lattice polytopes cut out by root systems and the koszul property we show that lattice polytopes cut out by root systems of classical type are normal and koszul generalizing a well known result of bruns gubeladze and trung in type a we prove similar results for cayley sums of collections of polytopes whose minkowski sums are cut out by root systems the proofs are based on a combinatorial characterization of diagonally split toric varieties

higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1d abstract we study the generalized finite element methods gfems for the second order elliptic eigenvalue problem with an interface in 1d the linear stable generalized finite element methods sgfem were recently developed for the elliptic source problem with interfaces we first generalize sgfem to arbitrary order elements and establish the optimal error convergence of the approximate solutions for the elliptic source problem with an interface we then apply the abstract theory of spectral approximation of compact operators to establish the error estimation for the eigenvalue problem with an interface the error estimations on eigenpairs strongly depend on the estimation of the discrete solution operator for the source problem we verify our theoretical findings in various numerical examples including both source and eigenvalue problems

spectral approximation properties of isogeometric analysis with variable continuity abstract we study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh which allows for large computational savings during the solution of the resulting linear system the continuity reduction results in extra degrees of freedom that modify the approximation properties of the method the convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis we show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra such as stopping bands and outliers and present a unified description of these effects in finite element method isogeometric analysis and refined isogeometric analysis accuracy of the refined isogeometric analysis approximations can be improved by using non standard quadrature rules in particular optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence as it occurs in standard isogeometric analysis

multigraded regularity and the koszul property abstract we give a criterion for the section ring of an ample line bundle to be koszul in terms of multigraded regularity we discuss applications to adjoint bundles on toric varieties as well as to polytopal semigroup rings

feedback equivalence for general control systems abstract we discuss the solution to the problem of local equivalence of control systems with n states and p controls in a neighbourhood of a generic point under the lie pseudo group of local time independent feedback transformations we have shown earlier that this problem is identical with the problem of simple equivalence of the time optimal variational problem here we indicate the way in which this identification may be used to obtain closed loop time critical controls for general systems we show that the classification of general nonlinear systems depends on the classification of all n p dimensional affine subspaces of the space of symmetric forms in p variables and that the case of control linear systems depends on the classification of all n p dimensional affine subspaces of the space of skew forms in p variables we show that in the latter case the g structure is the prolongation of one determined by a state space transformation group we give a complete list of normal forms for control linear systems in the case p n 1

eigenvalue approximation by mixed and hybrid methods note univ maryland dept math college pk md 20742 ecole polytech ctr math appl f 91128 palaiseau france univ paris 6 f 75230 paris 05 france mercier b cea limeil f 94190 villeneuve france isi document delivery no lq584times cited 29cited reference count 44 reference asn article 1981 005doi 10 2307 2007651 record created on 2006 08 24 modified on 2017 05 12

dispersion optimized quadratures for isogeometric analysis abstract we develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span the schemes yield two extra orders of convergence superconvergence on the eigenvalue errors while the eigenfunction errors are of optimal convergence order both dispersion and spectrum analysis are unified in the form of a taylor expansion for eigenvalue errors the resulting schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems we also derive an a posteriori error estimator for the eigenvalue error based on the superconvergence result we verify with numerical examples the analysis of the performance of the proposed schemes

hp adaptive composite discontinuous galerkin methods for elliptic eigenvalue problems on complicated domains in this paper we develop the a posteriori error estimation of hp adaptive discontinuous galerkin composite finite element methods dgfems for the discretization of second order elliptic eigenvalue problems dgfems allow for the approximation of problems posed on computational domains which may contain local geometric features the dimension of the composite finite element space is independent of the number of geometric features this is in contrast with standard finite element methods as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space computable upper bounds on the error for both eigenvalues and eigenfunctions are derived numerical experiments highlighting the practical application of the proposed estimators within an automatic hp adaptive refinement procedure will be presented

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

stable generalized finite element method and associated iterative schemes application to interface problems abstract the generalized finite element method gfem is an extension of the finite element method fem where the standard finite element space is augmented with a space of non polynomial functions called the enrichment space the functions in the enrichment space mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully applied to a wide range of problems however it often suffers from bad conditioning i e its conditioning may not be robust with respect to the mesh and in fact the conditioning could be much worse than that of the standard fem in this paper we present a numerical study that shows that if the angle between the finite element space and the enrichment space is bounded away from 0 uniformly with respect to the mesh then the gfem is stable i e the conditioning of gfem is not worse than that of the standard fem a gfem with this property is called a stable gfem sgfem the last part of the paper is devoted to the derivation of a robust iterative solver exploiting this angle condition it is shown that the required wall clock time is greatly reduced compared to popular gfems used in the literature

spectral approximations by the hdg method we consider the numerical approximation of the spectrum of a secondorder elliptic eigenvalue problem by the hybridizable discontinuous galerkin hdg method we show for problems with smooth eigenfunctions that the approximate eigenvalues and eigenfunctions converge at the rate 2k 1 and k 1 respectively here k is the degree of the polynomials used to approximate the solution its flux and the numerical traces our numerical studies show that a rayleigh quotient like formula applied to certain locally postprocessed approximations can yield eigenvalues that converge faster at the rate 2k 2 for the hdg method as well as for the brezzidouglas marini bdm method we also derive and study a condensed nonlinear eigenproblem for the numerical traces obtained by eliminating all the other variables

feedback classification of multi input nonlinear control systems we study the feedback group action on multi input nonlinear control systems with uncontrollable mode we follow slightly an approach proposed in kang and krener w kang and a j krener siam j control optim 30 1992 pp 1319 1337 which consists of analyzing the system and the feedback group step by step we construct a normal form which generalizes on one hand the results obtained in the single input case and on the other hand those recently obtained by the same author in the controllable case we illustrate our results by studying the caltech multi vehicle wireless testbed mvwt and the prototype of planar vertical takeoff and landing aircraft pvtol we also study the notion of bifurcation of controllability for systems with one nonzero uncontrollable mode we first show that the equilibria for those systems is a p dimensional submanifold p equals number of inputs provided that one term in their normal form is nonzero we show that these systems are linearly controllable hence stabilizable at any nearby equilibrium point of the origin

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

koszul homology and syzygies of veronese subalgebras a graded k algebra r has property np if it is generated in degree 1 has relations in degree 2 and the syzygies of order p on the relations are linear the green lazarsfeld index of r is the largest p such that it satisfies the property np our main results assert that under a mild assumption on the base field the cth veronese subring of a polynomial ring has green lazarsfeld index c 1 the same conclusion also holds for an arbitrary standard graded algebra provided c gg 0

strongly stable generalized finite element method application to interface problems abstract in this paper we present the theoretical justification for the stable generalized finite element method sgfem when applied to smooth interface problems we prove that a generalized finite element method gfem is stable i e its conditioning is a not worse than that of the finite element method and b robust with respect to the mesh if the enrichment space of the gfem satisfies two axioms we provide element based sufficient conditions on the enrichment space that will guarantee that the axioms are satisfied we show that the enrichment space of the gfem used to address the interface problems in 2d satisfies the sufficient conditions and thus the two axioms yielding an sgfem the idea of a strongly stable gfem associated with one of the two axioms has been introduced strong stability of the gfem is important for designing efficient iterative methods to solve the underlying linear system a proof of the optimal convergence of the gfem for the interface problem has also been derived in this paper the work in this paper is the continuation of kergrene et al 2016 where the stability of the gfem when applied to interface problems was established through numerical experiments the numerical results in kergrene et al 2016 indicated that the gfem is indeed strongly stable

discontinuous galerkin approximation of the laplace eigenproblem in this paper we analyse the problem of computing eigenvalues and eigenfunctions of the laplace operator by means of discontinuous galerkin dg methods it results that several dg methods actually provide a spectrally correct approximation of the laplace operator we present here the convergence theory which applies to a wide class of dg methods as well as numerical tests demonstrating the theoretical results

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

singular trajectories and the contrast imaging problem in nuclear magnetic resonance in this article the contrast imaging problem in nuclear magnetic resonance is modeled as a mayer problem in optimal control whose solutions can be parameterized using pontryagin s maximum principle and analyzed using geometric optimal control in particular the optimal problem can be mainly reduced to the analysis of the hamiltonian dynamics describing the singular trajectories and encoding their optimality status

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

dispersion minimized mass for isogeometric analysis abstract we introduce the dispersion minimized mass for isogeometric analysis to approximate the structural vibration which we model as a second order differential eigenvalue problem the dispersion minimized mass reduces the eigenvalue error significantly from the optimum order of 2 p to the superconvergence order of 2 p 2 for the p th order isogeometric elements with maximum continuity which in return leads to a more accurate method we first establish the dispersion error where the leading error term is explicitly written in terms of the stiffness and mass entries for arbitrary polynomial order isogeometric elements we derive the dispersion minimized mass in one dimension by solving a p dimensional local matrix problem for the p th order approximation and then extend it to multiple dimensions on tensor product grids we show that the dispersion minimized mass can also be obtained by approximating the mass matrix using optimally blended quadratures we generalize the lower order quadrature blending results to arbitrary polynomial order isogeometric approximations as well as to arbitrary quadrature rules various numerical examples validate the eigenvalue and eigenfunction error estimates

feedback equivalence for nonlinear systems and the time optimal control problem this article relates the classification of affine control systems under the action of the feedback group with a differential classification of a set of constrained hamiltonian vector fields arising from pontryagin s maximum principle for the time minimal control problem they represent the singularities of the input state mapping this relation provides a method to compute feedback invariants

optimal spectral approximation of 2n order differential operators by mixed isogeometric analysis abstract we approximate the spectra of a class of 2 n order differential operators using isogeometric analysis in mixed formulations this class includes a wide range of differential operators such as those arising in elliptic biharmonic cahn hilliard swift hohenberg and phase field crystal equations the spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations which require auxiliary parameters the mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying b spline space we improve this order to be 2 p 2 by applying optimally blended quadrature rules developed in puzyrev et al 2017 caloet al 0000 and this order is an optimum in the view of dispersion error we also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations

finite element and nurbs approximations of eigenvalue boundary value and initial value problems abstract we study the spectral approximation properties of finite element and nurbs spaces from a global perspective we focus on eigenfunction approximations and discover that the l 2 norm errors for finite element eigenfunctions exhibit pronounced spikes about the transition points between branches of the eigenvalue spectrum this pathology is absent in nurbs approximations by way of the pythagorean eigenvalue error theorem we determine that the squares of the energy norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the l 2 norm eigenfunction errors the spurious behavior of the higher eigenvalues for standard finite elements is well known and therefore inherited by the energy norm errors along with the spikes in the l 2 norm of the eigenfunction errors the eigenvalue pathology is absent for nurbs the implications of these results to the corresponding elliptic boundary value problem and parabolic and hyperbolic initial value problems are discussed

convex normality of rational polytopes with long edges abstract we introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths guaranteeing the property as an application we show that if every edge of a lattice d polytope p has lattice length 4 d d 1 then p is normal this answers in the positive a question raised in 2007 if p is a lattice simplex whose edges have lattice lengths d d 1 then p is even covered by lattice parallelepipeds for the approach developed here it is necessary to involve rational polytopes even for the application to lattice polytopes

high order extended finite element method for cracked domains the aim of the paper is to study the capabilities of the extended finite element method xfem to achieve accurate computations in non smooth situations such as crack problems although the xfem method ensures a weaker error than classical nite element methods the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity the di culty can be overcome by modifying the enrichment of the nite element basis with the asymptotic crack tip displacement solutions as well as with the heaviside function numerical simulations show that the modi ed xfem method achieves an optimal rate of convergence i e like in a standard nite element method for a smooth problem

stable generalized finite element method sgfem abstract the generalized finite element method gfem is a partition of unity method pum where the trial space of standard finite element method fem is augmented with non polynomial shape functions with compact support these shape functions which are also known as the enrichments mimic the local behavior of the unknown solution of the underlying variational problem gfem has been successfully used to solve a variety of problems with complicated features and microstructure however the stiffness matrix of gfem is badly conditioned much worse compared to the standard fem and there could be a severe loss of accuracy in the computed solution of the associated linear system in this paper we address this issue and propose a modification of the gfem referred to as the stable gfem sgfem we show that sgfem retains the excellent convergence properties of gfem does not require a ramp function in the presence of blending elements and the conditioning of the associated stiffness matrix is not worse than that of the standard fem moreover sgfem is very robust with respect to the parameters of the enrichments we show these features of sgfem on several examples

normality and covering properties of affine semigroups s x gp s mx s for some m 0 one calls s normal if s s for simplicity we will often assume that gp s z this is harmless because we can replace z by gp s if necessary the rank of s is the rank of gp s we will only be interested in the case in which s s 0 such affine semigroups will be called positive the positivity of s is equivalent to the pointedness of the cone c s r s generated by s in r one has ranks dimc s all the cones appearing in this paper have their apex at the origin the normality of s can now be characterized geometrically one obviously has s c s z and so s is normal if and only if s c s z conversely every convex pointed finitely generated rational cone c r yields a normal affine semigroup s c c z this semigroup is finitely generated by gordan s lemma if we simply speak of a cone c in the following then it is always assumed that c is convex pointed finitely generated and rational since dimc s ranks we often call the dimension of a cone its rank each positive affine semigroup s can be embedded into z m ranks one uses m linearly independent integral linear forms representing support hyperplanes of c s therefore every element can be written as a sum of irreducible elements and the set of irreducible elements of s is finite since s is finitely generated for a cone c the set of irreducible elements of s c is often called the hilbert basis hilb c of c in the combinatorial literature a convention we will follow more generally we also call the set of irreducible elements of a positive affine semigroup s its hilbert basis and denote it by hilb s this paper is devoted to a discussion of sufficient and potentially necessary conditions for the normality of s in terms of combinatorial properties of hilb s as far as the necessity is concerned these conditions can of course be formulated in terms of hilbert bases of rational cones the first property under consideration is the existence of a unimodular hilbert cover

higher order stable generalized finite element method the generalized finite element method gfem is a galerkin method where the trial space is obtained by augmenting the trial space of the standard finite element method fem by non polynomial functions called enrichments that mimic the local behavior of the unknown solution of the underlying variational problem the gfem has excellent approximation properties but its conditioning could be much worse than that of the fem however if the enrichments satisfy certain properties then the conditioning of the gfem is not worse than that of the standard fem and the gfem is referred to as the stable gfem sgfem in this paper we address the higher order sgfem that yields higher order convergence and suggest a specific modification of the enrichment function that guarantees the required conditioning yielding a robust implementation of the higher order sgfem

feedback classification of nonlinear single input control systems with controllable linearization normal forms canonical forms and invariants we study the feedback group action on single input nonlinear control systems we follow an approach of kang and krener based on analyzing step by step the action of homogeneous transformations on the homogeneous part of the same degree of the system we construct a dual normal form and dual invariants with respect to those obtained by kang we also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms resp their dual canonical forms coincide we give an explicit construction of transformations bringing the system to its normal dual normal canonical and dual canonical forms we illustrate our results by simple examples on mathbb r 3 and mathbb r 4

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

optimally blended spectral finite element scheme for wave propagation and nonstandard reduced integration we study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent finite element mass matrix and lumped spectral element mass matrix for the small wave number limit we find and prove that for the optimum blending the resulting scheme a provides 2p 4 order accuracy for p th order method two orders more accurate compared with finite and spectral element schemes b has an absolute accuracy which is mathcal o p 3 and mathcal o p 2 times better than that of the pure finite and spectral element schemes respectively c tends to exhibit phase lag moreover we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived