a hybrid discontinuous galerkin method for computing the ground state solution of bose einstein condensates a numerical method for computing the ground state solution of bose einstein condensates modeled by the gross pitaevskii equation is presented in this method the three dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions both polynomial and plane wave bases are considered for this purpose and lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution the ground state is computed by an iterative procedure for minimizing the energy the performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher order finite element method

weighted inductive means abstract in this paper we present a unified framework for weighted inductive means on the cone p of positive definite hermitian matrices as natural multivariable extensions of two variable weighted means particularly of metric midpoint operations on p it includes some well known multivariable weighted matrix means the weighted arithmetic harmonic resolvent sturm s inductive geometric mean on the riemannian manifold p equipped with the trace metric log euclidean and spectral geometric means a recursion or weight additive formula is derived and applied to find a closed form and basic properties for a weighted inductive mean an upper bound on the sensitivity a metric characterization and min and max optimization problems over permutations for the inductive geometric mean are presented moreover we apply the obtained results to a class of midpoint operations of the non positively curved hadamard metrics on p parameterized over hermitian unitary matrices

a family of symplectic integrators stability accuracy and molecular dynamics applications the following integration methods for special second order ordinary differential equations are studied leapfrog implicit midpoint trapezoid stormer verlet and cowell numerov we show that all are members or equivalent to members of a one parameter family of schemes some methods have more than one common form and we discuss a systematic enumeration of these forms we also present a stability and accuracy analysis based on the idea of modified equations and a proof of symplecticness it follows that cowell numerov and lim2 a method proposed by zhang and schlick are symplectic a different interpretation of the values used by these integrators leads to higher accuracy and better energy conservation hence we suggest that the straightforward analysis of energy conservation is misleading

simultaneous finite and infinite zero assignments of linear systems abstract a simultaneous finite and infinite zero assignment problem via sensor selection for linear multivariable systems is proposed by sensor selection we mean an appropriate choice of the output matrix c here by utilizing the well known burnovsky canonical form for a linear system characterized by the matrix pair a b we obtain an explicit construction algorithm that generates a non empty set c of output matrices such that for any member c of this set the corresponding system characterized by the triple a b c has the prescribed finite and infinite zero structures two examples are also given to illustrate our results

dimension reduction of the schrodinger equation with coulomb and anisotropic confining potentials we consider dimension reduction for the three dimensional 3d schrodinger equation with the coulomb interaction and an anisotropic confining potential to lower dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model sam or surface density model sdm in two dimensions 2d and the line adiabatic model lam in one dimension 1d efficient and accurate numerical methods for computing ground states and dynamics of the sam sdm and lam models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower dimensional models they are applied to find numerically convergence and convergence rates for the dimension reduction from 3d to 2d and 3d to 1d in terms of ground state and dynamics which confirm some existing analytical results for the dimension reduction in the literature in particular we explain and demonstrate that the standard schrodinger poisson system in 2

equilibria of runge kutta methods it is known that certain runge kutta methods share the property that in a constant step implementation if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system such methods are calledregular in the present paper we provide a recursive test to check whether given method is regular moreover by examining solution trajectories of linear equations we prove that the order of ans stage regular method may not exceed 2 s 2 2 and that the maximal order of regular runge kutta method with an irreducible stability function is 4

an extended lie trotter formula and its applications abstract in this paper we present a class of lie trotter formulae for hermitian operators including the formulae derived by hiai petz and furuta a lie trotter formula for weighted log euclidean geometric means of several positive definite operators is given in terms of sagae tanabe geometric and spectral geometric means

dissipativity of θ methods for a class of nonlinear neutral delay differential equations abstract this paper is concerned with the dissipativity of theoretical solution and numerical solution of a class of nonlinear neutral delay differential equations at first a sufficient condition for the dissipativity of theoretical solution of the mentioned problem above is given then the dissipativity results are obtained for one leg and linear methods when they are applied to these problems

finite element approximation of the cahn hilliard equation on surfaces abstract in this paper we consider the phase separation on general surfaces by solving the nonlinear cahn hilliard equation using a finite element method a fully discrete approximation scheme is introduced and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time this in turn leads to convergence and optimal error estimates of the discretization scheme numerical examples are also provided to substantiate the theoretical results

strong stability for additive runge kutta methods space discretization of some time dependent partial differential equations gives rise to ordinary differential equations containing additive terms with different stiffness properties in these situ

an explicit one step multischeme sixth order method for systems of special structure abstract structure based partitioning of a system of ordinary differential equations is considered a general form of the explicit multischeme runge kutta type method for such systems is presented order conditions and simplifying conditions are written down an algorithm of derivation of the sixth order method with seven stages and reuse with two free parameters is given it embeds a fourth order error estimator numerical comparison to the dormand prince method with the same computation cost but of lower order is performed

construction of a multirate rodas method for stiff odes multirate time stepping is a numerical technique for efficiently solving large scale ordinary differential equations odes with widely different time scales localized over the components this technique enables one to use large time steps for slowly varying components and small steps for rapidly varying ones multirate methods found in the literature are normally of low order one or two focusing on stiff odes in this paper we discuss the construction of a multirate method based on the fourth order rodas method special attention is paid to the treatment of the refinement interfaces with regard to the choice of the interpolant and the occurrence of order reduction for stiff linear systems containing a stiff source term we propose modifications for the treatment of the source term which overcome order reduction originating from such terms and which we can implement in our multirate method

conjugate gradient method for finding fundamental solitary waves abstract the conjugate gradient method cgm is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices in this paper we modify this method so that it can find fundamental solitary waves of nonlinear hamiltonian equations the main obstacle that such a modified cgm overcomes is that the operator of the equation linearized about a solitary wave is not sign definite instead it has a finite number of eigenvalues on the opposite side of zero than the rest of its spectrum we present versions of the modified cgm that can find solitary waves with prescribed values of either the propagation constant or power we also extend these methods to handle multi component nonlinear wave equations convergence conditions of the proposed methods are given and their practical implications are discussed we demonstrate that our modified cgms converge much faster than say petviashvili s or similar methods especially when the latter converge slowly

analysis and computation for ground state solutions of bose fermi mixtures at zero temperature previous numerical studies on the ground state structure of bose fermi mixtures mostly relied on thomas fermi tf approximation for the fermi gas in this paper we establish the existence and uniqueness of ground state solutions of bose fermi mixtures at zero temperature for both a coupled gross pitaevskii gp equations model and a model with tf approximation for fermions to prove the uniqueness the key is to estimate the l infty bounds of the ground state solution by implementing an efficient method gradient flow with discrete normalization with backward euler finite difference discretization to compute the coupled gp equations we report extensive numerical results in one and two dimensions the numerical experiments show that we can also extract many interesting phenomena without reference to tf approximation for the fermions finally we numerically compare the ground state solutions for the coupled gp equations model and the model with tf approximation for fermions as well as for the

lie trotter means of positive definite operators abstract as extension of the lie trotter product formula we define the two variable and multivariate lie trotter means with several examples including the sagae tanabe and hansen inductive means we show that the weighted means satisfying the arithmetic geometric harmonic mean inequalities are the multivariate lie trotter means we finally give a few interesting versions of the extended lie trotter formula derived from multivariate lie trotter means

strict contractions of symmetric cones let v be a euclidean jordan algebra with an associative inner product hji and let be the corresponding symmetric cone then the linear automorphism group h of the symmetric cone acts transitively on it let t v i be the symmetric tube domain associated to the symmetric cone and let g t be the biholomorphic automorphism group of t there exists a conformal compactification m g t p of v with respect to a parabolic subgroup p a lie semigroup that is naturally related to the symmetric cone occurs as the semigroup of compressionsof in the lie group g t fg 2 g t j g g it turns out that admits an ol shanskii decomposition h expc for an ad h invariant convex cone c in the lie algebra g t of g t a triple decomposition h 8 9 and it determines a globally hyperbolic causal order on the homogeneous space g t h defined by ah bh if and only if a 1 b 2 the family of bilinear forms x u v hp x 1 ujvi defines a hinvariant riemannian metric on where the map p is the quadratic representation of the jordan algebra v in 8 koufany showed that the elements of are contractions of for the riemannian metric the main result of this paper is that the elements of the interior of called strict compressions are strict contractions of for the riemannian distance induced by

comparison of the asymptotic stability properties for two multirate strategies this paper contains a comparison of the asymptotic stability properties for two multirate strategies for each strategy the asymptotic stability regions are presented for a 2x2 test problem and the differences between the results are discussed the considered multirate schemes use rosenbrock type methods as the main time integration method and have one level of temporal local refinement some remarks on the relevance of the results for 2x2 test problems are presented

time scale structure assignment in linear multivariable systems using high gain feedback abstract a systematic design method for assigning a required time scale structure to a given multivariable system is developed the time scale structure assignment is carried out using high gain feedback of either the state or the output the method developed decomposes a given multivariable system into several single input single output subsystems each of which is designed separately a standard method of design for single input single output systems is developed and this forms a building block for the multivariable system design the state feedback design depends only on the stage sequence of the given system while the output feedback design depends only on the set of integers that specifies the order of infinite zeros in this way our design is made robust with respect to certain parameter uncertainties high gain parameter s can be adjusted on line and this provides a tool for the fine tuning of the desired performance characteristics of a control system

the effective stability of adaptive timestepping ode solvers we consider the behavior of certain adaptive timestepping methods based upon embedded explicit runge kutta pairs when applied to dissipative odes it has been observed numerically that the standard local error controls can impart desirable stability properties but this has been rigorously verified only for very special low order runge kutta pairs the rooted tree expansion of a certain quadratic form central to the stability theory of runge kutta methods is derived this together with key assumptions on the sequence of accepted time steps and the local error estimate provides a general explanation for the observed stability of such algorithms on dissipative problems under these assumptions which are expected to hold for typical numerical trajectories two different results are proved first for a large class of embedded runge kutta pairs of order 1 2 controlled on an error per unit step basis all such numerical trajectories will eventually enter a particular bounded set this occurs for sufficiently small tolerances independent of the initial conditions second for pairs of arbitrary orders p 1 p operating under either error per step or error per unit step control similar results are obtained when an additional structural assumption that should be valid for many cases of interest is imposed on the dissipative vector field numerical results support both the analysis and the assumptions made

an energy stable runge kutta method for convex gradient problems abstract we propose a class of runge kutta methods which provide a simple unified framework to solve the gradient flow of a convex functional in an unconditionally energy stable manner stiffly accurate runge kutta methods are high order accurate in terms of time and also assure the energy stability for any time step size when they satisfy the positive definite condition we provide a detailed proof of the unconditional energy stability as well as unique solvability of the proposed scheme we demonstrate the accuracy and stability of the proposed methods using numerical experiments for a specific example

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

multirate generalized additive runge kutta methods this work constructs a new class of multirate schemes based on the recently developed generalized additive runge kutta gark methods sandu and gunther siam j numer anal 53 1 17 42 2015 multirate schemes use different step sizes for different components and for different partitions of the right hand side based on the local activity levels we show that the new multirate gark family includes many well known multirate schemes as special cases the order conditions theory follows directly from the gark accuracy theory nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

properties of runge kutta summation by parts methods abstract we review and extend the list of stability and convergence properties satisfied by runge kutta rk methods that are associated with summation by parts sbp operators herein called rk sbp methods the analysis covers classical generalized as well as upwind sbp operators previous work on the topic has relied predominantly on energy estimates in contrast we derive all results using a purely algebraic approach that rests on the well established theory of rk methods the purpose of this paper is to provide a bottom up overview of stability and convergence results for linear and non linear problems that relate to general rk sbp methods to this end we focus on the rk viewpoint since this perspective so far is largely unexplored this approach allows us to derive all results as simple consequences of the properties of sbp methods combined with well known results from rk theory in this way new proofs of known results such as a l and b stability are given additionally we establish previously unreported results such as strong s stability dissipative stability and stiff accuracy of certain rk sbp methods further it is shown that a subset of methods are b convergent for strictly contractive non linear problems and convergent for non linear problems that are both contractive and dissipative

convex splitting rungekutta methods for phase field models in this paper we present the convex splitting rungekutta csrk methods which provide a simple unified framework to solve phase field models such as the allencahn cahnhilliard and phase field crystal equations the core idea of the csrk methods is the combination of convex splitting methods and multi stage implicitexplicit rungekutta methods our csrk methods are high order accurate in time and we investigate the energy stability numerically we present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third order accuracy

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

dissipativity of linear θ methods for integro differential equations this paper is concerned with the analytic and numerical dissipativity of nonlinear integro differential equations ides a dissipativity criteria for ides is given it is shown that any a stable linear 0 method for the systems is dissipative numerical examples are given to confirm the theoretical results

some geometric interpretations of quantum fidelity abstract we consider quantum fidelity between two states and where we fix and allow to be sent through a quantum channel we determine the minimal fidelity where one minimizes over a all unital channels b all mixed unitary channels and c arbitrary channels we derive results involving the minimal eigenvalue of which we can interpret as a convex combination coefficient as a consequence we give a new geometric interpretation of the minimal fidelity with respect to the closed convex set of density matrices and with respect to the closed convex set of quantum channels we further investigate the geometric nature of fidelity by considering density matrices arising as normalized projections onto subspaces in this way fidelity can be viewed as a geometric measure of distance between two spaces we give a connection between fidelity and the canonical principal angles between the subspaces

unconditionally stable methods for gradient flow using convex splitting runge kutta scheme abstract we propose a convex splitting runge kutta csrk scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner the key feature of the scheme is a combination of a convex splitting method and a specially designed multi stage two additive runge kutta method our methods are high order accurate in time and assure the gradient energy stability for any time step size we provide detailed proof of the unconditional energy stability and present issues on the practical implementations we demonstrate the accuracy and stability of the proposed methods using numerical experiments of the cahn hilliard equation

two level discretization techniques for ground state computations of bose einstein condensates this work presents a new methodology for computing ground states of bose einstein condensates based on finite element discretizations on two different scales of numerical resolution in a preprocessing step a low dimensional coarse generalized finite element space is constructed it is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties the nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low dimensional space without significant loss of accuracy when compared with the solution of the full fine scale problem the preprocessing step is independent of the types and numbers of bosons a postprocessing step further improves the accuracy of the method we present rigorous a priori error estimates that predict convergence rates h 3 for the ground state eigenfunction and h 4 for the corresponding eigenvalue without pre asymptotic effects h being the coarse scale discretization parameter numerical experiments indicate that these high rates may still be pessimistic

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

an analysis of schwarz waveform relaxation domain decomposition methods for the imaginary time linear schrodinger and gross pitaevskii equations the aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of classical and quasi optimal schwarz waveform relaxation swr domain decomposition methods applied to the computation of the stationary states of the one dimensional linear and nonlinear schrodinger equations with a space dependent potential although swr methods are currently used for efficiently solving high dimensional partial differential equations their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension except in simple cases in this work we tackle this problem for linear and nonlinear one dimensional schrodinger equations by developing techniques which can be extended to higher dimensional problems and other types of pdes the approach combines the method developed in gander and halpern siam j num anal 45 2 666 697 2007 for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolic asymptotic expansions for computing the stationary states we consider the imaginary time formulation of the schrodinger equation based on the normalized gradient flow ngf method and use a semi implicit euler scheme for the discretization some numerical results in the one dimensional case illustrate the analysis for both the linear schrodinger and gross pitaevskii equations

uniform ultimate boundedness of numerical solutions to nonlinear neutral delay differential equations this paper is concerned with the long time behaviour of the numerical solutions generated by runge kutta rk methods for nonlinear neutral delay differential equations nddes it is proved that the numerical solutions produced by k l algebraically stable rk methods are uniformly ultimately bounded some examples reveal that some rk methods completely preserve the long time behaviour of the exact solutions to nddes for sufficiently small time stepsize h as a comparison with the previous results a numerical example which further illustrates our theoretical results is provided

a new sobolev gradient method for direct minimization of the gross pitaevskii energy with rotation in this paper we improve traditional steepest descent methods for the direct minimization of the gross pitaevskii gp energy with rotation at two levels we first define a new inner product to equip the sobolev space h 1 and derive the corresponding gradient second for the treatment of the mass conservation constraint we use a projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods the descent method with these two new ingredients is studied theoretically in a hilbert space setting and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the gp energy the new method is implemented in both finite difference and finite element two dimensional settings and is used to compute various complex configurations with vortices of rotating bose einstein condensates the new sobolev gradient method shows better numerical performances compared to classical l 2 or h 1 gradient methods especially when high rotation rates are considered

a new positive definite geometric mean of two positive definite matrices abstract we introduce and study a new positive definite in certain singular cases positive semidefinite geometric mean of two positive definite under certain conditions positive semidefinite matrices

a generalized structure approach to additive runge kutta methods this work considers a general structure of the additively partitioned runge kutta methods by allowing for different stage values as arguments of different components of the right hand side an order conditions theory is developed for the new formulation of generalized additive methods and stability and monotonicity investigations are carried out the paper discusses the construction and properties of implicit explicit and implicit implicit methods in the new framework the new approach named gark introduces additional flexibility when compared to traditional partitioned runge kutta formalism and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales or driven by multiple physical processes

the essential stability of local error control for dynamical systems although most adaptive software for initial value problems is designed with an accuracy requirement control of the local error it is frequently observed that stability is imparted by the adaptation this relationship between local error control and numerical stability is given a firm theoretical underpinning the dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations dissipative contractive and gradient systems dissipative dynamical systems are characterised by having a bounded absorbing set mathcal b which all trajectories eventually enter and remain inside the exponentially contractive problems studied have a unique globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set mathcal b may be chosen to be a ball of arbitrarily small radius around the equilibrium point the gradient systems studied are those for which the set of equilibria comprises isolated points and all traje

factorizations and geometric means of positive definite matrices abstract in this paper we provide a new class of metric geometric means of positive definite matrices varying over hermitian unitary matrices we show that each hermitian unitary matrix induces a factorization of the cone p m of m m positive definite hermitian matrices into geodesically convex subsets and a hadamard metric structure on p m an explicit formula for the corresponding metric midpoint operation is presented in terms of the geometric and spectral geometric means and show that the resulting two variable mean is different to the standard geometric mean some basic properties comparable to those of the geometric mean and its extensions to finite number of positive definite matrices are studied

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

strong majorization for hermitian matrices we introduce and study an additive relation and a multiplicative relation for hermitian matrices the relation g a f means that f is the real part of some matrix similar to g the relation g m f expresses the property that the positive definite matrix f is the modulus of some matrix similar to the positive definite matrix g there is a close connection with the additive and multiplicative majorization relations for the corresponding spectra

decentralization of large scale systems a new canonical form for linear multivariable systems this note presents a new canonical form for linear multivariable systems this canonical form casts and partitions the system to the so called component connection model ccm the application of this canonical form to the problem of decentralized stabilization and estimation is discussed

equilibrium attractive properties of a class of multistep runge kutta methods the main purpose of this paper is to discuss the equilibrium attractive properties of a class of multistep runge kutta methods for initial value problems of ordinary differential equations some algebraic conditions insuring the equilibrium attractivity are given and some methods satisfying these algebraic conditions are constructed some numerical examples confirm our results

euler like discrete models of the logistic differential equation abstract to understand the behavior of difference schemes on nonlinear differential equations it seems desirable to extend the standard linear stability theory into a nonlinear theory as a step in that direction we investigate the stability properties of euler related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other we find that these schemes are superior to explicit schemes when they are stable and the blow up time has not passed for these h values they are dynamically faithful when these schemes turn unstable however they have much less desirable properties than explicit or fully implicit schemes they become simultaneously superstable and unstable this is explained by the fact that these schemes are not self adjoint the linearly implicit self adjoint scheme is dynamically faithful in an euler typical range of step sizes and gives correct stability for all step sizes

multilevel preconditioning technique for schwarz waveform relaxation domain decomposition method for real and imaginary time nonlinear schrodinger equation abstract this paper is dedicated to the derivation of multilevel schwarz waveform relaxation swr domain decomposition methods ddm in real and imaginary time for the nonlinear schrodinger equation nlse in imaginary time it is shown that the multilevel swr ddm accelerates the convergence compared to the one level swr ddm resulting in an important reduction of the computational time and memory storage in real time the method requires in addition the storage of the solution in overlapping zones at any time but on coarser discretization levels the method is numerically validated on the classical swr and robin based swr methods but can however be applied to any swr ddm approach

γ commuting triples of positive definite matrices and midpoint equations on unitary matrices triples of positive definite matrices that can be transformed to a commuting triple by a congruence transformation are called commuting in this paper we give various descriptions and characterizations of commuting triples and particularly focus on those involving the geometric mean and midpoint equations on a generating set of the special unitary group

a fourth order time splitting laguerre hermite pseudospectral method for bose einstein condensates a fourth order time splitting laguerre hermite pseudospectral method is introduced for bose einstein condensates becs in three dimensions with cylindrical symmetry the method is explicit time reversible and time transverse invariant it conserves the position density and is spectral accurate in space and fourth order accurate in time moreover the new method has two other important advantages i it reduces a three dimensional 3 d problem with cylindrical symmetry to an effective two dimensional 2 d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain the method is applied to vector gross pitaevskii equations vgpes for multicomponent becs extensive numerical tests are presented for the one dimensional 1 d gpe the 2 d gpe with radial symmetry the 3 d gpe with cylindrical symmetry as well as 3 d vgpes for two component becs to show the efficiency and accuracy of the new numerical method

dissipativity of runge kutta methods for neutral delay integro differential equations this paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro differential equations the dissipativity results are obtained for algebraically stable runge kutta methods when they are applied to above problems

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

energy stable compact scheme for cahn hilliard equation with periodic boundary condition abstract we present a compact scheme to solve the cahn hilliard equation with a periodic boundary condition which is fourth order accurate in space we introduce schemes for two and three dimensions which are derived from the one dimensional compact stencil the energy stability is completely proven for the proposed scheme based on the application of the compact method and well known convex splitting methods detailed proofs of the mass conservation and unique solvability are also established numerical experiments are presented to demonstrate the accuracy and stability of the proposed methods

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

linearly parameterized bandits we consider bandit problems involving a large possibly infinite collection of arms in which the expected reward of each arm is a linear function of an r dimensional random vector z rr where r 2 the objective is to minimize the cumulative regret and bayes risk when the set of arms corresponds to the unit sphere we prove that the regret and bayes risk is of order r t by establishing a lower bound for an arbitrary policy and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases the phase based policy is also shown to be effective if the set of arms satisfies a strong convexity condition for the case of a general set of arms we describe a near optimal policy whose regret and bayes risk admit upper bounds of the form o r t log3 2t

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

a finite element method with mesh adaptivity for computing vortex states in fast rotating bose einstein condensates numerical computations of stationary states of fast rotating bose einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices in this paper we propose a low order finite element method with mesh adaptivity by metric control as an alternative approach to the commonly used high order finite difference or spectral approximation methods the mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states an imaginary time propagation method and a sobolev gradient descent method we first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state considerable computational time saving is obtained compared to uniform mesh computations the new method is applied to compute difficult cases relevant for physical experiments large nonlinear interaction constant and high rotation rates

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

computing the ground state and dynamics of the nonlinear schrodinger equation with nonlocal interactions via the nonuniform fft we present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear schrodinger equation nlse with nonlocal interactions based on a fast and accurate evaluation of the long range interactions via the nonuniform fast fourier transform nufft we begin with a review of the fast and accurate nufft based method in 29 for nonlocal interactions where the singularity of the fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates we then extend the method to compute other nonlocal interactions whose fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform many of these interactions do not decay at infinity in the physical space which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain the performance of our method against other existing methods is illustrated numerically with particular attention on the effect of the size of the computational domain in the physical space finally to study the ground state and dynamics of the nlse we propose efficient and accurate numerical methods by combining the nufft method for potential evaluation with the normalized gradient flow using backward euler fourier pseudospectral discretization and time splitting fourier pseudospectral method respectively extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the nlse with various nonlocal interactions numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency

multirate runge kutta schemes for advection equations explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena the wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps resulting in a moderate to substantial reduction of the computational cost depending on the scenario to simulate s osher r sanders numerical approximations to nonlinear conservation laws with locally varying time and space grids math comput 41 1983 321 336 h tang g warnecke a class of high resolution schemes for hyperbolic conservation laws and convection diffusion equations with varying time and pace grids siam j sci comput 26 4 2005 1415 1431 e constantinescu a sandu multirate timestepping methods for hyperbolic conservation laws siam j sci comput 33 3 2007 239 278 in air pollution modeling the advection part is usually integrated explicitly in time where the time step is constrained by a locally varying courant friedrichs lewy cfl number multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction therefore it is of major interest to apply multirate schemes to the advection equation we introduce a generic recursive multirate runge kutta scheme that can be easily adapted to an arbitrary number of refinement levels it preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit runge kutta methods as base method

trust region algorithms and timestep selection unconstrained optimization problems are closely related to systems of ordinary differential equations odes with gradient structure in this work we prove results that apply to both areas we analyze the convergence properties of a trust region or levenberg marquardt algorithm for optimization the algorithm may also be regarded as a linearized implicit euler method with adaptive timestep for gradient odes from the optimization viewpoint the algorithm is driven directly by the levenberg marquardt parameter rather than the trust region radius this approach is discussed for example in r fletcher practical methods of optimization 2nd ed john wiley new york 1987 but no convergence theory is developed we give a rigorous error analysis for the algorithm establishing global convergence and an unusual extremely rapid type of superlinear convergence the precise form of superlinear convergence is exhibited the ratio of successive displacements from the limit point is bounded above and below by geometrically decreasing sequences we also show how an inexpensive change to the algorithm leads to quadratic convergence from the ode viewpoint this work contributes to the theory of gradient stability by presenting an algorithm that reproduces the correct global dynamics and gives very rapid local convergence to a stable steady state

an adaptive multigrid scheme for bose einstein condensates in a periodic potential we present a novel multigrid continuation method for treating parameter dependent problems the proposed algorithm which can be flexibly implemented is a generalization of the two grid discretization schemes c s chien b w jeng a two grid discretization scheme for semilinear elliptic eigenvalue problems siam j sci comput 27 2006 1287 1304 that is approximating points on a solution curve do not necessarily lie on the same fine grid we apply the algorithm to compute energy levels and superfluid densities of bose einstein condensates bec in a periodic potential both positive and negative scattering lengths are considered in our numerical experiments for positive scattering length if the chemical potential is large enough and the domain is properly chosen the results show that the number of peaks of the first few energy states of the 2d bec in a periodic potential depends on the wave number of the periodic potential moreover for bright solitons the number of peaks of the ground state solutions is 1d 1 2 and 1d 2 where the periodic potential is expressed in terms of the sine or the cosine functions respectively however these formulae do not hold if the scattering length is negative the numerical study is extended to the two component 1d and 2d bec in a periodic potential

structure preservation for constrained dynamics with super partitioned additive runge kutta methods a broad class of partitioned differential equations with possible algebraic constraints is considered including hamiltonian and mechanical systems with holonomic constraints for mechanical systems a formulation eliminating the coriolis forces and closely related to the euler lagrange equations is presented a new class of integrators is defined the super partitioned additive runge kutta spark methods this class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms a linear stability and convergence analysis of these methods is given spark methods allowing the direct preservation of certain properties are characterized different structures and invariants are considered the manifold of constraints symplecticness reversibility contractivity dilatation energy momentum and quadratic invariants with respect to linear stability and structure preservation the class of s stage lobatto iiia b c c spark methods is of special interest controllable numerical damping can be introduced by the use of additional parameters some issues related to the implementation of a reversible variable stepsize strategy are discussed

dissipativity of θ methods for nonlinear volterra delay integro differential equations this paper deals with the analytic and numerical dissipativity of nonlinear volterra delay integro differential equations vdides a dissipativity criteria for vdides is given we investigate the dissipativity properties of one leg q methods and linear q methods for the underlying systems it is shown that for q 12 1 any one leg q method and linear q method can inherit the dissipativity of the underlying systems

a continuation bsor lanczos galerkin method for positive bound states of a multi component bose einstein condensate we develop a continuation block successive over relaxation bsor lanczos galerkin method for the computation of positive bound states of time independent coupled gross pitaevskii equations cgpes which describe a multi component bose einstein condensate bec a discretization of the cgpes leads to a nonlinear algebraic eigenvalue problem naep the solution curve with respect to some parameter of the naep is then followed by the proposed method for a single component bec we prove that there exists a unique global minimizer the ground state which is represented by an ordinary differential equation with the initial value for a multi component bec we prove that m identical ground bound states will bifurcate into m different ground bound states at a finite repulsive inter component scattering length numerical results show that various positive bound states of a two three component bec are solved efficiently and reliably by the continuation bsor lanczos galerkin method

congruence of hermitian matrices by hermitian matrices two hermitian matrices a b mn c are said to be hermitian congruent if there exists a nonsingular hermitian matrix c mn c such that b cac in this paper we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable hermitian matrices a and b to be hermitian congruent moreover when a and b are hermitian congruent we describe the possible inertias of the hermitian matrices c that carry the congruence we also give necessary and sufficient conditions for any 2 by 2 nonsingular hermitian matrices to be hermitian congruent in both of the studied cases we show that if a and b are real and hermitian congruent then they are congruent by a real symmetric matrix finally we note that if a and b are 2 by 2 nonsingular real symmetric matrices having the same sign pattern then there is always a real symmetric matrix c satisfying b cac moreover if both matrices are positive then c can be picked with arbitrary inertia

the inverse mean problem of geometric mean and contraharmonic means abstract in this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices proposed in w n anderson m e mays t d morley g e trapp the contraharmonic mean of hsd matrices siam j algebra disc meth 8 1987 674 682 a x y 2 x 1 y 1 1 b x y by proving its equivalence to the well known nonlinear matrix equation x t bx 1b where t 1 2 a a a 8 ba 1 b is the unique positive definite solution of x a 2bx 1b the inverse mean problem is solvable if and only if b a