonomous version is achieved by the reflection equation the corresponding quadratic algebra generalizes the sklyanin algebra as by product we define integrable xyz spin chain on a finite lattice with new boundary conditions

twisted wess zumino witten models on elliptic curves investigated is a variant of the wess zumino witten model called a twisted wzw model which is associated to a certain lie group bundle on a family of elliptic curves the lie group bundle is a non trivial bundle with flat connection and related to the classical elliptic r matrix the usual non twisted wzw model is associated to a trivial group bundle with trivial connection on a family of compact riemann surfaces and a family of its principal bundles the twisted wzw model on a fixed elliptic curve at the critical level describes the xyz gaudin model the elliptic knizhnik zamolodchikov equations associated to the classical elliptic r matrix appear as flat connections on the sheaves of conformal blocks in the twisted wzw model

survey on symplectic finite difference time domain schemes for maxwell s equations to discretize maxwell s equations a variety of high order symplectic finite difference time domain schemes which use th order symplectic integration time stepping and th order staggered space differencing are surveyed first the order conditions for the symplectic integrators are derived second the comparisons of numerical stability dispersion and energy conservation are provided between the high order symplectic schemes and other high order time approaches finally these symplectic schemes are studied by using different space and time strategies according to our survey high order time schemes for matching high order space schemes are required for optimum electromagnetic simulation numerical experiments have been conducted on radiation of electric dipole and wideband s parameter extraction of dielectric filled waveguide the results demonstrate that the high order symplectic scheme can obtain satisfying numerical solutions under high courant friedrichs levy number and coarse grid conditions

development of a 3d staggered fdtd scheme for solving maxwell s equations in drude medium an explicit finite difference scheme is developed to solve the three dimensional maxwell s equations in drude medium our aim of developing this scheme in time domain is to compute solutions in staggered grids from the faraday s and ampere s equations the electric and magnetic field solutions are sought subject to the discrete zero divergence condition gauss s law the local conservation laws in ideal maxwell s equations are also numerically preserved all the time using the explicit second order accurate symplectic partitioned runge kutta temporal scheme the spatial derivative terms in the faraday s and ampere s equations are discretized to obtain fourth order accuracy using the developed scheme underlying the concept of minimizing the discrepancy between the exact and the derived numerical phase velocities dispersion and anisotropy errors have been much reduced through the procedure of minimizing phase velocity error in addition to performing the fundamental analysis on the proposed scheme the computational efficiency and long term accurate properties embedded in the proposed symplectic dispersion error reduction centered scheme are numerically demonstrated through several test problems investigated in ideal and drude media

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

rational top and its classical r matrix we construct a rational integrable system the rational top on a co adjoint orbit of sl n lie group it is described by the lax operator with spectral parameter and classical non dynamical skew symmetric r matrix in the case of the orbit of minimal dimension the model is gauge equivalent to the rational calogero moser cm system to obtain the results we represent the lax operator of the cm model in two different factorized forms without spectral parameter related to the spinless case and another one with the spectral parameter the latter gives rise to the rational top while the first one is related to generalized cremmer gervais r matrices the gauge transformation relating the rational top and cm model provides the classical rational version of the irf vertex correspondence from the geometrical point of view it describes the modification of bundles over degenerated elliptic curve in view of the symplectic hecke correspondence the rational top is related to the rational spin cm model possible applications and generalizations of the suggested construction are discussed in particular the obtained r matrix defines a class of kzb equations

time integration methods for finite element discretisations of the second order maxwell equation this article deals with time integration for the second order maxwell equations with possibly non zero conductivity in the context of the discontinuous galerkin finite element method dg fem and the h curl conforming fem for the spatial discretisation hierarchic h curl conforming basis functions are used up to polynomial order p 3 over tetrahedral meshes meaning fourth order convergence rate a high order polynomial basis often warrants the use of high order time integration schemes but many well known high order schemes may suffer from a severe time step stability restriction owing to the conductivity term we investigate several possible time integration methods from the point of view of accuracy stability and computational work we also carry out a numerical fourier analysis to study the dispersion and dissipation properties of the semi discrete dg fem scheme as well as the fully discrete schemes with several of the time integration methods the dispersion and dissipation properties of the spatial discretisation and those of the time integration methods are investigated separately providing additional insight into the two discretisation steps

high order interface treatment techniques for modeling curved dielectric objects three modified conformal techniques combined with high order finite difference time domain method fdtd 2 4 are proposed to investigate the interaction of electromagnetic waves with three dimensional 3 d curved dielectric surfaces these proposed conformal techniques utilize an effective average dielectric constant to modify the update equations in the fdtd 2 4 scheme which is derived by area linear and volume averages of different dielectric regions in twenty seven spatial discrete cells at most respectively some numerical results are presented to show the accuracies of linear and area average techniques good agreements are obtained with those of the method of moments and other analytical ones even with coarse meshes adopted for handling electrically large or complex geometries effectively further using our proposed techniques the rcs of some typical 3 d objects are predicted and studied in detail

numerical integration of damped maxwell equations we study the numerical time integration of maxwell s equations from electromagnetism following the method of lines approach we start from a general semidiscrete maxwell system for which a number of time integration methods are considered these methods have in common an explicit treatment of the curl terms central in our investigation is the question how to efficiently raise the temporal convergence order beyond the standard order of two in particular in the presence of an explicitly or implicitly treated damping term which models conduction

the optimal force gradient symplectic finite difference time domain scheme for electromagnetic wave propagation in this communication we design an optimal force gradient symplectic finite difference time domain sfdtd scheme for numerically solving maxwell s equations based on the additional constraint of the minimized norm of fourth order truncation errors the numerical dissipation can be minimized and the optimal time coefficients for the third order force gradient symplectic method can be obtained due to the second order accurate adopted in space domain in general the proposed optimal sfdtd method is still second order accurate the numerical stability and numerical dispersion analysis indicate that the optimal force gradient sfdtd method has higher accuracy and stability compared with the nonoptimal force gradient sfdtd method moreover in order to verify the validity of the presented method the scattering parameters of a line fed rectangular microstrip antenna are calculated

spectral embedded clustering a framework for in sample and out of sample spectral clustering spectral clustering sc methods have been successfully applied to many real world applications the success of these sc methods is largely based on the manifold assumption namely that two nearby data points in the high density region of a low dimensional data manifold have the same cluster label however such an assumption might not always hold on high dimensional data when the data do not exhibit a clear low dimensional manifold structure e g high dimensional and sparse data the clustering performance of sc will be degraded and become even worse than k means clustering in this paper motivated by the observation that the true cluster assignment matrix for high dimensional data can be always embedded in a linear space spanned by the data we propose the spectral embedded clustering sec framework in which a linearity regularization is explicitly added into the objective function of sc methods more importantly the proposed sec framework can naturally deal with out of sample data we also present a new laplacian matrix constructed from a local regression of each pattern and incorporate it into our sec framework to capture both local and global discriminative information for clustering comprehensive experiments on eight real world high dimensional datasets demonstrate the effectiveness and advantages of our sec framework over existing sc methods and k means based clustering methods our sec framework significantly outperforms sc using the nystrom algorithm on unseen data

numerical dispersion analysis of a multi symplectic scheme for the three dimensional maxwell s equations in this paper we study a multi symplectic scheme for three dimensional maxwell s equations in a simple medium this is a system of pdes with multi symplectic structures we prove that this multi symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence furthermore we extend the discussion to several dispersion properties of the multi symplectic scheme including the numerical dispersion relation the numerical group velocity the effect of large time steps and the cfl condition

the lobatto cell robust explicit higher order fdtd that handles inhomogeneous media to improve numerical dispersion and second order convergence behavior of the traditional yee finite difference time domain fdtd algorithm various higher order finite difference implementations have been proposed many have difficulty handling interfaces between distinct dielectric media or metallic sheets because the high order taylor expansions on which these codes are based are invalidated by field or derivative discontinuities one sided extensions extrapolations are a common approach to remedy interface discontinuities here another approach based upon whitney form mass lumped finite elements is pursued to develop the ldquolobatto cell rdquo a high order replacement for the yee cell with appropriate continuity behavior at contrasting media interfaces the lobatto cell retains many robust features of the yee method discrete conservation laws a guarantee of conditional stability and straightforward explicit updating but provides higher order interpolation and dispersion error convergence a notable drawback of the method is larger modeling granularity staircasing which is managed here by combining the lobatto cell method with preexisting robust subgridding techniques

explicit local time stepping methods for maxwell s equations explicit local time stepping methods are derived for time dependent maxwell equations in conducting and non conducting media by using smaller time steps precisely where smaller elements in the mesh are located these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators when combined with a finite element discretisation in space with an essentially diagonal mass matrix the resulting discrete time marching schemes are fully explicit and thus inherently parallel in a non conducting source free medium they also conserve a discrete energy which provides a rigorous criterion for stability starting from the standard leap frog scheme local time stepping methods of arbitrarily high accuracy are derived for non conducting media numerical experiments with a discontinuous galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes

spectral curve and hamiltonian structure of isomonodromic su 2 calogero gaudin system this paper presents a new approach to the hamiltonian structure of isomonodromic deformations of a matrix system of ordinary differential equations odes on a torus an isomonodromic analogue of the su 2 calogero gaudin system is used for a case study of this approach a clue of this approach is a mapping to a finite number of points on the spectral curve of the isomonodromic lax equation the coordinates of these moving points give a new set of darboux coordinates called the spectral darboux coordinates the system of isomonodromic deformations is thereby converted to a nonautonomous hamiltonian system in the spectral darboux coordinates the hamiltonians turn out to resemble those of a previously known isomonodromic system of a second order scalar ode the two isomonodromic systems are shown to be linked by a simple relation

kernel based hard clustering methods with kernelization of the metric and automatic weighting of the variables this paper presents kernel based hard clustering methods with kernelization of the metric and automatic weighting of the variables the proposed methodology is supported by the fact that a kernel function can be written as a sum of kernels evaluated separately on each variable thus in the proposed algorithms dissimilarity measures are obtained as sums of euclidean distances between patterns and centroids computed individually for each variable by means of kernels the main advantage of this approach over the conventional approach is that it allows the use of kernelized adaptive distances which are suitable to learn the weights of the variables dynamically improving the performance of the algorithms moreover various partition and cluster interpretation tools are introduced experiments with a number of benchmark datasets corroborate the usefulness of the proposed algorithms and the merit of the partition and cluster interpretation tools highlightspresents kernel based clustering methods with automatic weighting of the variables kernelized local and global adaptive distances are introduced the proposed algorithms are suitable to learn the weights of the variables partition and cluster interpretation tools are given experiments with several benchmark data sets corroborate the proposed methods

a new solution of schrodinger equation based on symplectic algorithm an explicit three dimensional symplectic finite difference time domain method is introduced to solve the schrodinger equation the method is obtained by using a symplectic scheme with a propagator of exponential differential operators for the time direction approximation and fourth order collocated finite differences for the space discretization firstly a high order symplectic framework for discretizing the schrodinger equation is presented secondly the comparisons on numerical stability dispersion are also provided between the new symplectic scheme and other commonly used methods finally numerical examples are given to evaluate the performance of the proposed method and the advantages on the accuracy and stability are also further demonstrated

a general approach for the stability analysis of the time domain finite element method for electromagnetic simulations this paper presents a general approach for the stability analysis of the time domain finite element method tdfem for electromagnetic simulations derived from the discrete system analysis the approach determines the stability by analyzing the root locus map of a characteristic equation and evaluating the spectral radius of the finite element system matrix the approach is applicable to the tdfem simulation involving dispersive media and to various temporal discretization schemes such as the central difference forward difference backward difference and newmark methods it is shown that the stability of the tdfem is determined by the material property and by the temporal and spatial discretization schemes the proposed approach is applied to a variety of tdfem schemes which include 1 time domain finite element modeling of dispersive media 2 time domain finite element boundary integral method 3 higher order tdfem and 4 orthogonal tdfem numerical results demonstrate the validity of the proposed approach for stability analysis

backlund transformations for the elliptic gaudin model and a clebsch system a two parameters family of backlund transformations for the classical elliptic gaudin model is constructed the maps are explicit symplectic preserve the same integrals as for the continuous flows and are a time discretization of each of these flows the transformations can map real variables into real variables sending physical solutions of the equations of motion into physical solutions the starting point of the analysis is the integrability structure of the model it is shown how the analogue transformations for the rational and trigonometric gaudin model are a limiting case of this one an application to a particular case of the clebsch system is given

application of the symplectic finite difference time domain scheme to electromagnetic simulation an explicit fourth order finite difference time domain fdtd scheme using the symplectic integrator is applied to electromagnetic simulation a feasible numerical implementation of the symplectic fdtd sfdtd scheme is specified in particular new strategies for the air dielectric interface treatment and the near to far field nff transformation are presented by using the sfdtd scheme both the radiation and the scattering of three dimensional objects are computed furthermore the energy conserving characteristic hold for the sfdtd scheme is verified under long term simulation numerical results suggest that the sfdtd scheme is more efficient than the traditional fdtd method and other high order methods and can save computational resources

optimization of the artificially anisotropic parameters in wcs fdtd method for reducing numerical dispersion a 3 d artificial anisotropy weakly conditionally stable finite difference time domain aa wcs fdtd method is presented to reduce the dispersion error without introducing additional computational cost and complexity the motivation is to equal the speed of light in the numerical simulation to the physical speed through manipulating the permittivity and permeability the proposed aa wcs fdtd algorithm achieves the reduced dispersion error for the arbitrary incident angles different time step sizes and wideband frequency the analyses of the stability and dispersion indicate that the proposed algorithm is wcs and the courant friedrich levy condition is slightly changed the numerical example demonstrates that the proposed method has a reduced dispersion error and high accuracy as compared to the wcs fdtd method

three dimensional orthogonal vector basis functions for time domain finite element solution of vector wave equations three dimensional 3 d orthogonal vector basis functions are developed for the time domain finite element solution of vector wave equations these basis functions enforce both the tangential continuity of the electric field and the normal continuity of the electric flux the stability of the resulting time domain finite element schemes is investigated and demonstrated to be guaranteed the use of the proposed basis functions completely eliminates the matrix solution at each time step required by the time domain finite element solution of vector wave equations the computational cost thereby scales as o n sub t n with n sub t and n denoting the number of time steps and the number of unknowns respectively defined over tetrahedral elements the proposed basis functions increase the solution efficiency without compromising the geometry modeling flexibility both numerical results and comparison with traditional vector basis functions demonstrate the accuracy as well as the efficiency of the proposed three dimensional orthogonal vector bases

toward the optimization of normalized graph laplacian normalized graph laplacian has been widely used in many practical machine learning algorithms e g spectral clustering and semisupervised learning however all of them use the euclidean distance to construct the graph laplacian which does not necessarily reflect the inherent distribution of the data in this brief we propose a method to directly optimize the normalized graph laplacian by using pairwise constraints the learned graph is consistent with equivalence and nonequivalence pairwise relationships and thus it can better represent similarity between samples meanwhile our approach unlike metric learning automatically determines the scale factor during the optimization the learned normalized laplacian matrix can be directly applied in spectral clustering and semisupervised learning algorithms comprehensive experiments demonstrate the effectiveness of the proposed approach

an explicit fourth order staggered finite difference time domain method for maxwell s equations a new explicit fourth order accurate staggered finite difference time domain fdtd scheme is proposed and applied to electromagnetic wave problems it is fourth order accurate in both space and time conditionally stable and highly efficient with respect to yee s scheme and still retains much of the original simplicity of yee s scheme both extension to perfectly matched layers and modification to deal with dielectric interfaces and perfectly conducting boundaries of the scheme have also been presented numerical examples are shown to illustrate the efficiency of the method