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

linear pdes and numerical methods that preserve a multisymplectic conservation law multisymplectic methods have recently been proposed as a generalization of symplectic ode methods to the case of hamiltonian pdes their excellent long time behavior for a variety of hamiltonian wave equations has been demonstrated in a number of numerical studies a theoretical investigation and justification of multisymplectic methods is still largely missing in this paper we study linear multisymplectic pdes and their discretization by means of numerical dispersion relations it is found that multisymplectic methods in the sense of bridges and reich phys lett a 284 2001 pp 184 193 and reich j comput phys 157 2000 pp 473 499 such as gauss legendre runge kutta methods possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity a certain cfl type restriction on delta t delta x might be required for methods higher than second order in time it is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of delta t delta x despite being multisymplectic in the sense of discrete variational mechanics j e marsden g p patrick and s shkoller commun math phys 199 1999 pp 351 395

an algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds this paper gives an o n algorithm for a singly constrained convex quadratic program using binary search to solve the kuhn tucker system computational results indicate that a randomized version of this algorithm runs in expected linear time and is suitable for practical applications for the nonconvex case ane approximate algorithm is proposed which is based on convex and piecewise linear approximations of the objective function

dispersive properties of multisymplectic integrators multisymplectic ms integrators i e numerical schemes which exactly preserve a discrete space time symplectic structure are a new class of structure preserving algorithms for solving hamiltonian pdes in this paper we examine the dispersive properties of ms integrators for the linear wave and sine gordon equations in particular a leapfrog in space and time scheme a member of the lobatto runge kutta family of methods and the preissman box scheme are considered we find the numerical dispersion relations are monotonic and that the sign of the group velocity is preserved the group velocity dispersion gvd is found to provide significant information and succinctly explain the qualitative differences in the numerical solutions obtained with the different schemes further the numerical dispersion relations for the linearized sine gordon equation provides information on the ability of the ms integrators to capture the sine gordon dynamics we are able to link the numerical dispersion relations to the total energy of the various methods thus providing information on the coarse grid behavior of ms integrators in the nonlinear regime

unstable eigenvalues and the linearization about solitary waves and fronts with symmetry the linear stability of solitary wave or front solutions of hamiltonian evolutionary equations which are equivariant with respect to a lie group is studied the organizing centre for the analysis is a multisymplectic formulation of hamiltonian pdes where a distinct symplectic operator is assigned for time and space this separation of symplectic structures leads to new characterizations of the following components of the analysis the states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure the momentum of the connecting orbit or shape of the solitary wave considered as a heteroclinic orbit in a phase space representation is given a new characterization as a one form on the tangent space to the heteroclinic manifold and this one form is a restriction of the temporal symplectic structure for the linear stability analysis a new symplectic characterization of the evans function and its derivatives are obtained leading to an abstract geometric proof of instability for a large class of solitary wave states of equivariant hamiltonian evolutionary pdes the theory sheds new light on several well known models the gkdv equation a boussinesq system and a nonlinear wave equation the generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed

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

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

linear time algorithms for testing the realisability of line drawings of curved objects this paper shows that the semantic labelling of line drawings of curved objects with piecewise c 3 surfaces is solvable in linear time this result is robust to changes in the assumptions on object shape when all vanishing points are known a different linear time algorithm exists to solve the labelling problem furthermore in both cases all legally labelled line drawings of curved objects are shown to be physically realisable however when some but not all of the vanishing points are known when the drawing is an orthographic projection of a scene containing parallel lines or when we wish to minimise the number of phantom junctions the labelling problem becomes np hard the introduction of collinearity constraints also renders the labelling problem np complete except in the case when all vanishing points are known 1999 elsevier science b v all rights reserved

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

wireframe projections physical realisability of curved objects and unambiguous reconstruction of simple polyhedra the reconstruction of an object from a single 2d projection of a 3d wireframe model is a vision problem with applications in cad cam and computer graphics we propose an algorithm for the interpretation of wireframe projections based on assigning semantic and numerical depth labels to lines this method allows us to state necessary and sufficient conditions for the physical realisability of a wireframe projection of a curved object the presence of linear features provides further constraints on the positions of object vertices for example each straight line gives rise to a coplanarity constraint between a set of object vertices we show that extra information such as vanishing points parallel lines or user entered depth parity information is sufficient to uniquely determine the face circuits in wireframe projections of polyhedra with simple trihedral vertices in fact a polyhedron with simple trihedral vertices can be unambiguously reconstructed from its 3d wireframe model

order stars and stability for delay differential equations we consider runge kutta methods applied to delay differential equations y t ay t by t 1 with real a and b if the numerical solution tends to zero whenever the exact solution does the method is called tau 0 stable using the theory of order stars we characterize high order symmetric methods with this property in particular we prove that all gauss methods are tau 0 stable furthermore we present sufficient conditions and we give evidence that also the radau methods are tau 0 stable we conclude this article with some comments on the case where a andb are complex numbers

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

a new multisymplectic scheme for generalized kadomtsev petviashvili equation from the multisymplectic euler box scheme we derive a new multisymplectic 16 point scheme for the generalized kadomtsev petviashvili equation by eliminating the auxiliary variables the new scheme is an explicit scheme in sense that it does not need iteration the results of backward error analysis for the euler box scheme show that the modified equation associated with the new scheme is not multisymplectic but of a high order approximation to a multisymplectic partial differential equations with a modified structure numerical results on one soliton and lump type solitary waves are reported to illustrate the efficiency of the multisymplectic scheme

the nonlinear resource allocation problem in this paper we study the nonlinear resource allocation problem defined as the minimization of a convex function over one convex constraint and bounded integer variables this problem is encountered in a variety of applications including capacity planning in manufacturing and computer networks production planning capital budgeting and stratified sampling despite its importance to these and other applications the nonlinear resource allocation problem has received little attention in the literature therefore we develop a branch and bound algorithm to solve this class of problems first we present a general framework for solving the continuous variable problem then we use this framework as the basis for our branch and bound method we also develop reoptimization procedures and a heuristic that significantly improve the performance of the branch and bound algorithm in addition we show how the algorithm can be modified to solve nonconvex problems so that a concave objective function can be handled

on multisymplecticity of partitioned runge kutta methods previously it has been shown that discretizing a multi hamiltonian pde in space and time with partitioned runge kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law however these previous studies use the same partitioning of the variables into two parts in both space and time this gives rise to a large number of cases to be considered each with its own set of conditions to be satisfied we present here a much simpler set of conditions covering all of these cases where the variables are partitioned independently in space and time into an arbitrary number of parts in general it is not known when such a discretization of a multi hamiltonian pde will give rise to a well defined numerical integrator however a numerical integrator that is explicit will typically be well defined in this paper we give sufficient conditions on a multi hamiltonian pde for a lobatto iiia iiib discretization in space to give rise to explicit odes and an algorithm for constructing these odes

constraints between distant lines in the labelling of line drawings of polyhedral scenes the machine interpretation of line drawings has applications both in vision and geometric modelling this paper extends the classic technique of assigning semantic labels to lines subject to junction constraints by introducing new constraints often between distant lines these include generic constraints between lines lying on a path in the drawing as well as preference constraints between the labellings of pairs of junctions lying on parallel lines such constraints are essential to avoid an exponential number of legal labellings of drawings of objects with non trihedral vertices the strength of these constraints is demonstrated by their ability to identify the unique correct labelling of many drawings of polyhedral objects with tetrahedral vertices these new constraints also allowed us to deduce a general polyhedral junction constraint for the case when there is no limit on the number of faces which can meet at a junction

overcoming superstrictness in line drawing interpretation presents an algorithm for correcting incorrect line drawings incorrect projections of a polyhedral scene such incorrect drawings arise e g when an image of a polyhedral world is taken the edges and vertices are extracted and a drawing is synthesized along the way the true positions of the vertices in the 2d projection are perturbed due to digitization errors and the preprocessing as most available algorithms for interpreting line drawings are superstrict they judge these noisy inputs as incorrect and fail to reconstruct a three dimensional scene from them the presented method overcomes this problem by moving the positions of all vertices until a very close correct drawing is found the closeness criterion is to minimize the sum of squared distances from each vertex in the input drawing to its corrected position with this tool any superstrict method for line drawing interpretation is now practical as it can be applied to the corrected version of the input drawing

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

nonlinear integer programming for optimal allocation in stratified sampling a stratified random sampling plan is one in which the elements of the population are first divided into nonoverlapping groups and then a simple random sample is selected from each group in this paper we focus on determining the optimal sample size of each group we show that various versions of this problem can be transformed into a particular nonlinear program with a convex objective function a single linear constraint and bounded variables two branch and bound algorithms are presented for solving the problem the first algorithm solves the transformed subproblems in the branch and bound tree using a variable pegging procedure the second algorithm solves the subproblems by performing a search to identify the optimal lagrange multiplier of the single constraint we also present linearization and dynamic programming methods that can be used for solving the stratified sampling problem computational testing indicates that the pegging branch and bound algorithm is fastest for some classes of problems and the linearization method is fastest for other classes of problems

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

a survey of methods for pure nonlinear integer programming the subject of this paper is a classification and discussion of algorithms for solution of nonlinear pure integer programming problems the survey is organized by characterizing the mathematical form of the nonlinear optimization problems addressed by the various algorithms if any method can be used without changes to solve mixed integer nonlinear programming problems that fact is mentioned in the description of the algorithm however algorithms which must be applied only to mixed integer problems are not surveyed

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

plane based optimization for 3d object reconstruction from single line drawings in previous optimization based methods of 3d planar faced object reconstruction from single 2d line drawings the missing depths of the vertices of a line drawing and other parameters in some methods are used as the variables of the objective functions a 3d object with planar faces is derived by finding values for these variables that minimize the objective functions these methods work well for simple objects with a small number tv of variables as n grows however it is very difficult for them to find the expected objects this is because with the nonlinear objective functions in a space of large dimension the search for optimal solutions can easily get trapped into local minima in this paper we use the parameters of the planes that pass through the planar faces of an object as the variables of the objective function this leads to a set of linear constraints on the planes of the object resulting in a much lower dimensional null space where optimization is easier to achieve we prove that the dimension of this null space is exactly equal to the minimum number of vertex depths that define the 3d object since a practical line drawing is usually not an exact projection of a 3d object we expand the null space to a larger space based on the singular value decomposition of the projection matrix of the line drawing in this space robust 3d reconstruction can be achieved compared with the two most related methods our method not only can reconstruct more complex 3d objects from 2d line drawings but also is computationally more efficient

some new structure preserving algorithms for general multi symplectic formulations of hamiltonian pdes many partial differential equations pdes can be written as a multi symplectic hamiltonian system which has three local conservation laws namely multi symplectic conservation law local energy conservation law and local momentum conservation law in this paper we give several systematic methods for discretizing general multi symplectic formulations of hamiltonian pdes including a local energy preserving algorithm a class of global energy preserving methods and a local momentum preserving algorithm the methods are illustrated by the nonlinear schrodinger equation and the korteweg de vries equation numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods

a geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities action symplecticity signature and complex instability are fundamental concepts in hamiltonian dynamics which can be characterized in terms of the symplectic structure in this paper hamiltonian pdes on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two form is assigned to each space direction and time this leads to a new geometric formulation of the conservation of wave action for linear and nonlinear hamiltonian pdes and via stokes s theorem a conservation law for symplecticity each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode the first invariant in this family is classical krein signature or energy sign when the energy is time independent and the other spatial signatures are energy flux signs leading to a classification of instabilities that includes information about directional spatial spreading of an instability the theory is applied to several examples the boussinesq equation the water wave equations linearized about an arbitrary stokes s wave rotating shallow water flow and flow past a compliant surface some implications for non conservative systems are also discussed

high order multisymplectic runge kutta methods we study the spatial semidiscretizations obtained by applying runge kutta rk and partitioned runge kutta prk methods to multisymplectic hamiltonian pdes these methods can be regarded as multisymplectic hp finite element methods for wave equations all the methods we consider are multisymplectic we determine their properties with regard to existence of solutions dispersion and order the lobatto iiia iiib prk method can lead to explicit odes and exhibits a surprisingly high order of 2 higher than the number of degrees of freedom per cell

multi conformal symplectic pdes and discretizations past work on integration methods that preserve a conformal symplectic structure focuses on hamiltonian systems with weak linear damping in this work systems of pdes that have conformal symplectic structure in time and space are considered meaning conformal symplecticity is fully generalized for pdes using multiple examples it is shown that pdes with this particular structure have interesting applications what it means to preserve a multi conformal symplectic conservation law numerically is explained along with presentation of two numerical methods that preserve such properties then the advantages of the methods are briefly explored through applications to linear equations consideration of momentum and energy dissipation and backward error analysis numerical simulations for two pdes illustrate the properties of the methods as well as the advantages over other standard methods

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

explicit multi symplectic extended leap frog methods for hamiltonian wave equations in this paper we study the integration of hamiltonian wave equations whose solutions have oscillatory behaviors in time and or space we are mainly concerned with the research for multi symplectic extended runge kutta nystrom erkn discretizations and the corresponding discrete conservation laws we first show that the discretizations to the hamiltonian wave equations using two symplectic erkn methods in space and time respectively lead to an explicit multi symplectic integrator eleap frogi then we derive another multi symplectic discretization using a symplectic erkn method in time and a symplectic partitioned runge kutta method which is equivalent to the well known stormer verlet method in space eleap frogii these two new multi symplectic schemes are extensions of the leap frog method the numerical stability and dispersive properties of the new schemes are analyzed numerical experiments with comparisons are presented where the two new explicit multi symplectic methods and the leap frog method are applied to the linear wave equation and the sine gordon equation the numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation

multivariate stratified sampling by optimization an important recurring problem in statistics involves the determination of strata boundaries for use in stratified sampling this paper describes a practical method for stratifying a population of observations based on optimal cluster analysis the goal of stratification is constructing a partition such that observations within a stratum are homogeneous as defined by within cluster variances for attributes that are deemed important while observations between strata are heterogeneous the problem is defined as a deterministic optimization model with integer variables and is solved by means of a subgradient method computational tests with several examples show that the within strata variances and thus the accompanying standard errors can be substantially reduced since the proposed model strives to minimize standard error it is applicable to situations where a precise sample is essential for example microeconomic simulation studies

conformal conservation laws and geometric integration for damped hamiltonian pdes conformal conservation laws are defined and derived for a class of multi symplectic equations with added dissipation in particular the conservation laws of energy and momentum are considered along with those that arise from linear symmetries numerical methods that preserve these conformal conservation laws are presented in detail providing a framework for proving a numerical method exactly preserves the dissipative properties considered the conformal methods are compared analytically and numerically to standard conservative methods which includes a thorough inspection of numerical solution behavior for linear equations damped klein gordon and sine gordon equations and a damped nonlinear schrodinger equation are used as examples to demonstrate the results

multi symplectic spectral discretizations for the zakharov kuznetsov and shallow water equations abstract the time evolution of multi symplectic pdes with periodic boundary conditions in one and two space dimensions is considered we introduce the idea of a multi symplectic fourier transform for systems on periodic domains leading to a semi discretization on fourier space and to the concept of multi symplecticity on fourier space the spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it a discrete wave model a by product of the semi discretization is that it leads automatically to a system of hamiltonian odes in time when truncated we show that the one dimensional shallow water equations and the two dimensional zakharov kuznetsov equation are multi symplectic and derive spectral discretizations for these systems and present numerical experiments

the interpretation of line drawings with contrast failure and shadows in line drawings derived from real images lines may be missing due to contrast failure and objects with curved surfaces may cast shadows from multiple light sources this paper shows that it is the presence of shadows rather than contrast failure that renders the line drawing labelling problem np complete however shadows are a valuable visual cue since their presence is formally shown to reduce the average ambiguity of drawings this is especially true when constraints concerning shadow formation are employed to differentiate shadow and non shadow lines the extended junction constraint concerning straight lines colinear with junctions compensates the loss of information caused by contrast failure in fact we observe the contrast failure paradox a drawing is sometimes less ambiguous when lines are partly missing due to contrast failure it is known that the coplanarity of sets of object vertices can be deduced from the presence of straight lines in the drawing this paper shows that these coplanarity constraints are robust to the presence of contrast failure

on the multi symplectic structure of boussinesq type systems i derivation and mathematical properties the boussinesq equations are known since the end of the xixst century however the proliferation of various boussinesq type systems started only in the second half of the xxst century today they come under various flavours depending on the goals of the modeller at the beginning of the xxist century an effort to classify such systems at least for even bottoms was undertaken and developed according to both different physical regimes and mathematical properties with special emphasis in this last sense on the existence of symmetry groups and their connection to conserved quantities of particular interest are those systems admitting a symplectic structure with the subsequent preservation of the total energy represented by the hamiltonian in the present paper a family of boussinesq type systems with multi symplectic structure is introduced some properties of the new systems are analyzed their relation with already known boussinesq models the identification of those systems with additional hamiltonian structure as well as other mathematical features like well posedness and existence of different types of solitary wave solutions the consistency of multi symplectic systems with the full euler equations is also discussed

conservation of wave action under multisymplectic discretizations in this paper we discuss the conservation of wave action under numerical discretization by variational and multisymplectic methods both the abstract wave action conservation defined with respect to a smooth periodic one parameter ensemble of flow realizations and the specific wave action based on an approximated and averaged lagrangian are addressed in the numerical context it is found that the discrete variational formulation gives rise in a natural way not only to the discrete wave action conservation law but also to a generalization of the numerical dispersion relation to the case of variable coefficients indeed a fully discrete analogue of the modulation equations arises on the other hand the multisymplectic framework gives easy access to the conservation law for the general class of multisymplectic runge kutta methods a numerical experiment confirms conservation of wave action to machine precision and suggests that the solution of the discrete modulation equations approximates the numerical solution to order cal o varepsilon on intervals of cal o varepsilon 1

on the continuous quadratic knapsack problem we introduce a new algorithm for the continuous bounded quadratic knapsack problem this algorithm is motivated by the geometry of the problem is based on the iterative solution of a series of simple projection problems and is easy to understand and implement in practice the method compares favorably to other well known algorithms some of which have superior worst case complexity on problem sizes up ton 4000

a discrete calculus analysis of the keller box scheme and a generalization of the method to arbitrary meshes abstract the keller box scheme is a face based method for solving partial differential equations that has numerous attractive mathematical and physical properties it is shown that these attractive properties collectively follow from the fact that the scheme discretizes partial derivatives exactly and only makes approximations in the algebraic constitutive relations appearing in the pde the exact discrete calculus associated with the keller box scheme is shown to be fundamentally different from all other mimetic physics capturing numerical methods this suggests that a unique exact discrete calculus does not exist it also suggests that existing analysis techniques based on concepts in algebraic topology in particular the discrete de rham complex are unnecessarily narrowly focused since they do not capture the keller box scheme the discrete calculus analysis allows a generalization of the keller box scheme to non simplectic meshes to be constructed analysis and tests of the method on the unsteady advection diffusion equations are presented

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

linear constraints for the interpretation of line drawings of curved objects abstract drawings of curved objects often contain many linear features straight lines colinear or coplanar points parallel lines and vanishing points these linear features give rise to linear constraints on the 3d position of scene points the resulting problem can be solved by standard linear programming techniques an important characteristic of this approach is that instead of making a strong assumption such as all surfaces are planar only a very weak assumption which disallows coincidences and highly improbable objects needs to be made to be able to deduce planarity the linear constraints combined with junction labelling constraints are a powerful means of discriminating between possible and impossible line drawings they provide an important tool for the machine reconstruction of a 3d scene from a human entered line drawing

interpreting line drawings of curved objects in this thesis we study the problem of interpreting line drawings of scenes composed or opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture on them it is assumed that the line drawing has been formed by orthographic projection of such a scene under general viewpoint that the line drawing is error free and that there are no lines due to shadows or specularities our definition implicitly excludes laminae wires and the apices of cones a major component of the interpretation of line drawings is line labelling by line labelling we mean a classification of each image curve as corresponding to either a depth or orientation discontinuity in the scene and b further subclassification of each kind of discontinuity for a depth discontinuity we determine whether it is a limb a locus of points on the surface where the line of sight is tangent to the surface or an occluding edge a tangent plane discontinuity of the surface for an orientation discontinuity we determine whether it corresponds to a convex or concave edge this thesis presents the first mathematically rigorous scheme for labelling line drawings of the class of scenes described by analysing the projection of the neighborhoods of different kinds of points on a piecewise smooth surface we are able to catalog all local labelling possibilities for the different types of junctions in a line drawing an algorithm is developed which utilizes this catalog to determine all legal labellings of the line drawing a local minimum complexity rule at each vertex select those labellings which correspond to the minimum number of faces meeting at the vertex is used in order to prune highly counter intuitive interpretations the labelling scheme was implemented and tested on a number of line drawings the labellings obtained are few and by and large in accordance with human interpretations

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

on the complexity of labeling perspective projections of polyhedral scenes abstract this paper investigates the computational time complexity of the labeling problem for line drawings of trihedral scenes it is shown that the class of problems having polynomial complexity is larger than the simple case of line drawings of legoland scenes kirousis and papadimitriou 1988 once the location of the vanishing points in the image plane is known the labeling problem can be solved in time o nn where n is the number of segments and n is the number of vanishing points the vanishing points can be given a priori otherwise can in many cases be detected by standard techniques from the line drawing itself the np completeness of the labeling problem for line drawings of trihedral scenes kirousis and papadimitriou 1988 is then due to the lack of knowledge about the vanishing points which is equivalent to the knowledge of the possible directions for the edges these results help draw a more accurate boundary between the problems in the interpretation of line drawings that are polynomially solvable and those that are np complete

disaggregation and resource allocation using convex knapsack problems with bounded variables the allocation of a specific amount of a given resource among competitive alternatives can often be modelled as a knapsack problem this model formulation is extremely efficient because it allows convex cost representation with bounded variables to be solved without great computational efforts practical applications of this problem abound in the fields of operations management finance manpower planning marketing etc in particular knapsack problems emerge in hierarchical planning systems when a first level of decisions need to be further allocated among specific activities which have been previously treated in an aggregate way in this paper we provide a recursive procedure to solve such problems the method differs from classical optimization algorithms of convex programming in that it determines at each iteration the optimal value of at least one variable applications and computational results are presented

identification of faces in line drawings by edge decomposition this paper presents a method to find the faces of objects in 2d line drawings using an approach totally different from existing ones it consists of two stages decomposition and face forming in decomposition a drawing is decomposed into chains of connected edges each chain belongs to one face only but may not yet form a closed loop the decomposition is local to a vertex and takes no account of the overall structure of the drawing the algorithm ensures that there is a chain for every face in the object the face forming stage completes the loop for each chain to form a face our method can deliver both the real faces and the internal faces separately it depends only on the topology of the drawing and not on the geometry and is therefore applicable to line drawings with curves and straight lines and 3d wireframes we implemented our algorithm and tested it with hundreds of objects including objects used in previous publications it always delivers the correct result comparison with the latest methods shows that it is faster often by tens of times and in some cases hundreds of times decompose a drawing into sub loops of edges that belong to only one face form face by finding the missing edges of a sub loop in a limited search space low computational complexity more efficient than all existing methods handle manifolds and non manifolds handle line drawings and 3d wireframes with straight and curved edges

multisymplectic box schemes for the complex modified korteweg de vries equation in this paper two multisymplectic integrators an eight point preissman box scheme and a narrow box scheme are considered for numerical integration of the complex modified korteweg de vries equation energy and momentum preservation of both schemes and their dispersive properties are investigated the performance of both methods is demonstrated through numerical tests on several solitary wavesolutions