backward error analysis of a full discretization scheme for a class of semilinear parabolic partial differential equations a numerical method for reaction diffusion equations with analytic nonlinearity is presented for which a rigorous backward error analysis is possible we construct a modified equation which describes the behavior of the full discretization scheme up to exponentially small errors in the step size in the construction the numerical scheme is first exactly embedded into a nonautonomous equation this equation is then averaged with only exponentially small remainder terms the long time behavior near hyperbolic equilibria the persistence of homoclinic orbits and regularity properties are analyzed

the fast multipole method fmm for electromagnetic scattering problems the fast multipole method fmm developed by v rokhlin 1990 to efficiently solve acoustic scattering problems is modified and adapted to the second kind integral equation formulation of electromagnetic scattering problems in two dimensions the present implementation treats the exterior dirichlet problem for two dimensional closed conducting objects of arbitrary geometry the fmm reduces the operation count for solving the second kind integral equation from o n sup 3 for gaussian elimination to o n sup 4 3 per conjugate gradient iteration where n is the number of sample points on the boundary of the scatterer a sample technique for accelerating convergence of the iterative method termed complexifying k the wavenumber is also presented this has the effect of bounding the condition number of the discrete system consequently the operation count of the entire fmm all iterations becomes o n sup 4 3 computational results for moderate values of ka where a is the characteristic size of the scatterer are given

existence of a tree of stieltjes strings corresponding to two given spectra transversal vibrations of a plane rooted tree of stieltjes strings are considered with dirichlet boundary conditions at each pendant vertex continuity and kirchhoff conditions are imposed at each interior vertex except the root the dirichlet problem is that with the dirichlet condition at the root and the neumann problem is that with continuity and kirchhoff conditions at the root it is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of neumann and dirichlet problems generated by stieltjes string recurrent relations on any prescribed tree

shadow hybrid monte carlo an efficient propagator in phase space of macromolecules shadow hybrid monte carlo shmc is a new method for sampling the phase space of large molecules particularly biological molecules it improves sampling of hybrid monte carlo hmc by allowing larger time steps and system sizes in the molecular dynamics md step the acceptance rate of hmc decreases exponentially with increasing system size n or time step t this is due to discretization errors introduced by the numerical integrator shmc achieves an asymptotic o n1 4 speedup over hmc by sampling from all of phase space using high order approximations to a shadow or modified hamiltonian exactly integrated by a symplectic md integrator shmc satisfies microscopic reversibility and is a rigorous sampling method shmc requires extra storage modest computational overhead and a reweighting step to obtain averages from the canonical ensemble this is validated by numerical experiments that compute observables for different molecules ranging from a small n alkane butane with four united atoms to a larger solvated protein with 14 281 atoms in these experiments shmc achieves an order magnitude speedup in sampling efficiency for medium sized proteins sampling efficiency is measured by monitoring the rate at which different conformations of the molecules dihedral angles are visited and by computing ergodic measures of some observables

electromagnetic analysis of large scale periodic arrays using a two level cbfs method accelerated with fmm fft to realize efficient analysis of large scale planar periodic arrays a two level characteristic basis functions method accelerated with fast multipole method fast fourier transform fmm fft is developed in this paper the characteristic basis functions cbfs for each element are constructed in a two level framework in which the cbfs construction based on plane wave derivation for single element at child level and the cbfs construction based on local interaction between array elements at father level are combined the number of unknowns is reduced significantly by the two level cbfs further fmm fft is applied to expedite the computation of interaction between elements numerical results of large scale planar periodic arrays are given to demonstrate the validity and efficiency of the present method

error propagation when approximating multi solitons the kdv equation as a case study abstract the paper studies the influence of the time discretizations when simulating some phenomena involving more than one solitary wave taking the kdv equation as a case study we obtain some conditions on the numerical method in order to get a more correct simulation of multi soliton solutions they are related to the evolution of the conserved quantities of the problem through the numerical integration it is shown that when approximating n solitons a method that preserves n invariants of the problem shows a better time propagation of the error than that of a general scheme as a consequence of this the simulation of some physical parameters that characterize the waves is more suitable when using conservative integrators we also show how these results can be extended to the approximation to multi solitons of any equation of the kdv hierarchy and more generally other integrable equations

construction of multisymplectic schemes of any finite order for modified wave equations in this paper we analyze several symplectic discretizations of linear wave equations with periodic boundary conditions and show the discretizations are multisymplectic of arbitrary finite order for wave equations when the periodic boundary condition vanishes

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

the discrete fourier transform method of solving differential integral equations in scattering theory an accurate and efficient numerical method is presented for solving many differential integral equations arising from electromagnetic scattering theory it uses the discrete fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations as a consequence this method is extremely simple to implement uses less computer memory than comparable methods and yields accurate predictions the differential integral equation is recast into a periodic form conducive to application of the discrete fourier convolution theorem the differential operators are approximated by appropriate finite difference and discrete convolution operators all these quantities are computed by using the fast fourier transform an approximate solution is obtained by using the conjugate gradient method results are compared to experimental data or analytical solutions for a 3 lambda 3 lambda metal plate where lambda is the wavelength a homogeneous and a layered infinite circular dielectric cylinder and a dielectric sphere the accuracy of the method is further illustrated by comparing predictions with independent measurements by r a ross 1966 on a 2 lambda 1 lambda metal plate at grazing incidence in all cases agreement is excellent

the nose poincare method for constant temperature molecular dynamics we present a new extended phase space method for constant temperature canonical ensemble molecular dynamics our starting point is the hamiltonian introduced by nose to generate trajectories corresponding to configurations in the canonical ensemble using a poincare time transformation we construct a hamiltonian system with the correct intrinsic timescale and show that it generates trajectories in the canonical ensemble our approach corrects a serious deficiency of the standard change of variables nose hoover dynamics which yields a time reversible system but simultaneously destroys the hamiltonian structure a symplectic discretization method is presented for solving the nose poincare equations the method is explicit and preserves the time reversal symmetry in numerical experiments it is shown that the new method exhibits enhanced stability when the temperature fluctuation is large extensions are presented for nose chains holonomic constraints and rigid bodies

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

non ergodicity of the nose hoover thermostatted harmonic oscillator the nose hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical gibbs distribution newton s equations are modified by coupling an additional reservoir variable to the physical variables the correct sampling of the phase space according to the gibbs measure is dependent on the nose hoover dynamics being ergodic hoover presented numerical experiments to show that the nose hoover dynamics are non ergodic when applied to the harmonic oscillator in this article we prove that the nose hoover thermostat does not give an ergodynamical system for the one dimensional harmonic oscillator when the mass of the reservoir is large our proof of non ergodicity uses kam theory to demonstrate the existence of invariant tori for the nose hoover dynamical system that separate phase space into invariant regions we present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

a time reversible variable stepsize integrator for constrained dynamics this article considers the design and implementation of variable timestep methods for simulating holonomically constrained mechanical systems symplectic variable stepsizes are briefly discussed and we consider time reparameterization techniques employing a time reversible symmetric integration method to solve the equations of motion we give several numerical examples including a simulation of an elastic inextensible unshearable rod undergoing large deformations and collisions with the sides of a bounding box numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation

on higher order semi explicit symplectic partitioned runge kutta methods for constrained hamiltonian systems in this paper we generalize the class of explicit partitioned runge kutta prk methods for separable hamiltonian systems to systems with holonomic constraints for a convenient analysis of such schemes we first generalize the backward error analysis for systems in bbb r m to systems on manifolds embedded in bbb r m by applying this analysis to constrained prk methods we prove that such methods will in general suffer from order reduction as well known for higher index differential algebraic equations however this order reduction can be avoided by a proper modification of the standard prk methods this modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant we also give a numerical comparison of several second fourth and sixth order methods

an efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse matrix canonical grid method an efficient algorithm for wave scattering from two dimensional lossy rough surfaces is proposed it entails the use of a single magnetic field integral equation smfie in conjunction with a multilevel sparse matrix canonical grid msmcg method the rao wilton glisson rwg triangular discretization is adopted to better model the rough surface than the pulse basis functions used in the well established smcg method using the smfie formulation only one unknown per interior edge of the triangular mesh approximating the rough surface is required and the iterative solution to the moment equation converges more rapidly than that of the conventional coupled equations for dielectric rough surfaces the msmcg method extends the applicability of the smcg method to rougher surfaces parallel implementation of the proposed method enables us to model dielectric surfaces up to a few thousand square wavelengths simulation results are presented as bistatic scattering coefficients for gaussian randomly rough surfaces

computing global orbits of the forced spherical pendulum abstract orbits of the spherical pendulum with time periodic forcing are considered a numerical framework is developed which allows orbits to explore the entire globe the spherical pendulum is considered as an invariant manifold in an ambient six dimensional euclidean space the numerical integrator is the second order stormer verlet method coupled with the shake rattle algorithm the algorithm preserves numerically the phase space of the sphere which is a manifold to machine accuracy poincare sections restricted to the configuration space are used to illustrate the transition from oscillatory behavior to chaotic solutions as the amplitude of the pivot motion is changed the qualitative change in the poincare sections from regular to chaotic behavior is in excellent qualitative agreement with corresponding computations of the lyapunov exponents les the les are also computed using a novel variant of the shake rattle algorithm the results show that irregular behavior can explore the entire sphere even at low forcing amplitudes and therefore local methods which parameterize only part of the sphere are inadequate in general and may lead to spurious dynamics the numerical framework provides a tool for detailed investigation of the symmetric chaos of the forced spherical pendulum

scattering by infinite one dimensional rough surfaces we consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane this problem models time harmonic electromagnetic scattering by a one dimensional infinite rough perfectly conducting surface the same problem arises in acoustic scattering by a sound soft surface chandler wilde and zhang have suggested a radiation condition for this problem a generalization of the rayleigh expansion condition for diffraction gratings and uniqueness of solution has been established recently an integral equation formulation of the problem has also been proposed and in the special case when the whole boundary is both lyapunov and a small perturbation of a flat boundary the unique solvability of this integral equation has been shown by chandler wilde and ross by operator perturbation arguments in this paper we study the general case with no limit on surface amplitudes or slopes and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers as an important corollary we prove that for a variety of incident fields including the incident plane wave the dirichlet boundary value problem for the scattered field has a unique solution

calculation of acoustical scattering from a cluster of scatterers the problem of determining the field scattered by a cluster of scatterers when they are insonified by a known acoustical field is addressed the problem is formulated by using the t matrix method and the resulting system of linear equations is solved by using the multilevel fast multipole algorithm mlfma and the fast multipole method fast fourier transform fmmfft method and the efficiency of the two methods is compared it was observed that in general the mlfma performs better than the fmmfft algorithm however when the scatterers are distributed uniformly on a rectangular grid the fmmfft algorithm performs as good as the mlfma the accuracy of the methods is evaluated by modeling a spherical scatterer as composed of many small spheres

error propagation in numerical approximations near relative equilibria we study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of hamiltonian differential equations with symmetries first it is shown that taking an initial perturbation of a relative equilibrium the corresponding solution is related in a first approximation to another relative equilibrium with the parameters perturbed from the original then this is used to prove that for stable relative equilibria error growth with respect to the perturbed solution is in general quadratic but only linear for schemes that preserve the invariant quantities of the problem in this sense the conclusion is similar to the one obtained when integrating unperturbed relative equilibria numerical experiments illustrate the results

scattering from two dimensional objects of varying shape combining the method of moments with the stochastic galerkin method in this communication the combined field integral equation for perfect electrically conducting scatterers is combined with the stochastic galerkin method sgm to model the impact of stochastic variations of the shape of the scatterer on the radar cross section and on the induced current distribution the sgm is compared to the stochastic collocation method scm and it is shown that for a modest number of random variables the sgm is a good alternative to the scm

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

spectral collocation variational integrators abstract spectral methods are a popular choice for constructing numerical approximations for smooth problems as they can achieve geometric rates of convergence and have a relatively small memory footprint in this paper we introduce a general framework to convert a spectral collocation method into a shooting based variational integrator for hamiltonian systems we also compare the proposed spectral collocation variational integrators to spectral collocation methods and galerkin spectral variational integrators in terms of their ability to reproduce accurate trajectories in configuration and phase space their ability to conserve momentum and energy as well as the relative computational efficiency of these methods when applied to some classical hamiltonian systems in particular we note that spectrally accurate variational integrators such as the galerkin spectral variational integrators and the spectral collocation variational integrators combine the computational efficiency of spectral methods together with the geometric structure preserving and long time structural stability properties of symplectic integrators

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

an inverse eigenvalue problem for pseudo jacobi matrices abstract in this paper the theory on direct and inverse spectral problems for jacobi matrices is revisited in a kind of pseudo jacobi matrices j n r with a mixed path as its graph in the non self adjoint setting in this context a sign change in one of the nondiagonal entries of the matrix yields strong perturbations in its spectral properties the reconstruction of a pseudo jacobi matrix from its spectrum and the spectra of two complementary principal matrices is investigated an algorithm for the reconstruction of matrices from prescribed spectral data is provided and illustrative numerical experiments are performed

one can hear the composition of a string experiments with an inverse eigenvalue problem to what extent do the vibrations of a mechanical system reveal its composition despite innumerable applications and mathematical elegance this question often slips through those cracks that separate courses in mechanics differential equations and linear algebra we address this omission by detailing a classical finite dimensional example the use of frequencies of vibration to recover positions and masses of beads vibrating on a string first we derive the equations of motion then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory s monochord more challenging is the recovery of masses and positions of the beads from spectral data a problem for which a variety of elegant algorithms exist after presenting one such method based on orthogonal polynomials in a manner suitable for advanced undergraduates we confirm its efficacy through physical experiment we encourage readers to conduct their own explorations using the numerous data sets we provide

aspects of backward error analysis of numerical odes abstract in this paper some preliminary cases are studied to see how the idea of backward error analysis carries over to numerical solution of ordinary differential equations the interest here is in one step methods and infinite time intervals

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

conservation of energy momentum and actions in numerical discretizations of non linear wave equations for classes of symplectic and symmetric time stepping methods trigonometric integrators and the stormer verlet or leapfrog method applied to spectral semi discretizations of semilinear wave equations in a weakly non linear setting it is shown that energy momentum and all harmonic actions are approximately preserved over long times for the case of interest where the cfl number is not a small parameter such results are outside the reach of standard backward error analysis here they are instead obtained via a modulated fourier expansion in time

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

on manifolds of connecting orbits in discretizations of dynamical systems it is shown that one step methods when applied to a one parametric dynamical system with a homoclinic orbit exhibit a closed loop of discrete homoclinic orbits on this loop the parameter varies periodically while the orbit shifts its index after one revolution we show that at least two homoclinic tangencies occur on this loop our approach works for systems with finite smoothness and also applies to general connecting orbits it provides an alternative to the interpolation approach by fiedler and scheurle mem amer math soc 119 570 1996 and it allows to recover some of their results on exponentially small splittings of separatrices by using some recent backward error analysis for the analytic case

discretization errors in molecular dynamics simulations with deterministic and stochastic thermostats we investigate the influence of numerical discretization errors on computed averages in a molecular dynamics simulation of tip4p liquid water at 300k coupled to different deterministic nose hoover and nose poincare and stochastic langevin thermostats we propose a couple of simple practical approaches to estimating such errors and taking them into account when computing the averages we show that it is possible to obtain accurate measurements of various system quantities using step sizes of up to 70 of the stability threshold of the integrator which for the system of tip4p liquid water at 300k corresponds to the step size of about 7fs

a technique to construct symmetric variable stepsize linear multistep methods for second order systems some previous works show that symmetric fixed and variable stepsize linear multistep methods for second order systems which do not have any parasitic root in their first characteristic polynomial give rise to a slow error growth with time when integrating reversible systems in this paper we give a technique to construct variable stepsize symmetric methods from their fixed stepsize counterparts in such a way that the former have the same order as the latter the order and symmetry of the integrators obtained is proved independently of the order of the underlying fixed stepsize integrators as this technique looks for efficiency we concentrate on explicit linear multistep methods which just make one function evaluation per step and we offer some numerical comparisons with other one step adaptive methods which also show a good long term behaviour

on the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata we consider the nonnegative inverse eigenvalue problem with partial eigendata which aims to find a nonnegative matrix such that it is nearest to a pre estimated nonnegative matrix and satisfies the prescribed eigendata in this paper we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem we also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries numerical tests including a practical engineering application in vibrations show the efficiency of the proposed iterative schemes

monte carlo simulations of large scale problems of random rough surface scattering and applications to grazing incidence with the bmia canonical grid method scattering of a te incident wave from a perfectly conducting one dimensional random rough surface is studied with the banded matrix iterative approach canonical grid bmia cag method the bmia cag is an improvement over the previous bmia the key idea of bmia cag is that outside the near field interaction the rest of the interactions can be translated to a canonical grid by taylor series expansion the use of a flat surface as a canonical grid for a rough surface facilitates the use of the fast fourier transform for nonnear field interaction the method can be used for monte carlo simulations of random rough surface problems with a large surface length including all the coherent wave interactions within the entire surface we illustrate results up to a surface length of 2500 wavelengths with 25000 surface unknowns the method is also applied to study scattering from random rough surfaces at near grazing incidence the numerical examples illustrate the importance of using a large surface length for some backscattering problems

reversible long term integration with variable stepsizes the numerical integration of reversible dynamical systems is considered a backward analysis for variable stepsize one step methods is developed and it is shown that the numerical solution of a symmetric one step method implemented with a reversible stepsize strategy is formally equal to the exact solution of a perturbed differential equation which again is reversible this explains geometrical properties of the numerical flow such as the nearby preservation of invariants in a second part the efficiency of symmetric implicit runge kutta methods linear error growth when applied to integrable systems is compared with explicit nonsymmetric integrators quadratic error growth

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

an inverse eigenvalue problem for a vibrating string with two dirichlet spectra a classical inverse problem is can you hear the density of a string clamped at both ends the mathematical model gives rise to an inverse sturm liouville problem for the unknown density rho and it is well known that the answer is negative the dirichlet spectrum from the clamped end point conditions is insufficient there are many known ways to add additional information to gain a positive answer and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode the so called norming constants we make the assumption that neither of these changes are possible instead we will add known mass densities to the string in a way we can prescribe and remeasure the dirichlet spectrum we will not be able to answer the uniqueness question in its most general form but will give some insight to what added masses should be chosen and how this can lead to a reconstruction of the original string density

electromagnetic scattering from perfectly conducting rough surfaces a new full wave method a method that does not make use of the telegraphist s equations and takes into account the two dimensional roughness of the surface from the start is developed it is shown that the scattering coefficients obtained agree with those given in earlier work by e bahar 1973 1987 the method is based on reducing the three dimensional scattering problem to a two dimensional problem by expanding each rectangular component of maxwell s equations in terms of local basis functions along the perpendicular direction to the mean surface the transformed two dimensional field equations are solved using fourier transforms the full wave solutions are also compared with the first order perturbation solutions the kirchhoff type solutions and integral equation results

accurate and efficient simulation of rigid body rotations abstract this paper introduces efficient and accurate algorithms for simulating the rotation of a three dimensional rigid object and compares them to several prior methods first we introduce a second order accurate method that incorporates a third order correction then we introduce a third order accurate method and finally we give a fourth order accurate method these methods are single step and the update operation is only a single rotation the algorithms are derived in a general lie group setting second we introduce a near optimal energy correction method which allows exact conservation of energy this algorithm is faster and easier to implement than implicit methods for exact energy conservation our third order method with energy conservation is experimentally seen to act better than a fourth order accurate method these new methods are superior to naive runge kutta or predictor corrector methods which are only second order accurate for sphere valued functions the second order symplectic mclachlan reich methods are observed to be excellent at approximate energy conservation but are not as good at long term accuracy as our best methods finally we present comparisons with fourth order accurate symplectic methods which have good accuracy but higher computational cost

geometric integrators for multiple time scale simulation in this paper we review and extend recent research on averaging integrators for multiple time scale simulation such as are needed for physical n body problems including molecular dynamics materials modelling and celestial mechanics a number of methods have been proposed for direct numerical integration of multiscale problems with special structure such as the mollified impulse method garcia archilla sanz serna and skeel 1999 siam j sci comput 20 930 63 and the reversible averaging method leimkuhler and reich 2001 j comput phys 171 95 114 features of problems of interest such as thermostatted coarse grained molecular dynamics require extension of the standard framework at the same time in some applications the computation of averages plays a crucial role but the available methods have deficiencies in this regard we demonstrate that a new approach based on the introduction of shadow variables which mirror physical variables has promised for broadening the usefulness of multiscale methods and enhancing accuracy of or simplifying computation of averages the shadow variables must be computed from an auxiliary equation while a geometric integrator in the extended space is possible in practice we observe enhanced long term energy behaviour only through use of a variant of the method which controls drift of the shadow variables using dissipation and sacrifices the formal geometric properties such as time reversibility and volume preservation in the enlarged phase space stabilizing the corresponding properties in the physical variables the method is applied to a gravitational three body problem as well as a partially thermostatted model problem for a dilute gas of diatomic molecules

inverse problems for r s symmetric matrices in structural dynamic model updating let r s c n n be nontrivial involutions i e r r 1 i n and s s 1 i n a matrix a c n n is referred to as r s symmetric if and only if r a s a the set of all r s symmetric matrices of order n is denoted by c s n n r s given a full column rank matrix x c n m a matrix b c m m and a matrix a c n n in structural dynamic model updating we usually consider the sets s 1 a a c s n n r s x h a x b and s 2 a a c s n n r s x h a x b min in the frobenius norm sense where the superscript h denotes conjugate transpose then we characterize the unique matrices a arg min a s 1 a a and a arg min a s 2 a a when r r h and s s h by using the generalized singular value decomposition gsvd the necessary and sufficient conditions for the non emptiness of s 1 and the general representations of the elements in s 1 and a are derived respectively the analytical expressions of a s 2 and a are also obtained by using the gsvd the canonical correlation decomposition ccd and the projection theorem finally a corresponding numerical algorithm and some illustrated examples are presented

a liouville operator derived measure preserving integrator for molecular dynamics simulations in the isothermal isobaric ensemble the constant pressure constant temperature npt molecular dynamics approach is re examined from the viewpoint of deriving a new measure preserving reversible geometric integrator for the equations of motion the underlying concepts of non hamiltonian phase space analysis measure preserving integrators and the symplectic property for hamiltonian systems are briefly reviewed in addition current measure preserving schemes for the constant volume constant temperature ensemble are also reviewed a new geometric integrator for the npt method is presented is shown to preserve the correct phase space volume element and is demonstrated to perform well in realistic examples finally a multiple time step version of the integrator is presented for treating systems with motion on several time scales

on the rate of convergence to equilibrium of the andersen thermostat in molecular dynamics it has been shown in e and li comm pure appl math 2007 in press that the andersen dynamics is uniformly ergodic exponential convergence to the invariant measure is established with an error bound of the form mathit const cdot exp mathit const cdot kappa nu nu 2n t where n is the number of particles is the collision frequency and const as 0 in this article we study the dependence on of the rate of convergence to equilibrium in the one dimension and one particle case we improve the error bound to be mathit const cdot exp mathit const cdot kappa nu nu t in the d dimension n particle free streaming case it is proved that the optimal error bound is mathit const cdot exp biggl mathit const cdot frac nu n t biggr it is also shown that as on the diffusive time scale the andersen dynamics converges to a smoluchowski equation

some conservation issues for the dynamical cores of nwp and climate models the rationale for designing atmospheric numerical model dynamical cores with certain conservation properties is reviewed the conceptual difficulties associated with the multiscale nature of realistic atmospheric flow and its lack of time reversibility are highlighted a distinction is made between robust invariants which are conserved or nearly conserved in the adiabatic and frictionless limit and non robust invariants which are not conserved in the limit even though they are conserved by exactly adiabatic frictionless flow for non robust invariants a further distinction is made between processes that directly transfer some quantity from large to small scales and processes involving a cascade through a continuous range of scales such cascades may either be explicitly parameterized or handled implicitly by the dynamical core numerics accepting the implied non conservation an attempt is made to estimate the relative importance of different conservation laws it is argued that satisfactory model performance requires spurious sources of a conservable quantity to be much smaller than any true physical sources for several conservable quantities the magnitudes of the physical sources are estimated in order to provide benchmarks against which any spurious sources may be measured

electromagnetic scattering from rough surface using single integral equation and adaptive integral method an efficient algorithm for electromagnetic wave scattering from rough dielectric surfaces is developed the algorithm is based on the single magnetic field integral equation smfie and the surface is discretized using rao wilton glisson rwg triangular basis functions the new feature of the algorithm is the application of the adaptive integral method aim with smfie for speeding up the calculation the developed new method utilizes the flexibility of rwg functions to model arbitrary rough surface and the speed of fft enabling accurate simulations over large surfaces

symplectic numerical integrators in constrained hamiltonian systems recent work reported in the literature suggests that for the long time integration of hamiltonian dynamical systems one should use methods that preserve the symplectic or canonical structure of the flow here we investigate the symplecticness of numerical integrators for constrained dynamics such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies this leads to a constrained hamiltonian system of smaller dimension previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space here it is shown that the very popular verlet method with shake type constraints is equivalent to the same method with rattle type constraints and that the latter is symplectic and time reversible this assumes that the iteration is carried to convergence we also demonstrate the global convergence of the verlet scheme in the presence of shake type and rattle type constraints we perform numerical experiments to compare these methods with the second order backward differentiation method commonly recommended for ordinary differential equations with constraints

tapered wave with dominant polarization state for all angles of incidence typical applications of the method of moments mom to rough surface three dimensional 3 d electromagnetic scattering require a truncation of the surface considered and call for a tapered incident wave it is shown how such a wave can be constructed as a superposition of plane waves avoiding problems near both normal and grazing incidence and providing clean footprints and clear polarization at all angles of incidence the proposed special choice of polarization vectors removes an irregularity at the origin of the wavenumber space and leads to a least squared error property of the wave issues in the application to 3 d scattering from an object over a rough surface are discussed approximate 3 d scalar and vector tapered waves which can be evaluated without resorting to any numerical integrations are derived and important limitations to the accuracy and applicability of these approximations are pointed out

explicit symplectic approximation of nonseparable hamiltonians algorithm and long time performance explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable hamiltonians this article proposes for arbitrary hamiltonians similar integrators which are explicit of any even order symplectic in an extended phase space and with pleasant long time properties they are based on a mechanical restraint that binds two copies of phase space together using backward error analysis kolmogorov arnold moser theory and additional multiscale analysis an error bound of o t l is established for integrable systems where t l and are respectively the long simulation time step size integrator order and some binding constant for nonintegrable systems with positive lyapunov exponents such an error bound is generally impossible but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear schrodinger equation

on the stability of some algorithms for computing the action of the matrix exponential abstract this paper deals with the stability of some one step methods for computing the action of a matrix exponential on a vector i e e a b where a is an n by n matrix and b is a vector of dimension n the methods are based on certain assumptions made for the case that a is a matrix having sufficiently small norm if the norm of a is larger a fixed step size is used to perform a scaling of a an analysis of the roundoff error of the computed e a b shows that these methods are backward stable if a is hermitian and forward stable if for example a is skew hermitian or if a is real and essentially nonnegative and b is real and nonnegative in addition an upper bound on the forward error of the computed e a b is obtained for all a and b this bound which is expressed in terms of the condition number is evaluated for several examples of the data the results of this paper partly apply to the algorithm proposed by al mohy and higham 2011 for computing e a b

generalized mathematical homogenization from theory to practice a generalized mathematical homogenization gmh theory for coupling atomistic and continuum descriptions at finite temperature has been extended to account for many body potentials gmh gives rise to the constitutive law free coarse scale equations where coupled continuum thermo mechanical equations are directly derived from molecular dynamics equations an integrated computational approach which seamlessly integrates abaqus for coarse scale finite element method computations with a molecular dynamics code has been developed the accuracy and efficiency of the method has been studied for crystal silicon structures including failure predictions of nanowires

scattering from very large randomly rough surfaces using a markov random field equivalent current a markov random field mrf model is employed to learn the statistical properties of an equivalent current situated above a rough surface where this equivalent current represents near field electromagnetic scattered fields the mrf parameters are learned by considering numerical scattering from a relatively small rough surface which can be solved efficiently via existing numerical methods the mrf parameters may then be used to synthesize multiple realizations of the equivalent current associated with scattering from a much larger rough surface these equivalent currents may be used to efficiently calculate the far field scattered field statistics for large surfaces the advantages of this approach are manifested in the fact that once the mrf parameters are learned a forward model is no longer needed to generate random realizations of the equivalent currents and the mrf model can be employed to generate equivalent surfaces of size beyond the capability of existing numerical models to improve the accuracy of the mrf model a narrowband filter is used to preprocess the computed current distributions prior to mrf model training one dimensional rough surfaces are considered here to demonstrate the idea and the method of ordered multiple interactions momi is employed to perform the numerical forward modeling promising mrf generated results are demonstrated with comparison to direct momi solutions

monte carlo simulation of electromagnetic scattering from two dimensional random rough surfaces the fast multipole method fast fourier transform fmm fft method is developed to compute the scattering of an electromagnetic wave from a two dimensional 2 d rough surface the resulting algorithm computes a matrix vector multiply in o n log n operations this algorithm is shown to be more efficient than another o n log n algorithm the multilevel fast multipole algorithm mlfma for surfaces of small height for surfaces with larger roughness the mlfma is found to be more efficient using the mlfma monte carlo simulations are carried out to compute the statistical properties of the electromagnetic scattering from 2 d random rough surfaces using a workstation for the rougher surface backscattering enhancement is clearly observable as a pronounced peak in the backscattering direction of the computed bistatic scattering coefficient for the smoother surface the monte carlo results compare well with the results of the approximate kirchhoff theory

the geometric integration of scale invariant ordinary and partial differential equations this review paper examines a synthesis of adaptive mesh methods with the use of symmetry to solve ordinary and partial differential equations it looks at the effectiveness of numerical methods in preserving geometric structures of the underlying equations such as scaling invariance conservation laws and solution orderings studies are made of a series of examples including the porous medium equation and the nonlinear schrodinger equation

generalized dynamical thermostating technique we demonstrate that the nose method for constant temperature molecular dynamics simulation mol phys 52 255 1984 can be substantially generalized by the addition of auxiliary variables to encompass an infinite variety of hamiltonian thermostats such thermostats can be used to enhance ergodicity in systems such as the one dimensional harmonic oscillator or certain molecular systems for which the standard nose hoover methods fail to reproduce converged canonical distributions in this respect the method is similar in spirit to the method of nose hoover chains but is both more general and hamiltonian in structure which allows for the use of efficient symplectic integration schemes in particular we show that within the generalized nose formalism outlined herein any hamiltonian system can be thermostated with any other including a copy of itself this gives one an enormous flexibility in choosing the form of the thermostating bath numerical experiments are included in which a harmonic oscillator is thermostated with a collection of noninteracting harmonic oscillators as well as by a soft billiard system

dirichlet neumann inverse spectral problem for a star graph of stieltjes strings we solve two inverse spectral problems for star graphs of stieltjes strings with dirichlet and neumann boundary conditions respectively at a selected vertex called root the root is either the central vertex or in the more challenging problem a pendant vertex of the star graph at all other pendant vertices dirichlet conditions are imposed at the central vertex at which a mass may be placed continuity and kirchhoff conditions are assumed we derive conditions on two sets of real numbers to be the spectra of the above dirichlet and neumann problems our solution for the inverse problems is constructive we establish algorithms to recover the mass distribution on the star graph i e the point masses and lengths of subintervals between them from these two spectra and from the lengths of the separate strings if the root is a pendant vertex the two spectra uniquely determine the parameters on the main string i e the string incident to the root if the length of the main string is known the mass distribution on the other edges need not be unique the reason for this is the non uniqueness caused by the non strict interlacing of the given data in the case when the root is the central vertex finally we relate of our results to tree patterned matrix inverse problems

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

asynchronous variational integrators we describe a class of asynchronous variational integrators avi for nonlinear elastodynamics the avis are characterized by the following distinguishing attributes i the algorithms permit the selection of independent time steps in each element and the local time steps need not bear an integral relation to each other ii the algorithms derive from a spacetime form of a discrete version of hamilton s principle as a consequence of this variational structure the algorithms conserve local energy and momenta exactly subject to solvability of the local time steps numerical tests reveal that even when local energy balance is not enforced exactly the global and local energy behavior of the avis is quite remarkable a property which can probably be traced to the symplectic nature of the algorithm

an ergodic sampling scheme for constrained hamiltonian systems with applications to molecular dynamics this article addresses the problem of computing the gibbs distribu tion of a hamiltonian system that is subject to holonomic constraints in doing so we extend recent ideas of cances et al m2an vol 41 no 2 pp 351 389 2007 who could prove a law of large numbers for unconstrained molecular systems with a separable hamiltonian em ploying a discrete version of hamilton s principle studying ergodicity for constrained hamiltonian systems we specifically focus on the numerical discretization error even if the continuous system is perfectly ergodic this property is typically not preserved by the numerical discretization the discretization error is taken care of by means of a hybrid monte carlo al gorithm that allows for sampling bias free expectation values with respect to the gibbs measure independently of the stable step size we give a demonstration of the sampling algorithm by calculating the free energy profile of a small peptide

gshmc an efficient method for molecular simulation the hybrid monte carlo hmc method is a popular and rigorous method for sampling from a canonical ensemble the hmc method is based on classical molecular dynamics simulations combined with a metropolis acceptance criterion and a momentum resampling step while the hmc method completely resamples the momentum after each monte carlo step the generalized hybrid monte carlo ghmc method can be implemented with a partial momentum refreshment step this property seems desirable for keeping some of the dynamic information throughout the sampling process similar to stochastic langevin and brownian dynamics simulations it is however ultimate to the success of the ghmc method that the rejection rate in the molecular dynamics part is kept at a minimum otherwise an undesirable zitterbewegung in the monte carlo samples is observed in this paper we describe a method to achieve very low rejection rates by using a modified energy which is preserved to high order along molecular dynamics trajectories the modified energy is based on backward error results for symplectic time stepping methods the proposed generalized shadow hybrid monte carlo gshmc method is applicable to nvt as well as npt ensemble simulations

a transformation media based approach for efficient monte carlo analysis of scattering from rough surfaces with objects this paper presents a computational model that utilizes transformation based metamaterials to enhance the performance of numerical modeling methods for achieving the statistical characterization of two dimensional electromagnetic scattering from objects on or above one dimensional rough sea surfaces monte carlo simulation of the rough surface scattering problem by means of differential equation based finite methods such as finite element or finite difference methods usually places a heavy burden on computational resources because at each realization of the monte carlo technique a mesh must be generated anew for each surface realization the main purpose of the proposed approach in this paper is to create a single mesh without repeating mesh generation at each step by introducing a transformation medium above the rough surface in the computational domain of the finite methods material parameters of the medium are obtained by the coordinate transformation technique which is based on the form invariance property of maxwell s equations at each realization only the material parameters are modified with respect to the geometry of surface without changing the mesh in this manner a great reduction in cpu time is achieved the proposed technique is analyzed and validated via various finite element simulations

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

multilevel expansion of the sparse matrix canonical grid method for two dimensional random rough surfaces wave scattering from two dimensional 2 d random rough surfaces up to several thousand square wavelengths has been previously analyzed using the sparse matrix canonical grid smcg method the success of the smcg method highly depends on the roughness of the random surface for a given surface area we present a multilevel expansion algorithm to overcome this limitation the proposed algorithm entails the use of a three dimensional 3 d canonical grid this grid is generated by a uniform discretization of the vertical displacement along the height z axis of the rough surface in addition to the uniform sampling of the rough surface along the x y plane the green s function is expanded about the 3 d canonical grid for the far interactions the trade off in computer memory requirements and cpu time between the neighborhood distance and the number of discretization levels along the x axis are discussed for both perfectly electric conducting pec and lossy dielectric random rough surfaces ocean surfaces of the durden vesecky 1985 spectrum with various bandlimits are also studied