orientational ordering in two dimensional systems with long range interaction a variational method for calculating gaussian fluctuations of two dimensional 2d moment orientations in lattice systems with arbitrary anisotropic interaction is proposed it is asymptotically exact for very low temperatures and gives a good extrapolation for a certain low temperature range general relationships are obtained that enable one to represent from a unified standpoint and in some cases to refine the known results on 1 the existence of short range order in a degenerate 2d system of moments with isotropic short range interaction 2 long range order stabilization by weak local potentials and dipole interactions 3 the existence of long range order in a 2d system with interaction of the type v r r if 2 4 for systems with anisotropic long range dipole interaction of moments on arbitrary 2d lattices except square ones for which a short range phase takes place the existence of long range order is proved lower limits for temperatures of transitions into phases with long range order are evaluated the low values of the transition temperatures in the dipole systems can be explained due to the small values of interchain interaction in comparison with interchain ones the applicability area for the method used is discussed

information loss in coarse graining of stochastic particle dynamics recently a new class of approximating coarse grained stochastic processes and associated monte carlo algorithms were derived directly from microscopic stochastic lattice models for the adsorption desorption and diffusion of interacting particles 12 13 15 the resulting hierarchy of stochastic processes is ordered by the level of coarsening in the space time dimensions and describes mesoscopic scales while retaining a significant amount of microscopic detail on intermolecular forces and particle fluctuations here we rigorously compute in terms of specific relative entropy the information loss between non equilibrium exact and approximating coarse grained adsorption desorption lattice dynamics our result is an error estimate analogous to rigorous error estimates for finite element finite difference approximations of partial differential equations we prove this error to be small as long as the level of coarsening is small compared to the range of interaction of the microscopic model this result gives a first mathematical reasoning for the parameter regimes for which approximating coarse grained monte carlo algorithms are expected to give errors within a given tolerance

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

faber approximation of the mori zwanzig equation abstract we develop new approximations of the mori zwanzig equation based on operator series expansions of the orthogonal dynamics propagator in particular we study the faber polynomial series which yields asymptotically optimal approximations converging at least r superlinearly with the polynomial order numerical applications are presented for random wave propagation and harmonic chains of oscillators interacting on the bethe lattice and on graphs with arbitrary topology most of the analysis and numerical results presented in this paper are for linear dynamical systems evolving from random initial states the extension to nonlinear systems is discussed as well the challenges in the implementation of the proposed approximation methods

superconvergence of spectral collocation and p version methods in one dimensional problems superconvergence phenomenon of the legendre spectral collocation method and the p version finite element method is discussed under the one dimensional setting for a class of functions that satisfy a regularity condition m u k l cm k on a bounded domain it is demonstrated both theoretically and numerically that the optimal convergent rate is supergeometric furthermore at proper gaussian points or lobatto points the rate of convergence may gain one or two orders of the polynomial degree

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

an approximate orthogonal decomposition method for the solution of generalized liouville equations abstract an analytical numerical integration method for the generalized liouville equation t f q p t l f q p t is proposed and analyzed taking into account a cauchy condition f q p t t 0 f 0 q p for the phase space distribution function we constructed the problem solution as series expansion in time variable t using orthogonal polynomials and hermite function also we proved the corresponding convergence theorems under certain boundedness conditions upon a liouville operator

a nonequilibrium statistical model of spectrally truncated burgers hopf dynamics exact spectral truncations of the unforced inviscid burgers hopf equation are hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior for this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies in the present paper the burgers hopf dynamics truncated to n fourier modes is treated by a new statistical model reduction technique and a closed system of evolution equations for the mean values of the m lowest modes is derived for m n in the reduced model the m mode macrostates are associated with trial probability densities on the phase space of the n mode microstates and a cost functional is introduced to quantify the lack of fit of paths of these densities to the liouville equation the best fit macrodynamics is obtained by minimizing the cost functional over paths and the equations governing the closure are then derived from hamilton jacobi theory the resulting reduced equations have a fractional diffusion and modified nonlinear interactions and the explicit form of both are determined up to a single closure parameter the accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles and the predicted behavior is found to be well represented by the reduced equations 2014 wiley periodicals inc

convergence of a p version hp version method for fractional differential equations recently m zayernouri and g e karniadakis 2014 10 proposed a new spectral method for fractional differential equations and observed an exponential rate of convergence in this paper we will prove a convergence rate of their spectral method and thus provide an explanation for their observations

equation free modelling of evolving diseases coarse grained computations with individual based models we demonstrate how direct simulation of stochastic individualbased models can be combined with continuum numericalanalysis techniques to study the dynamics of evolving diseases sidestepping the necessity of obtaining explicit populationlevel models the approach analyses the unavailable in closed form coarse macroscopic equations estimating the necessary quantities through appropriately initialized short bursts of individualbased dynamic simulation we illustrate this approach by analysing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts building up from classical susceptibleinfectedrecovered and susceptibleinfectedrecoveredsusceptible models our example uses a onedimensional lattice for variant space and assumes a finite number of individuals macroscopic computational tasks enabled through this approach include stationarystate computation coarse projective integration parametric continuation and stability analysis

fitting sde models to nonlinear kac zwanzig heat bath models we study a class of particle in a heat bath models which are a generalization of the well known kac zwanzig class of models but where the coupling between the distinguished particle and the n heat bath particles is through nonlinear springs the heat bath particles have random initial data drawn from an equilibrium gibbs density the primary objective is to approximate the forces exerted by the heat bath which we do not want to resolve by a stochastic process by means of the central limit theorem for gaussian processes and heuristics based on linear response theory we demonstrate conditions under which it is natural to expect that the trajectories of the distinguished particle can be weakly approximated as n by the solution of a markovian sde the quality of this approximation is verified by numerical calculations with parameters chosen according to the linear response theory alternatively the parameters of the effective equation can be chosen using time series analysis this is done and agreement with linear response theory is shown to be good

applications of lobatto polynomials to an adaptive finite element method a posteriori error estimates for hp adaptivity and grid to grid interpolation hp adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids lobatto polynomial based techniques are presented for both an interpolation error based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution the estimates involve high order derivatives of the solution that must be approximated from the computed solution differentiating a taylor like series for error in the lobatto interpolant and using the weak form of the equations yields the correct derivative approximations this leads to a more robust order selection strategy interpolation between grids is done over each element using the lobatto interpolating polynomial explicit formulas for the inverse of the resulting lobatto interpolation matrices are given computational results illustrate the theory

models for stochastic climate prediction there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory this analysis demonstrates the potential need for more sophisticated models beyond stable linear langevin equations the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances the strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta plane with topography and a mean flow in this example the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom

fractal model for coarse grained nonlinear partial differential equations spatially coarse grained or effective versions of nonlinear partial differential equations must be closed with a model for the unresolved small scales for systems that are known to display fractal scaling we propose a model based on synthetically generating a scale invariant field at small scales using fractal interpolation and then analytically evaluating its effects on the large resolved scales the procedure is illustrated for the forced burgers equation solved numerically on a coarse grid detailed comparisons with direct simulation of the full burgers equation and with an effective viscosity model are presented s0031 9007 96 02215 6

elimination of fast chaotic degrees of freedom on the accuracy of the born approximation we apply standard projection operator techniques known from nonequilibrium statistical mechanics to eliminate fast chaotic degrees of freedom in a low dimensional dynamical system through the usual perturbative approach we end up in second order with a stochastic system where the fast chaotic degrees of freedom are modelled by gaussian white noise the accuracy of the perturbation expansion is analysed in detail by the discussion of an exactly solvable model

a comparative study of two stochastic mode reduction methods abstract we present a comparative study of two methods for the reduction of the dimensionality of a system of ordinary differential equations that exhibits time scale separation both methods lead to a reduced system of stochastic differential equations the novel feature of these methods is that they allow the use in the reduced system of higher order terms in the resolved variables the first method proposed by majda timofeyev and vanden eijnden is based on an asymptotic strategy developed by kurtz the second method is a short memory approximation of the mori zwanzig projection formalism of irreversible statistical mechanics as proposed by chorin hald and kupferman we present conditions under which the reduced models arising from the two methods should have similar predictive ability we apply the two methods to test cases that satisfy these conditions the form of the reduced models and the numerical simulations show that the two methods have similar predictive ability as expected

optimal prediction of underresolved dynamics abstract a method is presented for computing the average solution of problems that are too complicated for adequate resolution but where information about the statistics of the solution is available the method involves computing average derivatives by interpolation based on linear regression and an updating of a measure constrained by the available crude information examples are given

identification of almost invariant aggregates in reversible nearly uncoupled markov chains the topic of the present paper has been motivated by a recent computational approach to identify metastable chemical conformations and patterns of conformational changes within molecular systems after proper discretization such conformations show up as almost invariant aggregates in reversible nearly uncoupled markov chains numcs most of the former work on this subject treated the direct problem given the aggregates analyze the loose coupling in connection with the computation of the stationary distribution aggregation disaggregation techniques in contrast to that the present paper focuses on the inverse problem given the system as a whole identify the almost invariant aggregates together with the small probabilities of transitions between them a robust algorithm is worked out on the basis of some detailed perturbation analysis and illustrated at a simple molecular system

analysis of multiscale methods for stochastic differential equations we analyze a class of numerical schemes proposed in 26 for stochastic differential equations with multiple time scales both advective and diffusive time scales are considered weak as well as strong convergence theorems are proven most of our results are optimal they in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method c 2005 wiley periodicals inc

coarse projective kmc integration forward reverse initial and boundary value problems in equation free multiscale computation a dynamic model is given at a fine microscopic level yet we believe that its coarse grained macroscopic dynamics can be described by closed equations involving only coarse variables these variables are typically various low order moments of the distributions evolved through the microscopic model we consider the problem of integrating these unavailable equations by acting directly on kinetic monte carlo microscopic simulators thus circumventing their derivation in closed form in particular we use projective multi step integration to solve the coarse initial value problem forward in time as well as backward in time under certain conditions macroscopic trajectories are thus traced back to unstable source type and even sometimes saddle like stationary points even though the microscopic simulator only evolves forward in time we also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of coarse limit cycles of the macroscopic behavior and the approximation of their stability through estimates of the leading coarse floquet multipliers

stochastic modelling replacing fast degrees of freedom by noise in systems with timescale separation where the fast degrees of freedom exhibit chaotic motion the latter are replaced by suitable stochastic processes a projection technique is employed to derive equations of motion for the phase space density of the slow variables by eliminating the fast ones the resulting equations can be approximated in a controlled way by fokker planck equations or equivalently by stochastic differential equations for slow degrees of freedom we discuss some model situations and explore the accuracy of the approximations by numerical simulations

a mathematical framework for stochastic climate models there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction here a systematic mathematical strategy for stochastic climate modeling is developed and some of the new phenomena in the resulting equations for the climate variables alone are explored the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances all of these phenomena are derived from a systematic self consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit the theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated the feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography explicit concrete examples with the new phenomena are presented for the stochastically forced three mode interaction equations the conjecture of smith and waleffe phys fluids 11 1999 1608 1622 for stochastically forced three wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self consistency of the mathematical approach with the predictions of equilibrium statistical mechanics in particular for these examples the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables c 2001 john wiley sons inc

on the estimation of the mori zwanzig memory integral we develop a thorough mathematical analysis to deduce conditions for the accuracy and convergence of different approximations of the memory integral in the mori zwanzig mz equation in particular we derive error bounds and sufficient convergence conditions for short memory approximations the t model and hierarchical finite memory approximations in addition we derive useful upper bounds for the mz memory integral which allow us to estimate a priori the contribution of the mz memory to the dynamics such upper bounds are easily computable for systems with finite rank projections numerical examples are presented and discussed for linear and nonlinear dynamical systems evolving from random initial states we develop a thorough mathematical analysis to deduce conditions for the accuracy and convergence of different approximations of the memory integral in the mori zwanzig mz equation in particular we derive error bounds and sufficient convergence conditions for short memory approximations the t model and hierarchical finite memory approximations in addition we derive useful upper bounds for the mz memory integral which allow us to estimate a priori the contribution of the mz memory to the dynamics such upper bounds are easily computable for systems with finite rank projections numerical examples are presented and discussed for linear and nonlinear dynamical systems evolving from random initial states

unresolved computation and optimal predictions we present methods for predicting the solution of time dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically but when prior statistical information can be found the sparse numerical data are viewed as constraints on the solution and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future for linear equations we offer general recipes for advancing the constraints the methods are generalized to certain classes of nonlinear problems and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed our methods are related to certain data acquisition schemes in oceanography and meteorology c 1999 john wiley sons inc

convergence of optimal prediction for nonlinear hamiltonian systems optimal prediction is a computational method for systems that cannot be properly resolved in which the unresolved variables are viewed as random this paper presents a first analysis of optimal prediction as a numerical method we prove the convergence of the scheme for a class of equations of schrodinger type and derive error bounds for the mean error between the optimal prediction solution and the set of exact solutions with random initial data it is shown that optimal prediction is the scheme that minimizes the mean truncation error

stochastic homogenization for an energy conserving multi scale toy model of the atmosphere we study a hamiltonian toy model for a lagrangian fluid parcel in the semi geostrophic limit which exhibits slow and fast dynamics we first reinject unresolved fast dynamics into the deterministic equation through a stochastic parametrization that respects the conservation of the energy of the deterministic system in a second step we use stochastic singular perturbation theory to derive an effective reduced stochastic differential equation for the slow dynamics we verify the results in numerical simulations

data driven non markovian closure models abstract this paper has two interrelated foci i obtaining stable and efficient data driven closure models by using a multivariate time series of partial observations from a large dimensional system and ii comparing these closure models with the optimal closures predicted by the mori zwanzig mz formalism of statistical physics multilayer stochastic models msms are introduced as both a generalization and a time continuous limit of existing multilevel regression based approaches to closure in a data driven setting these approaches include empirical model reduction emr as well as more recent multi layer modeling it is shown that the multilayer structure of msms can provide a natural markov approximation to the generalized langevin equation gle of the mz formalism a simple correlation based stopping criterion for an emr msm model is derived to assess how well it approximates the gle solution sufficient conditions are derived on the structure of the nonlinear cross interactions between the constitutive layers of a given msm to guarantee the existence of a global random attractor this existence ensures that no blow up can occur for a broad class of msm applications a class that includes non polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants the emr msm methodology is first applied to a conceptual nonlinear stochastic climate model of coupled slow and fast variables in which only slow variables are observed it is shown that the resulting closure model with energy conserving nonlinearities efficiently captures the main statistical features of the slow variables even when there is no formal scale separation and the fast variables are quite energetic second an msm is shown to successfully reproduce the statistics of a partially observed generalized lotka volterra model of population dynamics in its chaotic regime the challenges here include the rarity of strange attractors in the model s parameter space and the existence of multiple attractor basins with fractal boundaries the positivity constraint on the solutions components replaces here the quadratic energy preserving constraint of fluid flow problems and it successfully prevents blow up

superconvergence of the h p version of the finite element method in one dimension in this paper we investigate the superconvergence properties of the h p version of the finite element method fem for two point boundary value problems a postprocessing technique for the h p finite element approximation is analyzed the analysis shows that the postprocess improves the order of convergence furthermore we obtain asymptotically exact a posteriori error estimators based on the postprocessing results numerical examples are included to illustrate the theoretical analysis

optimal prediction for hamiltonian partial differential equations optimal prediction methods compensate for a lack of resolution in the numerical solution of time dependent differential equations through the use of prior statistical information we present a new derivation of the basic methodology show that field theoretical perturbation theory provides a useful device for dealing with quasi linear problems and provide a nonlinear example that illuminates the difference between a pseudo spectral method and an optimal prediction method with fourier kernels along the way we explain the differences and similarities between optimal prediction the representer method in data assimilation and duality methods for finding weak solutions we also discuss the conditions under which a simple implementation of the optimal prediction method can be expected to perform well

from the perron frobenius equation to the fokker planck equation we show that for certain classes of deterministic dynamical systems the perron frobenius equation reduces to the fokker planck equation in an appropriate scaling limit by perturbative expansion in a small time scale parameter we also derive the equations that are obeyed by the first and second order correction terms to the fokker planck limit case in general these equations describe non gaussian corrections to a langevin dynamics due to an underlying deterministic chaotic dynamics for double symmetric maps the first order correction term turns out to satisfy a kind of inhomogeneous fokker planck equation with a source term for a special example we are able solve the first and second order equations explicitly

asymptotic and numerical analyses for mechanical models of heat baths a mechanical model of a particle immersed in a heat bath is studied in which a distinguished particle interacts via linear springs with a collection of n particles with variable masses and random initial conditions the jth particle oscillates with frequency jp where p is a parameter for p 1 2 the sequence of random processes that describe the trajectory of the distinguished particle tends almost surely as n to the solution of an integro differential equation with a random driving term the mean convergence rate is 1 np 1 2 we further investigate whether the motion of the distinguished particle can be well approximated by an integration scheme the symplectic euler scheme when the product of time step h and highest frequency np is of order 1 that is when high frequencies are underresolved for 1 2 p 1 the numerical solution is found to converge to the exact solution at a reduced rate of log h h2 1 p these results shed light on existing numerical data

the rate of convergence of fourier coefficients for entire functions of infinite order with application to the weideman cloot sinh mapping for pseudospectral computations on an infinite interval we analytically compute the asymptotic fourier coefficients for several classes of functions to answer two questions the numerical question is to explain the success of the weideman cloot algorithm for solving differential equations on an infinite interval their method combines fourier expansion with a change of coordinate using the hyperbolic sine function the sinh mapping transforms a simple function like exp z2 into an entire function of infinite order this raises the second analytical question what is the fourier rate of convergence for entire functions of an infinite order the answer is sometimes even slower than a geometric series in this case the fourier series converge only on the real axis even when the function u z being expanded is free of singularities except at infinity earlier analysis ignored stationary point contributions to the asymptotic fourier coefficients when u z had singularities off the real z axis but we show that sometimes these stationary point terms are more important than residues at the poles of u z

on exponential convergence of gegenbauer interpolation and spectral differentiation this paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the gegenbauer gauss and gegenbauer gauss lobatto points when the underlying function is analytic on and within an ellipse sharp error estimates in the maximum norm are derived

nonlinear manifold learning for model reduction in finite elastodynamics abstract model reduction in computational mechanics is generally addressed with linear dimensionality reduction methods such as principal components analysis pca hypothesizing that in many applications of interest the essential dynamics evolve on a nonlinear manifold we explore here reduced order modeling based on nonlinear dimensionality reduction methods such methods are gaining popularity in diverse fields of science and technology such as machine perception or molecular simulation we consider finite deformation elastodynamics as a model problem and identify the manifold where the dynamics essentially take place the slow manifold by nonlinear dimensionality reduction methods applied to a database of snapshots contrary to linear dimensionality reduction the smooth parametrization of the slow manifold needs special techniques and we use local maximum entropy approximants we then formulate the lagrangian mechanics on these data based generalized coordinates and develop variational time integrators our proof of concept example shows that a few nonlinear collective variables provide similar accuracy to tens of pca modes suggesting that the proposed method may be very attractive in control or optimization applications furthermore the reduced number of variables brings insight into the mechanics of the system under scrutiny our simulations also highlight the need of modeling the net effect of the disregarded degrees of freedom on the reduced dynamics at long times

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

variational boundary conditions for molecular dynamics simulations treatment of the loading condition this paper aims to extend the variational boundary conditions for molecular dynamics simulation x li w e variational boundary conditions for molecular dynamics simulations of solids at low temperature commun comp phys 1 2006 136 176 x li w e boundary conditions for molecular dynamics simulations at finite temperature treatment of the heat bath phys rev b 76 2007 104107 to take into account external loading conditions two derivations of the exact boundary conditions are presented one with mori zwanzig projection procedure and the other using lattice green s functions approximate boundary conditions which are more efficient in practice are then discussed finally several numerical experiments are presented to demonstrate the effectiveness of these methods

generalized langevin equations for systems with local interactions we present a new method to approximate the mori zwanzig mz memory integral in generalized langevin equations describing the evolution of smooth observables in high dimensional nonlinear systems with local interactions building upon the faber operator series we recently developed for the orthogonal dynamics propagator and an exact combinatorial algorithm that allows us to compute memory kernels from first principles we demonstrate that the proposed method is effective in computing auto correlation functions intermediate scattering functions and other important statistical properties of the observable we also develop a new stochastic process representation of the mz fluctuation term for systems in statistical equilibrium numerical applications are presented for the fermi pasta ulam model and for random wave propagation in homegeneous media

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

superconvergence points of polynomial spectral interpolation in this work we study superconvergence properties for some high order orthogonal polynomial interpolations at the superconvergent points the error is smaller by a factor of n alpha with alpha 0 where n is the polynomial degree the results are two fold when interpolating function values we identify those points where the first and second derivatives of the interpolant converge faster when interpolating the first derivative we locate those points where the function value of the interpolant superconverges for the earlier case we use various chebyshev polynomials and for the later case we also include the counterpart legendre polynomials

a priori tests of a stochastic mode reduction strategy several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors proc natl acad sci 96 1999 14687 commun pure appl math 54 2001 891 are developed here in this procedure reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables while the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big typically the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1 2 the examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated burgers hopf equations as a nonlinear heat bath systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables in an important illustrative example presented here the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises even for large value of the correlation time ratio of the order of 1 2 the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non gaussian statistics of the original nonlinear systems with 102 modes extremely well furthermore it is shown here that the standard regression fitting of the second order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example 2002 elsevier science b v all rights reserved

error analysis of coarse graining for stochastic lattice dynamics the coarse grained monte carlo cgmc algorithm was originally proposed in the series of works m a katsoulakis a j majda and d g vlachos j comput phys 186 2003 pp 250 278 m a katsoulakis a j majda and d g vlachos proc natl acad sci usa 100 2003 pp 782 787 m a katsoulakis and d g vlachos j chem phys 119 2003 pp 9412 9427 in this paper we further investigate the approximation properties of the coarse graining procedure and provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long time simulations we demonstrate that the numerical accuracy of the cgmc algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse graining ratio and that the natural small parameter is the coarse graining ratio over the range of particle particle interactions the error estimate is shown to hold in the weak convergence sense we employ the derived analytical results to guide cgmc algorithms and demonstrate a cpu speed up in demanding computational regimes that involve nucleation phase transitions and metastability

fast chaos versus white noise entropy analysis and a fokker planck model for the slow dynamics abstract we show in detail how methods of time series analysis such as dimension and entropy estimates support the idea that fast low dimensional chaos can be modelled properly by noise motivated by this observation we derive a formalism by which fast chaotic degrees of freedom in systems with time scale separation can be replaced by a suitable stochastic process a fokker planck equation for the phase space density of the slow degrees of freedom is derived from the full deterministic system by a projection operator technique together with a perturbation expansion we compare two different projection strategies and illustrate the resulting equations by specifying them for a simple model system

dynamical systems of langevin type abstract dynamical systems of langevin type are deterministic mappings that arise if the gaussian white noise of a langevin equation is replaced by a deterministic chaotic dynamics we describe the various types of mappings that can be obtained in this way and investigate typical transition scenarios from complicated non gaussian to langevin like behavior that occur if a time scale parameter is changed we define a new characteristic quantify fir ergodic mappings called effective langevin radius which is related to the critical time scale where the marginal invariant density of the dynamical system of langevin type loses its differentiability the invariant density is shown to contain useful information on the entire characteristic functional of the chaotic driving force some general symmetry properties of the invariant densities are discussed

ground state of an infinite two dimensional system of dipoles on a lattice with arbitrary rhombicity angle abstract a class of possible periodic orientational configurations of a system of dipole moments located at the sites of an infinite flat rhombic lattice with an arbitrary rhombicity angle is studied by using the luttinger and tisza method in the framework of the method it is obtained that for 80 the ground state is ferromagnetic at 60 being continuously degenerate in direction for 80 90 the ground state is antiferromagnetic at 90 being also continuously degenerate with respect to one parameter it is shown that the restriction of the interaction range leads to a change in the type of the ground state at 60 this takes place between the third and the fourth coordination spheres and at 30 at a distance of about 500 lattice constants from comparison with known results of numerical experiments on finite systems it is established that the type of the ground state of a finite and an infinite system may be essentially different

sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions this paper provides a rigorous and delicate analysis for exponential decay of jacobi polynomial expansions of analytic functions associated with the bernstein ellipse using an argument that can recover the best estimate for the chebyshev expansion we derive various new and sharp bounds of the expansion coefficients which are featured with explicit dependence of all related parameters and valid for degree n ge 1 we demonstrate the sharpness of the estimates by comparing with existing ones in particular the very recent results in siam j numer anal 50 2012 pp 1240 1263 we also extend this argument to estimate the gegenbauer gauss quadrature remainder of analytic functions which leads to some new tight bounds for quadrature errors

a posteriori error estimation for hp adaptivity for fourth order equations a posteriori error estimates developed to drive hp adaptivity for second order reaction diffusion equations are extended to fourth order equations a c 1 hierarchical finite element basis is constructed from hermite lobatto polynomials a priori estimates of the error in several norms for both the interpolant and finite element solution are derived in the latter case this requires a generalization of the well known aubin nitsche technique to time dependent fourth order equations we show that the finite element solution and corresponding hermite lobatto interpolant are asymptotically equivalent a posteriori error estimators based on this equivalence for solutions at two orders are presented both are shown to be asymptotically exact on grids of uniform order these estimators can be used to control various adaptive strategies computational results for linear steady state and time dependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings

classical dipoles on a finite triangular lattice the spherical model approximation abstract in the framework of the spherical model it is shown that a phase transition at finite temperature to a ferromagnetic phase in a two dimensional system with dipole dipole and exchange interactions on a lattice with arbitrary rhombicity angle exists if the ferromagnetic ground state is not continuously degenerate in spin direction when the degeneracy exists the system exhibits essential singularities in the thermodynamic functions at t 0 finite size effects in such systems are investigated in the example of the triangular latice under periodic boundary conditions the existence of only one relevant variable determining the crossover from infinite to finite size behavior is found as t 0 is approached despite that there are two characteristic lengths determining the scale of variation of the correlation function the validity of the general finite size scalling relations in terms of this variable is established

coarse graining molecular dynamics models using an extended galerkin projection method summary we present a new framework for coarse graining molecular dynamics models for crystalline solids the reduction method is based on a galerkin projection to a subspace whose dimension is much smaller than that of the full atomistic model to effectively reduce artificial reflections of phonons at the interface we construct extended subspaces with increasing accuracy by adding more coarse grained variables near the interface between lattice defects and the surrounding region this approach is equivalent to the generalized langevin model but it eliminates the need to precompute the memory function a well known practical difficulty further the variational formulation preserves the stability of the molecular models copyright 2014 john wiley sons ltd