physics of symplectic integrators perihelion advances and symplectic corrector algorithms symplectic integrators evolve dynamical systems according to modified hamiltonians whose error terms are also well defined hamiltonians the error of the algorithm is the sum of each error hamiltonian s perturbation on the exact solution when symplectic integrators are applied to the kepler problem these error terms cause the orbit to precess in this work by developing a general method of computing the perihelion advance via the laplace runge lenz vector even for nonseparable hamiltonians i show that the precession error in symplectic integrators can be computed analytically it is found that at leading order each paired error hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period hence symplectic corrector or process integrators which have equal coefficients for these paired error terms will have their precession errors cancel at that order after each period with the use of correctable algorithms both the energy and precession error are of effective order n 2 where n is the nominal order of the algorithm thus the physics of symplectic integrators determines the optimal algorithm for integrating long time periodic motions

quantum monte carlo methods recent developments abstract recent developments of the quantum monte carlo methods are reviewed briefly with emphasis on basic ideas namely the transformation of a d dimensional quantum system to the corresponding d 1 dimensional classical system some recent applications to quantum spin systems are summarized in particular numerical evidence of the haldane gap is given the quantum transfer matrix method is also presented for it is also based on the st transformation the quantum monte carlo cam is explained briefly fractal path integrals are also presented namely a general theory of decomposition of exponential operators and symplectic integrators are explained the negative sign problem is also discussed

computing the minimum distance between two bezier curves a sweeping sphere clipping method is presented for computing the minimum distance between two bezier curves the sweeping sphere is constructed by rolling a sphere with its center point along a curve the initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve the nearest point on a curve must be contained in the sweeping sphere along the other curve and all of the parts outside the sweeping sphere can be eliminated a simple sufficient condition when the nearest point is one of the two end points of a curve is provided which turns the curve curve case into a point curve case and leads to higher efficiency examples are shown to illustrate efficiency and robustness of the new method

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

the delta medial axis in this paper we present the delta medial axis dma a quasi linear algorithmic solution addressing several of the main concerns of discrete medial axes ma computation first its sensitivity to small shape perturbations is counterbalanced by a single parameter delta used in a pruning strategy that implicitly takes into account the local topology of the boundary second the discrete nature of images is addressed by approximating the original ma definition in continuous space third to allow real time performances a set of algorithmic optimizations is proposed we compare our approach both qualitatively and quantitatively to recent state of the art solutions and show that the dma presents an excellent choice for a wide range of applications finally to allow fast and efficient use of our algorithm we propose the complete pseudo code highlightsthe delta medial axis a fast and robust skeletonization algorithm is presented it includes a pruning parameter which considers the local topology of the boundary to allow real time performances optimizations are introduced with complete pseudo code a complete comparative study with four recent state of the art algorithms is proposed

study of correction terms for higher order decompositions of exponential operators many kinds of higher order decompositions are applied to an exactly solvable model and the trotter number dependence of the correction terms is evaluated systematically practically useful decompositions are explored

arbitrarily high order convected scheme solution of the vlasov poisson system abstract the convected scheme cs is a forward trajectory semi lagrangian method for solution of transport equations which has been most often applied to the kinetic description of plasmas and rarefied neutral gases in its simplest form the cs propagates the solution forward in time by advecting the so called moving cells along their characteristic trajectories and by remapping them on the mesh at the end of the time step the cs is conservative positivity preserving simple to implement and it is not subject to time step restriction to maintain stability recently guclu and hitchon 2012 1 a new methodology was introduced for reducing numerical diffusion based on a modified equation analysis the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping while the spatial accuracy was increased from 2nd to 4th order the new scheme retained the important properties of the original method and was shown to be extremely simple and efficient for constant advection problems here the cs is applied to the solution of the vlasov poisson system which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal coulomb interactions the vlasov equation is split into two constant advection equations one in configuration space and one in velocity space and high order time accuracy is achieved by proper composition of the operators the splitting procedure enables us to use the constant advection solver which we extend to arbitrarily high order of accuracy in time and space a new improved procedure is given which makes the calculation of the corrections straightforward focusing on periodic domains we describe a spectrally accurate scheme based on the fast fourier transform the proposed implementation is strictly conservative and positivity preserving the ability to correctly reproduce the system dynamics as well as resolving small scale features in the solution is shown in classical 1d 1v test cases both in the linear and the non linear regimes

generalized bracket formulation of constrained dynamics in phase space a generalized bracket formalism is used to define the phase space flow of constrained systems the generalized bracket naturally subsumes the approach to constrained dynamics given by dirac some time ago the dynamical invariant measure and the linear response of systems subjected to holonomic constraints are explicitly derived in light of previous results it is shown that generalized brackets provide a simple and unified view of the statistical mechanics of non hamiltonian phase space flows with a conserved energy

almost symplectic runge kutta schemes for hamiltonian systems symplectic runge kutta schemes for the integration of general hamiltonian systems are implicit in practice one has to solve the implicit algebraic equations using some iterative approximation method in which case the resulting integration scheme is no longer symplectic in this paper the preservation of the symplectic structure is analyzed under two popular approximation schemes fixed point iteration and newton s method respectively error bounds for the symplectic structure are established when n fixed point iterations or n iterations of newton s method are used the implications of these results for the implementation of symplectic methods are discussed and then explored through numerical examples numerical comparisons with non symplectic runge kutta methods and pseudo symplectic methods are also presented

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

finite dimensional lie poisson approximations to vlasov poisson equations many of the basic equations of conservative continuum mechanics euler vlasov poisson vlasov maxwell mhd etc are hamiltonian systems with respect to lie poisson brackets on dual spaces of infinite dimensional lie algebras the development of lie poisson integrators for finite dimensional lie poisson systems has shown that they are superior in the numerical simulations of these systems especially with regard to long term phenomena this paper shows how to truncate one of these systems the vlasov poisson equation of plasma physics to a finite dimensional lie poisson system this requires replacing the functions on single particle phase space with their poisson bracket lie algebra structure by a finite dimensional lie algebra replacing the densities by their moments of order up to k about a fixed reference point corresponds to replacing the functions by their taylor expansions up to order k unfortunately these truncated taylor expansions do not form a lie algebra since the functions which vanish through order k do not form an ideal under poisson bracket geometrically this corresponds to the fact that canonical transformations which fix the reference point do not form a normal subgroup introducing the location of a reference point in phase space as an extra variable and truncating with respect to this moving point turns out to decouple the location from shape coordinates of a lump of density in phase space as far as the poisson bracket is concerned one can then replace the shape coordinates by a finite number of moments the central result of the paper is a construction of the decoupling map described above in the general context of the decomposition of a lie group as a product of subgroups the main theorem is first proved by the general theory of poisson reduction then by explicit calculation and lastly by showing that the poisson isomorphism follows from the lift of a natural groupoid isomorphism the groupoid aspect of the theory also provides natural poisson maps useful in the application of ruth type integration techniques which do not seem easily derivable from the general theory of poisson reduction 1994 john wiley sons inc

composite integrators for bi hamiltonian systems abstract symmetric composition methods are applied to dynamical systems in bi hamiltonian form to lotka volterra equations to the completely integrable lorenz equation and to the periodic toda lattice the numerical results obtained show that the hamiltonians are preserved with high accuracy and the periodicity of the solutions are retained

energy preserving numerical schemes of high accuracy for one dimensional hamiltonian systems we present a class of non standard numerical schemes which are modifications of the discrete gradient method they preserve the energy integral exactly up to the round off error the considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order in numerical experiments we compare our integrators with some other numerical schemes including the standard discrete gradient method the leap frog scheme and a symplectic scheme of fourth order we study the error accumulation for a very long time and the conservation of the energy integral

semi continuity of skeletons in two manifold and discrete voronoi approximation the skeleton of a 2d shape is an important geometric structure in pattern analysis and computer vision in this paper we study the skeleton of a 2d shape in a two manifold mathcal m based on a geodesic metric we present a formal definition of the skeleton s omega for a shape omega in mathcal m and show several properties that make s omega distinct from its euclidean counterpart in mathbb r 2 we further prove that for a shape sequence lbrace omega i rbrace that converge to a shape omega in mathcal m the mapping omega rightarrow overline s omega is lower semi continuous a direct application of this result is that we can use a set p of sample points to approximate the boundary of a 2d shape omega in mathcal m and the voronoi diagram of p inside omega subset mathcal m gives a good approximation to the skeleton s omega examples of skeleton computation in topography and brain morphometry are illustrated

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

jolt factorization of pendulum map in this paper we apply the symplectic integration method using jolt factorization described in an earlier paper to the symplectic map describing the nonlinear pendulum hamiltonian we compare results obtained with this method with those obtained using nonsymplecticmethods and demonstrate that our results are much better

global large time self similarity of a thermal diffusive combustion system with critical nonlinearity we study the initial value problem of the thermal diffusive combustion system u 1 t u 1 x x u 1 u 2 2 u 2 t du 2 xx u 1 u 2 2 x is an element of r 1 for non negative spatially decaying initial data of arbitrary size and for any positive constant d we show that if the initial data decay to zero sufficiently fast at infinity then the solution u 1 u 2 converges to a self similar solution of the reduced system u 1 t u 1 xx u 1 u 2 2 u 2 t du 2 xx in the large time limit in particular u 1 decays to zero like o t 1 2 delta where delta 0 is an anomalous exponent depending on the initial data and u 2 decays to zero with normal rate o t 1 2 the idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self similarity of the solutions in the large time limit c 1996 academic press inc

a competition flow method for computing medial axis transform abstract medial axis computation mac has many applications in computer aided design computer graphics automated mesh generation solid modeling cnc tool path generation robotic motion planning bioinformatics and shape analysis there are three key issues that need to be further addressed on mac of a nurbs non uniform rational b spline curve boundary namely 1 computation of regular medial axis points 2 efficient detection of the existence of a branch point where two branches intersect and 3 convergence and correctness on computing the position of a branch point this paper presents a competition flow method for medial axis computation of a planar simply connected nurbs boundary different from previous methods which trace a voronoi diagram tree vdt in depth first order from a terminal point the competition flow method cfm traces the corresponding tree in breadth first order it has three main contributions firstly it utilizes an optimized method based on previous state of art algorithms for computing regular medial axis points which can achieve a higher computational efficiency secondly it implements a divide and conquer method in a natural way by proposing a simple tree technique in which a vdt is divided into several simple tree structures of level l l 0 1 for competitive medial axis computation finally by using the competition flow method possible places where branch points are located are intuitively and efficiently detected and bounded with high precision which ensures the convergence and the correctness of the computation of the position of a branch point and also ensures the correctness of the topological structure of the medial axis tree numerical examples show that the new method can achieve a better performance compared with several existing methods

convergence of unitary and complex decompositions of exponential operators the convergence of the fractal unitary and complex decompositions of exponential operators is proved for hermitian operators in a banach space

computation of nonlinear behavior of hamiltonian systems using lie algebraic methods lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general hamiltonian systems a procedure is presented for the computation of nonlinear effects of arbitrarily high degree and explicit formulas are given through effects of degree 5 expected applications include accelerator design charged particle beam and light optics other problems in the general area of nonlinear dynamics and perhaps with suitable modification the area of s matrix expansions in quantum field theory

construction of iso contours bisectors and voronoi diagrams on triangulated surfaces in the research of computer vision and machine perception 3d objects are usually represented by 2 manifold triangular meshes m in this paper we present practical and efficient algorithms to construct iso contours bisectors and voronoi diagrams of point sites on m based on an exact geodesic metric compared to euclidean metric spaces the voronoi diagrams on m exhibit many special properties that fail all of the existing euclidean voronoi algorithms to provide practical algorithms for constructing geodesic metric based voronoi diagrams on m this paper studies the analytic structure of iso contours bisectors and voronoi diagrams on m after a necessary preprocessing of model m practical algorithms are proposed for quickly obtaining full information about iso contours bisectors and voronoi diagrams on m the complexity of the construction algorithms is also analyzed finally three interesting applications surface sampling and reconstruction 3d skeleton extraction and point pattern analysis are presented that show the potential power of the proposed algorithms in pattern analysis

lie algebraic approach to fer s expansion for classical hamiltonian systems the so called fer s expansion is proposed as a solution for the time evolution operator of classical time dependent hamiltonian systems the quadratic hamiltonians treated as examples show that under very different regimes the second order approximation already gives extremely good results