derived category automorphisms from mirror symmetry inspired by the homological mirror symmetry conjecture of kontsevich 30 we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi projective variety msc 2000 18e30 14j32

adi scheme of a nonstandard fdtd method and its numerical properties an alternating direction implicit adi scheme for a nonstandard ns finite difference time domain fdtd method is proposed the ns fdtd method is derived by analyzing the dispersion relation two different groups of update coefficients are used to update the electric field and magnetic field respectively the dispersion property of the ns method is similar with that of the standard higher order method though it is only second order accurate in the space domain the total number of nonzero update coefficients for this method is fewer than that of the latter this small stencil property can be used to reduce computational burden in implicit formulation of this method the one step leapfrog adi scheme is adopted to examine this idea the detailed update equations of this adi scheme are derived the dispersion and stability properties are also analyzed and compared with the corresponding standard one the results show that the computational efficiency of the proposed method is higher than that of the standard one

mixed braid group actions from deformations of surface singularities we consider a set of toric calabi yau varieties which arise as deformations of the small resolutions of type a surface singularities by careful analysis of the heuristics of b brane transport in the associated gauged linear sigma models we predict the existence of a mixed braid group action on the derived category of each variety and then prove that this action does indeed exist this generalizes the braid group action found by seidel and thomas for the undeformed resolutions we also show that the actions for different deformations are related in a way that is predicted by the physical heuristics

convergence of time splitting energy conserved symplectic schemes for 3d maxwell s equations we propose two symplectic and two non symplectic schemes for 3d maxwell s equations based on the exponential operator splitting technique and fourier pseudo spectral method these schemes are efficient and unconditionally stable and also preserve four discrete energy conservation laws simultaneously the error estimates of the schemes are obtained by using some special techniques and the energy method numerical results confirm the theoretical analysis the numerical comparison with some existing methods show the good performance of the proposed schemes

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

analysis of iterated adi fdtd schemes for maxwell curl equations the convergence of the iterative adi fdtd method proposed by wang et al s wang f teixeira j chen an iterative adi fdtd with reduced splitting error ieee microwave wireless comp lett 15 2005 1531 1533 towards the classical implicit crank nicolson scheme when applied to maxwell curl equations and the accuracy stability and dispersion properties of the resulting iterated schemes are investigated the iterated schemes are shown both mathematically and numerically to be unconditionally stable for 2d wave problems in agreement with numerical experiments conducted in s wang f teixeira j chen an iterative adi fdtd with reduced splitting error ieee microwave wireless comp lett 15 2005 1531 1533 however these schemes lose their unconditional stability when applied to full 3d wave problems where te and tm modes do not decouple as illustrated by numerical experiments in a pec box

confidence intervals for multinomial logistic regression in sparse data logistic regression is one of the most widely used regression models in practice but alternatives to conventional maximum likelihood estimation methods may be more appropriate for small or sparse samples modification of the logistic regression score function to remove first order bias is equivalent to penalizing the likelihood by the jeffreys prior and yields penalized maximum likelihood estimates ples that always exist even in samples in which maximum likelihood estimates mles are infinite ples are an attractive alternative in small to moderate sized samples and are preferred to exact conditional mles when there are continuous covariates we present methods to construct confidence intervals ci in the penalized multinomial logistic regression model and compare ci coverage and length for the ple based methods to that of conventional mle based methods in trinomial logistic regressions with both binary and continuous covariates based on simulation studies in sparse data sets we recommend profile cis over asymptotic wald type intervals for the ples in all cases furthermore when finite sample bias and data separation are likely to occur we prefer ple profile cis over mle methods

a uniformly exponentially stable adi scheme for maxwell equations abstract a modified alternating direction implicit scheme for the time integration of linear isotropic maxwell equations with strictly positive conductivity on cuboids is constructed a key feature of the proposed scheme is its uniform exponential stability being achieved by coupling the maxwell system with an additional damped pde and adding artificial damping to the scheme the implicit steps in the resulting time integrator further decouple into essentially one dimensional elliptic problems requiring only linear complexity the convergence of the scheme to the solution of the original maxwell system is analyzed in the abstract time discrete setting providing an error bound in a space related to h 1

divergence preservation in the adi algorithms for electromagnetics the recent advances in alternating direct implicit adi methods promise important new capability for time domain plasma simulations namely the elimination of numerical stability limits on the time step but the utility of these methods in simulations with charge and current sources such as in electromagnetic particle in cell empic computations has been uncertain as the methods introduced so far do not have the property of divergence preservation this property is related to charge conservation and self consistency and is critical for accurate and robust empic simulation this paper contains a complete study of these adi methods in the presence of charge and current sources it is shown that there are four significantly distinct cases with four more related by duality of those only one preserves divergence and thus is guaranteed to be stable in the presence of moving charged particles computational verification of this property is accomplished by implementation in existing 3d empic simulation software of the other three cases two are verified unstable as expected and one remains stable despite the lack of divergence preservation this other stable algorithm is shown to be related to the divergence preserving case by a similarity transformation effectively providing the complement of the divergence preserving field in the finite difference energy quantity

finite element time domain solution of the vector wave equation in doubly dispersive media using mobius transformation technique several finite element time domain fetd formulations to model inhomogeneous and electrically magnetically doubly dispersive materials based on the second order vector wave equation discretized by the newmark scheme are developed in contrast to the existing formulations which employ recursive convolution rc approaches we use a mobius transformation method to derive our new formulations hence the obtained equations are not only simpler in form and easier to derive and implement but also do not suffer from the intrinsic limitations of the rc methods in modeling arbitrary high order media to obtain the formulations we first demonstrate that the update equation for the electric field strength e in the mixed crank nicolson cn fetd formulation which is based on expanding the electric and magnetic field in terms of the edge and face elements in space and discretizing the resultant first order differential equations using crank nicolson scheme in time is equivalent to the unconditionally stable us second order vector wave equation for the same variable e discretized by the newmark method with 1 4 in addition we show that the update equation for the magnetic flux density b in cn fetd is the same as the second order vector wave equation for b on the dual grid discretized again by a similar newmark method subsequently thanks to the mixed fetd formulation properties we derive update equations for the constitutive relations using a mobius transformation method separately in addition we use the shown equivalence to derive formulations based on the vector wave equation finally several numerical examples are solved to validate the developed formulations

autoequivalences of derived categories via geometric invariant theory abstract we study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of git quotient we show that these autoequivalences are spherical twists and describe how they result from mutations of semiorthogonal decompositions beyond the git setting we show that all spherical twist autoequivalences of a dg category can be obtained from mutation in this manner motivated by a prediction from mirror symmetry we refine the recent notion of grade restriction rules in equivariant derived categories we produce additional derived autoequivalences of a git quotient and propose an interpretation in terms of monodromy of the quantum connection we generalize this observation by proving a criterion under which a spherical twist autoequivalence factors into a composition of other spherical twists

high order unconditionally stable fc ad solvers for general smooth domains ii elliptic parabolic and hyperbolic pdes theoretical considerations a new pde solver was introduced recently in part i of this two paper sequence on the basis of two main concepts the well known alternating direction implicit adi approach on one hand and a certain fourier continuation fc method for the resolution of the gibbs phenomenon on the other unlike previous alternating direction methods of order higher than one which only deliver unconditional stability for rectangular domains the new high order fc ad fourier continuation alternating direction algorithm yields unconditional stability for general domains at an o nlog n cost per time step for an n point spatial discretization grid in the present contribution we provide an overall theoretical discussion of the fc ad approach and we extend the fc ad methodology to linear hyperbolic pdes in particular we study the convergence properties of the newly introduced fc gram fourier continuation method for both approximation of general functions and solution of the alternating direction odes we also present for parabolic pdes on general domains and thus for our associated elliptic solvers a stability criterion which when satisfied ensures unconditional stability of the fc ad algorithm use of this criterion in conjunction with numerical evaluation of a series of singular values of the alternating direction discrete one dimensional operators suggests clearly that the fifth order accurate class of parabolic and elliptic fc ad solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations to illustrate the fc ad methodology in the hyperbolic pde context finally we present an example concerning the wave equation demonstrating sixth order spatial and fourth order temporal accuracy as well as a complete absence of the debilitating dispersion error also known as pollution error that arises as finite difference and finite element solvers are applied to solution of wave propagation problems

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

an efficient 3 d hie fdtd method with weaker stability condition for analyzing the electromagnetic objects with the fine structures in one direction efficiently a three dimensional 3 d hybrid implicit explicit finite difference time domain hie fdtd method with weaker stability condition is proposed in this paper first by adopting the alternating direction implicit adi scheme the 3 d maxwell s equations are subdivided into two substep procedures and then by adding some disturbance terms into two substeps respectively the proposed method is derived compared with the existing hie fdtd methods the courant friedrichs lewy cfl stability condition of the proposed hie fdtd method is weaker which means the proposed method is more efficient due to the possibility of choosing the larger time step size in addition the relationships between the cfl stability conditions and the numerical dispersion errors of the traditional fdtd method the three hie fdtd methods and the adi fdtd method are summarized in order to provide a basis for the choice among them

te tm scheme for computation of electromagnetic fields in accelerators we propose a new two level economical conservative scheme for short range wake field calculation in three dimensions the scheme does not have dispersion in the longitudinal direction and is staircase free second order convergent unlike the finite difference time domain method fdtd it is based on a te tm like splitting of the field components in time additionally it uses an enhanced alternating direction splitting of the transverse space operator that makes the scheme computationally as effective as the conventional fdtd method unlike the fdtd adi and low order strang methods the splitting error in our scheme is only of fourth order as numerical examples show the new scheme is much more accurate on the long time scale than the conventional fdtd approach

an efficient method to reduce the numerical dispersion in the lod fdtd method based on the 2 4 stencil a parameter optimized 2 4 stencil based locally one dimensional lod finite difference time domain fdtd is presented with much reduced numerical dispersion errors the method is first proved to be unconditionally stable then by using different optimization schemes the method is optimized to satisfy different accuracy requirements such as minimum dispersion errors in the axial directions in the diagonal direction and in the specified angles performances of the parameter optimized lod fdtd with different time steps and frequencies are also studied it is found that the parameter optimization can significantly reduce numerical dispersion errors bringing them down to the level of the conventional fdtd but with the time step exceeding the cfl limit and without much additional computational cost in addition the optimized parameters are not sensitive to frequencies in particular the optimized parameters obtained at a higher frequency still present low numerical dispersion errors at a lower frequency

a stable fdtd method with embedded reduced order models the computational efficiency of the finite difference time domain fdtd method can be significantly reduced by the presence of complex objects with fine features small geometrical details impose a fine mesh and a reduced time step significantly increasing computational cost model order reduction has been proposed as a systematic way to generate compact models for complex objects by which one can then instantiate into a main fdtd mesh however the stability of fdtd with embedded reduced models remains an open problem we propose a systematic method to generate reduced models for fdtd domains and embed them into a main fdtd mesh with guaranteed stability up to the courant friedrichs lewy cfl limit of the fine mesh with a simple perturbation technique the cfl of the whole scheme can be further extended beyond the fine grid s cfl limit reduced models can be created for arbitrary domains containing inhomogeneous and lossy materials numerical tests confirm the stability of the proposed method and its potential to accelerate multiscale fdtd simulations

stability and accuracy of time extrapolated adi fdtd methods for solving wave equations some recent work on the adi fdtd method for solving maxwell s equations in 3 d have brought out the importance of extrapolation methods for the time stepping of wave equations such extrapolation methods have previously been used for the solution of odes the present context of wave equations brings up two main questions which have not been addressed previously 1 when will extrapolation in time of an unconditionally stable scheme for a wave equation again feature unconditional stability and 2 how will the accuracy and computational efficiency depend on how frequently in time the extrapolations are carried out we analyze these issues here

an updated review of general dispersion relation for conditionally and unconditionally stable fdtd algorithms we present a general numerical dispersion equation which is applicable to all known conditionally and unconditionally stable finite difference time domain fdtd algorithms on staggered rectangular grid including yee s fdtd wavelet based fdtd extended curl fdtd alternating direction implicit adi fdtd crank nicolson cn fdtd crank nicolson split step cnss fdtd and their modifications of higher order spatial stencils the real part of the complex eigenvalue of the total amplification matrix defines and distinguishes the dispersion relation for each individual scheme easy to check conditions are provided under which the numerical dispersion of a particular time domain scheme is governed by the proposed dispersion equation these conditions are on the amplification matrix eigenvalues the proposed dispersion equation includes each considered dispersion relation as a special case and presents itself a general governing equation to estimate 3 d numerical dispersion of the aforementioned schemes in the frame of plane waves

locally implicit discontinuous galerkin method for time domain electromagnetics in the recent years there has been an increasing interest in discontinuous galerkin time domain dgtd methods for the solution of the unsteady maxwell equations modeling electromagnetic wave propagation one of the main features of dgtd methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems such dgtd methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices however explicit dgtd methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation an implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step a more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part resulting in an hybrid explicit implicit or locally implicit time integration strategy in this paper we report on our recent efforts towards the development of such a hybrid explicit implicit dgtd method for solving the time domain maxwell equations on unstructured simplicial meshes numerical experiments for 3d propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes

numerical computations for n coupled nonlinear schrodinger equations by split step spectral methods in this paper various split step spectral sssp schemes are proposed for n coupled nonlinear schrodinger n cnls equations especially for the systems with n 3 for which numerical studies are few these schemes are spectrally accurate in space and sth s 1 2 4 6 8 order in time they are proved to be conservative and to admit the exact plane wave solution extensive numerical experiments are carried out for the 3 and 4 cnls systems to confirm the theoretical analysis half of the schemes are shown to possess wonderful ability of capturing high frequency waves however the eighth order schemes which seems to be optimal fail in this test accuracy and efficiency of the schemes are compared with each other and the high order schemes exhibit better since the eighth order schemes are not better than the sixth order ones it is believed that constructing too much high order schemes are unnecessary moreover interactions of two soliton solutions are also well simulated by the fourth order and the sixth order schemes so the two methods are sufficient for use

an efficient 3 d stochastic hie fdtd algorithm for investigation of statistical variation in electromagnetic field we propose a stochastic hybrid implicit explicit finite difference time domain method s hie fdtd to compute the mean and variance of the electromagnetic em fields using a single simulation given those of the conductivity and permittivity in the computation domain the mean and variance field update equations underlying the proposed method are derived from the field update equations of the traditional deterministic hie fdtd the courant friedrichs lewy condition of the s hie fdtd depends on the spatial discretization sizes only in two dimensions therefore for computation domains with fine geometric features only in the remaining dimension it uses a time step size that is larger than that of fully explicit schemes indeed numerical results demonstrate that the proposed method is faster than the previously developed stochastic fdtd in computing the mean and variance of the em fields in two different problems wave propagation through a multilayer human tissue and transmission through a frequency selective surface

a split step fdtd method for 3 d maxwell s equations in general anisotropic media a split step finite difference time domain fdtd method is presented for 3 d maxwell s equations in general anisotropic media the general anisotropic media can be characterized by full permittivity and permeability tensors stability analysis of the proposed split step fdtd method using the fourier method is presented the eigenvalues of the fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the courant limit time step thereby illustrating the stable and non dissipative nature of the split step fdtd method in general anisotropic media numerical results are presented to further validate the accuracy and stability of the proposed split step fdtd method in general anisotropic media

robust inference with gmm estimators the local robustness properties of generalized method of moments gmm estimators and of a broad class of gmm based tests are investigated in a unified framework gmm statistics are shown to have bounded influence if and only if the function defining the orthogonality restrictions imposed on the underlying model is bounded since in many applications this function is unbounded it is useful to have procedures that modify the starting orthogonality conditions in order to obtain a robust version of a gmm estimator or test we show how this can be obtained when a reference model for the data distribution can be assumed we develop a flexible algorithm for constructing a robust gmm rgmm estimator leading to stable gmm test statistics the amount of robustness can be controlled by an appropriate tuning constant we relate by an explicit formula the choice of this constant to the maximal admissible bias on the level or and the power of a gmm test and the amount of contamination that one can reasonably assume given some information on the data finally we illustrate the rgmm methodology with some simulations of an application to rgmm testing for conditional heteroscedasticity in a simple linear autoregressive model in this example we find a significant instability of the size and the power of a classical gmm testing procedure under a non normal conditional error distribution on the other side the rgmm testing procedures can control the size and the power of the test under non standard conditions while maintaining a satisfactory power under an approximatively normal conditional error distribution

higher order lod fdtd methods and their numerical dispersion properties higher order locally one dimensional finite difference time domain ho lod fdtd methods are proposed here to improve the accuracy of the four step lod fdtd five step lod fdtd and six step lod fdtd methods in the proposed ho lod fdtd methods higher order finite difference approximations are used for all the three space derivatives therefore they give improved accuracy in all the three directions in addition to ho lod fdtd methods hybrid ho lod fdtd hho lod fdtd methods are also proposed here in the hho lod fdtd methods higher order finite difference approximations are used for the desired direction s only as a result they give improved accuracy in the desired direction s and take less computational time than the ho lod fdtd methods moreover to validate the analytical results simulation results of the fourth order fo and hybrid fo lod fdtd methods are also discussed here

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

new energy analysis of adi fdtd for the two dimensional maxwell equations several new energy identities of the 2d maxwell equations with the perfectly electric conducting boundary conditions are first proposed and proved by the energy methods then the energy identities of the 2d adi fdtd method in the discrete l 2 and h 1 norms are derived by these identities it is proved that the 2d adi fdtd method is unconditionally stable and energy conserved in the discrete l 2 and h 1 norms numerical experiments are carried out and confirm the analysis on stability and energy conservation

an explicit and unconditionally stable fdtd method for the analysis of general 3 d lossy problems the root cause of the instability of an explicit finite difference time domain fdtd method is quantitatively identified for the analysis of general lossy problems where both dielectrics and conductors can be lossy and inhomogeneous based on the root cause analysis an efficient algorithm is developed to eradicate the root cause of instability and subsequently achieve unconditional stability in an explicit fdtd based simulation of general lossy problems numerical experiments have demonstrated the unconditional stability accuracy and efficiency of the proposed method

a new family of fourth order locally one dimensional schemes for the three dimensional wave equation in this paper we present a new family of locally one dimensional lod schemes with fourth order accuracy in both space and time for the three dimensional 3d acoustic wave equation it is well known that high order explicit schemes offer efficient time steppings but with restrictive cfl conditions implicit discretization gives unconditional stable schemes but with inefficient time steppings our lod schemes can be seen as a compromise between explicit and implicit schemes in the way that our schemes have more relax cfl conditions compared with traditional explicit schemes and only require solutions of tridiagonal systems of linear algebraic equations which is more efficient than implicit schemes moreover the new scheme is four layer in time and three layer in space so that boundary conditions can be imposed in the classical way the computations of the initial conditions for the three intermediate time layers are explicitly constructed furthermore the stability condition is derived and the restriction for the time step is given explicitly finally numerical examples are completed to show the performance of our new method

convergence of an adi splitting for maxwell s equations the convergence of an alternating direction implicit method for maxwell s equations on product domains is investigated unlike the classical yee scheme and most other integrators proposed in the literature this method is both unconditionally stable and computationally cheap we prove second order convergence of the time discretization in the framework of operator semigroup theory in contrast to formal considerations based on taylor expansions our convergence analysis respects the unboundedness of the involved differential operators the proofs are based on results concerning the regularity of the cauchy problems which then allow to apply an abstract convergence proof by hansen and ostermann numer math 108 557 570 2008

canonical runge kutta nystro m methods of orders five and six abstract in this paper we construct canonical explicit five stage and seven stage runge kutta nystrom methods of orders five and six respectively for hamiltonian dynamical systems

minimax aspects of bounded influence regression abstract bounded influence regression in the sense of hampel mallows krasker welsch is examined critically from the point of view of finite sample minimax robust estimation theory the main conclusion is that the influence of position of very high leverage points that is of points where the diagonal element of the hat matrix h x xtx 1 xt exceeds a certain bound should be cut down selectively this bound should be chosen roughly between 2 and 5 for the upper of these values the resulting estimate can be approximated very simply by scaling the residuals by their own estimated standard deviation a conflict between decision theoretic estimation and data analytic diagnostic viewpoints is pointed out and briefly discussed

an arbitrary order lod fdtd method and its stability and numerical dispersion an arbitrary order unconditionally stable three dimensional 3 d locally one dimensional finite difference time method fdtd lod fdtd method is proposed theoretical proof and numerical verification of the unconditional stability are shown and numerical dispersion is derived analytically effects of discretization parameters on the numerical dispersion errors are studied comprehensively it is found that the second order lod fdtd has the same level of numerical dispersion error as that of the unconditionally stable alternating direction implicit finite difference time domain adi fdtd method and other lod fdtd methods but with higher computational efficiency to reduce the dispersion errors either a higher order lod fdtd method or a denser grid can be applied but the choice has to be carefully made in order to achieve best trade off between the accuracy and computational efficiency the work presented in this paper lays the foundations and guidelines for practical uses of the lod method including the potential mixed order lod fdtd methods

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

an integral structure in quantum cohomology and mirror symmetry for toric orbifolds abstract we introduce an integral structure in orbifold quantum cohomology associated to the k group and the class in the case of compact toric orbifolds we show that this integral structure matches with the natural integral structure for the landau ginzburg model under mirror symmetry by assuming the existence of an integral structure we give a natural explanation for the specialization to a root of unity in y ruan s crepant resolution conjecture yongbin ruan the cohomology ring of crepant resolutions of orbifolds in contemp math vol 403 amer math soc providence ri 2006 pp 117 126

equivalences between git quotients of landau ginzburg b models we define the category of b branes in a not necessarily affine landau ginzburg b model incorporating the notion of r charge our definition is a direct generalization of the category of perfect complexes we then consider pairs of landau ginzburg b models that arise as different git quotients of a vector space by a one dimensional torus and show that for each such pair the two categories of b branes are quasi equivalent in fact we produce a whole set of quasi equivalences indexed by the integers and show that the resulting auto equivalences are all spherical twists

higher order a stable schemes for the wave equation using a successive convolution approach in several recent works we developed a new second order a stable approach to wave propagation problems based on the method of lines transpose mol t formulation combined with alternating direction implicit adi schemes because our method is based on an integral solution of the adi splitting of the mol t formulation we are able to easily embed non cartesian boundaries and include point sources with exact spatial resolution further we developed an efficient o n convolution algorithm for rapid evaluation of the solution which makes our method competitive with explicit finite difference e g finite difference time domain solvers in terms of both accuracy and time to solution even for courant numbers slightly larger than 1 we have demonstrated the utility of this method by applying it to a range of problems with complex geometry including cavities with cusps in this work we present several important modifications to our recently developed wave solver we obtain a family of wave solvers

unconditionally stable integration of maxwell s equations numerical integration of maxwell s equations is often based on explicit methods accepting a stability step size restriction in literature evidence is given that there is also a need for unconditionally stable methods as exemplified by the successful alternating direction implicit finite difference time domain scheme in this paper we discuss unconditionally stable integration for a general semi discrete maxwell system allowing non cartesian space grids as encountered in finite element discretizations such grids exclude the alternating direction implicit approach particular attention is given to the second order trapezoidal rule implemented with preconditioned conjugate gradient iteration and to second order exponential integration using krylov subspace iteration for evaluating the arising phi functions a three space dimensional test problem is used for numerical assessment and comparison with an economical second order implicit explicit integrator we further pay attention to the chebyshev series expansion for computing the exponential operator for skew symmetric matrices

examination clarification and simplification of stability and dispersion analysis for adi fdtd and cnss fdtd schemes we describe a rigorous analysis of unconditional stability for the alternating direction implicit finite difference time domain adi fdtd and crank nicholson split step cnss fdtd schemes avoiding use of the von neumann spectral criterion the proof is performed in the spectral domain and uses skew hermitivity of matrix terms in the adi and cnss total amplification matrices a bound for the total adi amplification matrix is provided while the cn and cnss fdtd amplification matrices are unitary the bound on the adi fdtd depends on the courant number importantly we have found that the adi fdtd amplification matrix is not normal i e the unit spectral radius alone cannot be used to prove the adi fdtd unconditional stability the paper also shows that the adi fdtd and cnss fdtd schemes share the same dispersion equation by a similarity of their total amplification matrices and the two schemes have identical numerical dispersion in the frame of plane waves

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

composition methods maxwell s equations and source terms this paper is devoted to high order numerical time integration of first order wave equation systems originating from spatial discretization of maxwell s equations the focus lies on the accuracy of high order composition in the presence of source functions source functions are known to generate order reduction and this is most severe for high order methods for two methods based on two well known fourth order symmetric compositions convergence results are given assuming simultaneous space time grid refinement herewith physical sources and source functions emanating from dirichlet boundary conditions are distinguished among other things it is shown that the reduction can cost two orders on the other hand when a certain perturbation of a source function is used the reduction is generally diminished by one order in that case reduction is absent for physical sources and for dirichlet sources the order is equal to at least three under stable simultaneous space time grid refinement

fundamental schemes for efficient unconditionally stable implicit finite difference time domain methods generalized formulations of fundamental schemes for efficient unconditionally stable implicit finite difference time domain fdtd methods are presented the fundamental schemes constitute a family of implicit schemes that feature similar fundamental updating structures which are in simplest forms with most efficient right hand sides the formulations of fundamental schemes are presented in terms of generalized matrix operator equations pertaining to some classical splitting formulae including those of alternating direction implicit locally one dimensional and split step schemes to provide further insights into the implications and significance of fundamental schemes the analyses are also extended to many other schemes with distinctive splitting formulae detailed algorithms are described for new efficient implementations of the unconditionally stable implicit fdtd methods based on the fundamental schemes a comparative study of various implicit schemes in their original and new implementations is carried out which includes comparisons of their computation costs and efficiency gains

preservation of physical properties of stochastic maxwell equations with additive noise via stochastic multi symplectic methods stochastic maxwell equations with additive noise are a system of stochastic hamiltonian partial differential equations intrinsically possessing the stochastic multi symplectic conservation law it is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic maxwell equations with additive noise preserves the divergence in the sense of expectation moreover we propose three novel stochastic multi symplectic methods to discretize stochastic maxwell equations in order to investigate the preservation of these properties numerically we make theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence meanwhile we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method especially the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case numerical experiments are performed to verify our theoretical results

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

a split step approach for the 3 d maxwell s equations split step procedures have previously been used successfully in a number of situations e g for hamiltonian systems such as certain nonlinear wave equations in this study we note that one particular way to write the 3 d maxwell s equations separates these into two parts requiring only the solution of six uncoupled 1 d wave equations the approach allows arbitrary orders of accuracy in both time and space and features in many cases unconditional stability

a three dimensional unconditionally stable five step lod fdtd method a three dimensional unconditionally stable five step locally one dimensional finite difference time domain lod fdtd method is presented unlike the two step lod fdtd and three step lod fdtd methods the proposed method has second order temporal accuracy hence it gives less numerical dispersion than the two step lod fdtd and three step lod fdtd methods it also gives less numerical dispersion than the alternating direction implicit finite difference time domain adi fdtd method moreover for every propagation angle it provides very small anisotropy error than the above mentioned fdtd methods effects of the time step and the mesh size on the performance of the proposed method are discussed in detail in this paper validation of the stability and the accuracy of the proposed method is done with the help of simulation results to further show the advantage of the proposed method performance of the proposed method with artificial coefficients control parameters is also discussed in this paper

an optimized one step leapfrog hie fdtd method with the artificial anisotropy parameters by introducing the artificial anisotropy aa parameters a 3 d one step leapfrog hybrid implicit explicit finite difference time domain hie fdtd method is proposed which can reduce the numerical dispersion error without increasing the computational cost the formulation of the aa one step leapfrog hie fdtd method is obtained by introducing the relative permittivity and permeability tensors in the original one step leapfrog hie fdtd method the courant friedrichs lewy cfl stability condition of the aa one step leapfrog hie fdtd method is close to that of the original one step leapfrog hie fdtd method in addition the proposed hie fdtd method has lower numerical dispersion error and higher calculation accuracy than that of the one step leapfrog hie fdtd method finally to testify the characteristics of the proposed hie fdtd method numerical simulation experiments are given

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

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

a particle in cell method for the simulation of plasmas based on an unconditionally stable field solver abstract we propose a new particle in cell pic method for the simulation of plasmas based on a recently developed unconditionally stable solver for the wave equation this method is not subject to a cfl restriction limiting the ratio of the time step size to the spatial step size typical of explicit methods while maintaining computational cost and code complexity comparable to such explicit schemes we describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases and present the results of several standard test problems showing good agreement with theory with time step sizes much larger than allowed by typical cfl restrictions

new robust statistical procedures for the polytomous logistic regression models this article derives a new family of estimators namely the minimum density power divergence estimators as a robust generalization of the maximum likelihood estimator for the polytomous logistic regression model based on these estimators a family of wald type test statistics for linear hypotheses is introduced robustness properties of both the proposed estimators and the test statistics are theoretically studied through the classical influence function analysis appropriate real life examples are presented to justify the requirement of suitable robust statistical procedures in place of the likelihood based inference for the polytomous logistic regression model the validity of the theoretical results established in the article are further confirmed empirically through suitable simulation studies finally an approach for the data driven selection of the robustness tuning parameter is proposed with empirical justifications

operator splitting methods for maxwell s equations in dispersive media with orientational polarization we present two operator splitting schemes for the numerical simulation of maxwell s equations in dispersive media of debye type that exhibit orientational polarization the maxwell debye model the splitting schemes separate the mechanisms of wave propagation and polarization to create simpler sub steps that are easier to implement in addition dimensional splitting is used to propagate waves in different axial directions we present a sequential operator splitting scheme and its symmetrized version for the maxwell debye system in two dimensions the splitting schemes are discretized using implicit finite difference methods that lead to unconditionally stable schemes we prove that the fully discretized sequential scheme is a first order time perturbation and the symmetrized scheme is a second order time perturbation of the crank nicolson scheme for discretizing the maxwell debye model numerical examples are presented that illustrate our theoretical results

high order split step unconditionally stable fdtd methods and numerical analysis high order split step unconditionally stable finite difference time domain fdtd methods in three dimensional 3 d domains are presented symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical maxwell s equations into four sub matrices accordingly the time step is divided into four sub steps in addition high order central finite difference operators based on the taylor central finite difference method are used to approximate the spatial differential operators first and then the uniform formulation of the proposed high order schemes is generalized subsequently the analysis shows that all the proposed high order methods are unconditionally stable the generalized form of the dispersion relations of the proposed high order methods is carried out moreover the effects of the mesh size the time step and the order of schemes on the dispersion are illustrated through numerical results specifically the normalized numerical phase velocity error nnpve and the maximum nnpve of the proposed second order scheme are lower than that of the alternating direction implicit adi fdtd method furthermore the analysis of the accuracy of the proposed methods is presented in order to demonstrate the efficiency of the proposed methods numerical experiments are presented

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

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

an unconditionally stable single field finite difference time domain method for the solution of maxwell equations in three dimensions the recently developed two directional unconditionally stable single field us sf finite difference time domain fdtd method is generalized to a 3 d the method is based on the application of the crank nicolson scheme to only one of maxwell curl equations which leads to an unconditionally stable finite difference solution of the 3 d vector wave equation in the time domain the method is designated as a single field because only the electric or magnetic field is updated in each time step the implicit equations for each time step can be solved using a tridiagonal matrix algorithm without a need for matrix inversion to achieve this we introduced a new modified time splitting scheme for the multilevel difference equations unlike the existing time splitting method used to solve such equations which results in instability when dealing with inhomogeneous media the presented method remains stable as an important feature of the proposed us sf fdtd method the updating of the three field components can be executed simultaneously in parallel by applying multithreading thereby significantly reducing runtime the unconditional stability of the proposed method is proved analytically the accuracy and computational efficiency of the proposed method are demonstrated by providing numerical examples and by comparison to other fdtd methods and to the analytic solution when available

newmark beta fdtd method for super resolution analysis of time reversal waves in this work a new unconditionally stable finite difference time domain fdtd method with the split field perfectly matched layer pml is proposed for the analysis of time reversal tr waves the proposed method is very suitable for multiscale problems involving microstructures the spatial and temporal derivatives in this method are discretized by the central difference technique and newmark beta algorithm respectively and the derivation results in the calculation of a banded sparse matrix equation since the coefficient matrix keeps unchanged during the whole simulation process the lower upper lu decomposition of the matrix needs to be performed only once at the beginning of the calculation moreover the reverse cuthillmckee rcm technique an effective preprocessing technique in bandwidth compression of sparse matrices is used to improve computational efficiency the super resolution focusing of tr wave propagation in two and three dimensional spaces is included to validate the accuracy and efficiency of the proposed method

unconditionally stable fundamental lod fdtd method with second order temporal accuracy and complying divergence an unconditionally stable fundamental locally one dimensional lod finite difference time domain fdtd method with second order temporal accuracy and complying divergence cd denoted as lod2 cd fdtd is presented for three dimensional 3 d maxwell s equations while the conventional lod fdtd method does not have complying divergence the lod2 cd fdtd method has complying divergence in a manner analogous to the conventional explicit fdtd method the update procedures for a family of lod fdtd methods that employ similar splitting matrix operators are presented by extending the previous concept of achieving second order temporal accuracy for the lod2 fdtd method via implicit output processing we hereby propose novel explicit output processing that not only retains second order temporal accuracy but also complying divergence for the lod2 cd fdtd method the current source implementation for the lod2 cd fdtd method involves source incorporation in only the first procedure to further enhance efficiency the lod2 cd fdtd method is formulated into the fundamental lod2 cd fdtd method with efficient matrix operator free right hand sides subsequently detailed implementation for the fundamental lod2 cd fdtd method is presented analytical proof is provided to ascertain the second order temporal accuracy of the lod2 cd fdtd method numerical results and examples are also presented to validate the divergence complying property of the lod2 cd fdtd method

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

massless d branes on calabi yau threefolds and monodromy we analyze the link between the occurrence of massless b type d branes for specific values of moduli and monodromy around such points in the moduli space this allows us to propose a classification of all massless b type d branes at any point in the moduli space of calabi yau s this classification then justifies a previous conjecture due to horja for the general form of monodromy our analysis is based on using monodromies around points in moduli space where a single d brane becomes massless to generate monodromies around points where an infinite number become massless we discuss the various possibilities within the classification

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

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

explicit time domain finite element method stabilized for an arbitrarily large time step the root cause of the instability is quantitatively identified for the explicit time domain finite element method that employs a time step beyond that allowed by the stability criterion with the identification of the root cause an unconditionally stable explicit time domain finite element method is successfully created which is stable and accurate for a time step solely determined by accuracy regardless of how large the time step is the proposed method retains the strength of an explicit time domain method in avoiding solving a matrix equation while eliminating its shortcoming in time step numerical experiments have demonstrated its superior performance in computational efficiency as well as stability compared with the conditionally stable explicit method and the unconditionally stable implicit method the essential idea of the proposed method for making an explicit method stable for an arbitrarily large time step irrespective of space step is also applicable to other time domain methods

complete characterization of fourth order symplectic integrators with extended linear coefficients the structure of symplectic integrators up to fourth order can be completely and analytically understood when the factorization split coefficients are related linearly but with a uniform nonlinear proportional factor the analytic form of these extended linear symplectic integrators greatly simplified proofs of their general properties and allowed easy construction of both forward and nonforward fourth order algorithms with an arbitrary number of operators most fourth order forward integrators can now be derived analytically from this extended linear formulation without the use of symbolic algebra

three new unconditionally stable fdtd methods with high order accuracy three novel finite difference time domain fdtd methods based on the split step ss scheme with high order accuracy are presented which are proven to be unconditionally stable in the first novel method symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical maxwell s equations into six sub matrices accordingly the time step is divided into six sub steps the second and third proposed methods are obtained by adjusting the sequence of the sub matrices deduced in the first method so all the novel methods presented in the paper have similar formulations of which the numerical dispersion errors and the anisotropic errors are lower than the alternating direction implicit finite difference time domain adi fdtd method the initial ss fdtd method and the modified ss fdtd method based on the strang splitting scheme specifically for the second method corresponding to a certain cell per wavelength cpw there is a courant number value making the numerical anisotropic error to be zero while in the third novel method corresponding to a certain courant number value there exists a cpw making the numerical anisotropic error to be zero in order to demonstrate the high order accuracy and efficiency of the proposed methods numerical results are presented