extracting macroscopic stochastic dynamics model problems the purpose of this work is to shed light on an algorithm designed to extract effective macroscopic models from detailed microscopic simulations the particular algorithm we study is a recently developed transfer operator approach due to schutte et al 20 the investigations involve the formulation and subsequent numerical study of a class of model problems the model problems are ordinary differential equations constructed to have the property that when projected onto a low dimensional subspace the dynamics is approximately that of a stochastic differential equation exhibiting a finite state space markov chain structure the numerical studies show that the transfer operator approach can accurately extract finite state markov chain behavior embedded within high dimensional ordinary differential equations in so doing the studies lend considerable weight to existing applications of the algorithm to the complex systems arising in applications such as molecular dynamics the algorithm is predicated on the assumption of markovian input data further numerical studies probe the role of memory effects although preliminary these studies of memory indicate interesting avenues for further development of the transfer operator methodology

structure and arithmetic complexity of products and inverses of toeplitz matrices we derive a formula for the product of two toeplitz matrices that is similar to the trench formula for the inverse of a toeplitz matrix we then derive upper and lower bounds for number of multiplications required to compute the inverse or the product of toeplitz matrices and consider several special cases e g symmetry as well the lower bounds for the general cases are in agreement with earlier results but the specialized lower bounds and all the upper bounds are new both upper and lower bounds are o n2 and differ only in lower order terms

the extended bethe ansatz for infinite s 1 2 quantum spin chains with non nearest neighbor interaction it is shown that the description of the states of infinites 1 2 interacting spin systems with the hamiltonian open image in new window can be performed by studying the hyperbolic calogero sutherland eigenvalue problem the construction of multimagnon wave functions in eachn magnon sector is based on solutions of the set of linear algebraic equations which determine also the structure of zonal spherical functions on symmetric spacesxn sl n h sp n of negative curvature the usual bethe ansatz for the xxx heisenberg model corresponds to asymptotic forms of these wave functions at small values ofa 1 or large distances between spins turned over the ferromagnetic ground state

stochastic mode reduction for the immersed boundary method we apply the formulation of a stochastic mode reduction method developed in a recent paper of majda timofeyev and vanden eijnden comm pure appl math 54 2001 pp 891 974 mtv to obtain simplified equations for the dynamics of structures immersed in a thermally fluctuating fluid at low reynolds or kubo number as simulated by a recent extension of the immersed boundary ib method by kramer and peskin proceedings of the second mit conference on computational fluid and solid mechanics elsevier science oxford uk 2003 pp 1755 1758 the effective dynamics of the immersed structures are not obvious in the primitive equations which involve both fluid and structure dynamics but the procedure of mtv allows the rigorous derivation of a reduced stochastic system for the immersed structures alone we find in the limit of small reynolds or kubo number that the lagrangian particle constituents of the immersed structures undergo a drift diffusive motion with several physically correct featur

on the connection between the one dimensional s 1 2 heisenberg chain and haldane shastry model extra integrals of motion and the lax representation are found for interacting spin systems with the hamiltonianh j 2 j k 1 j k n p j k j k where one of the periods of the weierstrassp function is equal ton the heisenberg and haldane shastry chains appear as limiting cases of these systems at some values of the second period the simplest eigenvectors and eigenvalues ofh corresponding to the scattering of two spin waves are presented explicitly for these finite dimensional systems and for their infinite dimensional version

a dynamically bi orthogonal method for time dependent stochastic partial differential equations i derivation and algorithms we propose a dynamically bi orthogonal method dybo to solve time dependent stochastic partial differential equations spdes the objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi orthogonal basis it is well known that the karhunen loeve expansion kle minimizes the total mean squared error and gives the sparsest representation of stochastic solutions however the computation of the kl expansion could be quite expensive since we need to form a covariance matrix and solve a large scale eigenvalue problem the main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the kl expansion unlike other reduced model methods our method constructs the reduced basis on the fly without the need to form the covariance matrix or to compute its eigendecomposition in the first part of our paper we introduce the derivation of the dynamically bi orthogonal formulation for spdes discuss several theoretical issues such as the dynamic bi orthogonality preservation and some preliminary error analysis of the dybo method we also give some numerical implementation details of the dybo methods including the representation of stochastic basis and techniques to deal with eigenvalue crossing in the second part of our paper 11 we will present an adaptive strategy to dynamically remove or add modes perform a detailed complexity analysis and discuss various generalizations of this approach an extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the dybo method

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state abstract order reduction schemes have been used successfully for the analysis and simplification of high dimensional systems exhibiting low dimensional dynamics in this work we first focus on presenting generic limitations of order reduction techniques in systems with stable mean state that exhibit irreducible high dimensional features such as non normal dynamics wide energy spectra or strong energy cascades between modes the reduced order framework that we consider to illustrate these limitations is the dynamically orthogonal do field equations this framework is applied to a series of examples with stable mean state including a linear non normal system and a nonlinear triad system in various dynamical configurations after illustrating the weaknesses and generic limitations of order reduction we develop a novel two way coupled blended approach based on the quasilinear gaussian qg closure and the do field equations the new method qg do overcomes the limitations of its two ingredients and achieves exceptional performance in the examples described previously as well as in other configurations with strongly transient character without using any tuned or adjustable parameters

a note on the correspondence of an immersed boundary method incorporating thermal fluctuations with stokesian brownian dynamics abstract in this paper a direct correspondence is made between the effective stochastic dynamics of elastic structures of an immersed boundary method incorporating thermal fluctuations and stokesian brownian dynamics the correspondence is made in the limit of small reynolds number in which the fluid relaxes rapidly on the time scale of the motion of the immersed structures by performing an averaging procedure directly on the stochastic equations of the immersed boundary method it is found that there is agreement with stokesian brownian dynamics for the far field hydrodynamic interactions and that a fluctuation dissipation relation is satisfied for the stochastic fluctuations of the effective equations

dynamic stochastic superresolution of sparsely observed turbulent systems real time capture of the relevant features of the unresolved turbulent dynamics of complex natural systems from sparse noisy observations and imperfect models is a notoriously difficult problem the resulting lack of observational resolution and statistical accuracy in estimating the important turbulent processes which intermittently send significant energy to the large scale fluctuations hinders efficient parameterization and real time prediction using discretized pde models this issue is particularly subtle and important when dealing with turbulent geophysical systems with an vast range of interacting spatio temporal scales and rough energy spectra near the mesh scale of numerical models here we introduce and study a suite of general dynamic stochastic superresolution dss algorithms and show that by appropriately filtering sparse regular observations with the help of cheap stochastic exactly solvable models one can derive stochastically superresolved velocity fields and gain insight into the important characteristics of the unresolved dynamics including the detection of the so called black swans the dss algorithms operate in fourier domain and exploit the fact that the coarse observation network aliases high wavenumber information into the resolved waveband it is shown that these cheap algorithms are robust and have significant skill on a test bed of turbulent solutions from realistic nonlinear turbulent spatially extended systems in the presence of a significant model error in particular the dss algorithms are capable of successfully capturing time localized extreme events in the unresolved modes and they provide good and robust skill for recovery of the unresolved processes in terms of pattern correlation moreover we show that dss improves the skill for recovering the primary modes associated with the sparse observation mesh which is equally important in applications the skill of the various dss algorithms depends on the energy spectrum of the turbulent signal and the observation time relative to the decorrelation time of the turbulence at a given spatial scale in a fashion elucidated here the dss technique exploiting a simple gaussian closure of the nonlinear stochastic forecast model emerges as the most suitable trade off between the superresolution skill and computational complexity associated with estimating the cross correlations between the aliasing modes of the sparsely observed turbulent signal such techniques offer a promising and efficient approach to constraining unresolved turbulent fluxes through stochastic superparameterization and a subsequent improvement in coarse grained filtering and prediction of the next generation atmosphere ocean system aos models

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

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

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

the weak and strong stability of algorithms in numerical linear algebra the stability of algorithms in numerical linear algebra is discussed the concept of stability is extended to notions of weak stability and strong stability justifications are given for these extensions and the implications of error analyses in terms of these definitions are discussed the concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for toeplitz systems

a stochastic immersed boundary method for fluid structure dynamics at microscopic length scales in modeling many biological systems it is important to take into account flexible structures which interact with a fluid at the length scale of cells and cell organelles thermal fluctuations of the aqueous environment become significant in this work it is shown how the immersed boundary method of c s peskin the immersed boundary method acta num 11 2002 1 39 for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations a stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps an important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved partially resolved or fully resolved while retaining a good level of accuracy error estimates in each of these regimes are given for the method a number of theoretical and numerical checks are furthermore performed to assess its physical fidelity for a conservative force the method is found to simulate particles with the correct boltzmann equilibrium statistics it is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters the method is also shown to reproduce a well known hydrodynamic effect of a brownian particle in which the velocity autocorrelation function exhibits an algebraic 3 2 decay for long times b j alder t e wainwright decay of the velocity autocorrelation function phys rev a 1 1 1970 18 21 a few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method specifically we present simulations of osmotic effects of molecular dimers worm like chain polymer knots and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force the theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales

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

a computational strategy for multiscale systems with applications to lorenz 96 model numerical schemes for systems with multiple spatio temporal scales are investigated the multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales the multiscale schemes are analyzed in general then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to lorenz issues like consistency accuracy and efficiency are discussed in detail the role of possible hidden slow variables as well as additional effects arising on the diffusive time scale are also investigated as a byproduct we obtain a rather complete characterization of the effective dynamics in lorenz 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

3d stochastic primitive equations of the large scale ocean global well posedness and attractors in this paper we consider the global well posedness and long time dynamics for the three dimensional viscous primitive equations describing the large scale oceanic motion under a random forcing which is an additive white in time noise we firstly prove the existence and uniqueness of global strong solutions to the initial boundary value problem for the stochastic primitive equations subsequently by studying the asymptotic behavior of strong solutions we obtain the existence of random attractors for the corresponding random dynamical system

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

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

hamiltonian structure and statistically relevant conserved quantities for the truncated burgers hopf equation in dynamical systems with intrinsic chaos many degrees of freedom and many conserved quantities a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes the galerkin truncation of the burgers hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments here it is established that the truncated burgers hopf model is a hamiltonian system with hamiltonian given by the integral of the third power this additional conserved quantity beyond the energy has been ignored in previous statistical mechanics studies of this equation thus the question arises of the statistical significance of the hamiltonian beyond that of the energy first an appropriate statistical theory is developed that includes both the energy and hamiltonian then a convergent monte carlo algorithm is developed for computing equilibrium statistical distributions the probability distribution of the hamiltonian on a microcanonical energy surface is studied through the monte carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the hamiltonian empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy the predictions of the theory for relevant and irrelevant values for the hamiltonian are confirmed through systematic numerical simulations for statistically relevant values of the hamiltonian these simulations show a surprising spectral tilt rather than equipartition of energy this spectral tilt is predicted and confirmed independently by monte carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt on the other hand the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the hamiltonian with excellent accuracy the results established here for the burgers hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications several interesting mathematical problems suggested by this study are mentioned in the final section 2002 wiley periodicals inc

instability in the solution of banded toeplitz systems the author considers some algorithms for the solution of banded toeplitz systems and calculates certain elements of the solution first and then the remaining elements by forward or backward substitution it is shown that for symmetric matrices this method is almost always highly unstable a numerical example is given to support the argument

fluctuation properties of steady state langevin systems motivated by stochastic models of climate phenomena the steady state of a linear stochastic model with additive gaussian white noise is studied fluctuation theorems for nonequilibrium steady states provide a constraint on the character of these fluctuations the properties of the fluctuations which are unconstrained by the fluctuation theorem are investigated and related to the model parameters the irreversibility of trajectory segments which satisfies a fluctuation theorem is used as a measure of nonequilibrium fluctuations the moments of the irreversibility probability density function pdf are found and the pdf is seen to be non gaussian the average irreversibility goes to zero for short and long trajectory segments and has a maximum for some finite segment length which defines a characteristic time scale of the fluctuations the initial average irreversibility growth rate is equal to the average entropy production and is related to noise amplification for systems with a separation of deterministic time scales modes with time scales much shorter than the trajectory time span and whose noise amplitudes are not asymptotically large do not to first order contribute to the irreversibility statistics providing a potential basis for dimensional reduction

on the stability of solution methods for band toeplitz systems abstract this paper is concerned with the numerical stability of inversion algorithms for banded toeplitz systems we analyze the numerical behavior of one algorithm due to b w dickinson ieee trans acoust speech signal process 27 421 423 1979 and two algorithms due to a k jain ieee trans acoust speech signal process 26 121 126 1978 we show that none of the three algorithms is weakly stable when used to invert symmetric positive definite systems one of jain s algorithms is shown to be weakly stable when used to invert a symmetric banded toeplitz matrix with a well conditioned positive definite infinite extension we present a new algorithm which is weakly stable under a more general condition and can be modified to invert certain toeplitz like matrices

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

predictions of critical transitions with non stationary reduced order models abstract here we demonstrate the ability of stochastic reduced order models to predict the statistics of non stationary systems undergoing critical transitions first we show that the reduced order models are able to accurately predict the autocorrelation function and probability density functions pdf of higher dimensional systems with time dependent slow forcing of either the resolved or unresolved modes second we demonstrate that whether the system tips early or repeatedly jumps between the two equilibrium points flickering depends on the strength of the coupling between the resolved and unresolved modes and the time scale separation both kinds of behaviour have been found to precede critical transitions in earlier studies furthermore we demonstrate that the reduced order models are also able to predict the timing of critical transitions the skill of various proposed tipping indicators are discussed

diffusion limit of 3d primitive equations of the large scale ocean under fast oscillating random force abstract the three dimensional 3d viscous primitive equations describing the large scale oceanic motions under fast oscillating random perturbation are studied under some assumptions on the random force the solution to the initial boundary value problem ibvp of the 3d random primitive equations converges in distribution to that of ibvp of the limiting equations which are the 3d stochastic primitive equations describing the large scale oceanic motions under a white in time noise forcing this also implies the convergence of the stationary solution of the 3d random primitive equations

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

linear theory for filtering nonlinear multiscale systems with model error in this paper we study filtering of multiscale dynamical systems with model error arising from limitations in resolving the smaller scale processes in particular the analysis assumes the availability of continuous time noisy observations of all components of the slow variables mathematically this paper presents new results on higher order asymptotic expansion of the first two moments of a conditional measure in particular we are interested in the application of filtering multiscale problems in which the conditional distribution is defined over the slow variables given noisy observation of the slow variables alone from the mathematical analysis we learn that for a continuous time linear model with gaussian noise there exists a unique choice of parameters in a linear reduced model for the slow variables which gives the optimal filtering when only the slow variables are observed moreover these parameters simultaneously give the optimal equilibrium statistical estimates of the underlying system and as a consequence they can be estimated offline from the equilibrium statistics of the true signal by examining a nonlinear test model we show that the linear theory extends in this non gaussian nonlinear configuration as long as we know the optimal stochastic parametrization and the correct observation model however when the stochastic parametrization model is inappropriate parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa this finding is based on analytical and numerical results on our nonlinear test model and the two layer lorenz 96 model finally even when the correct stochastic ansatz is given it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates in numerical experiments on the two layer lorenz 96 model we find that the parameters estimated online as part of a filtering procedure simultaneously produce accurate filtering and equilibrium statistical prediction in contrast an offline estimation technique based on a linear regression which fits the parameters to a training dataset without using the filter yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed this finding does not imply that all offline methods are inherently inferior to the online method for nonlinear estimation problems it only suggests that an ideal estimation technique should estimate all parameters simultaneously whether it is online or offline

normal forms for reduced stochastic climate models the systematic development of reduced low dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low frequency variability climate sensitivity and improved extended range forecasting here techniques from applied mathematics are utilized to systematically derive normal forms for reduced stochastic climate models for low frequency variables the use of a few empirical orthogonal functions eofs also known as principal component analysis karhunen loeve and proper orthogonal decomposition depending on observational data to span the low frequency subspace requires the assessment of dyad interactions besides the more familiar triads in the interaction between the low and high frequency subspaces of the dynamics it is shown below that the dyad and multiplicative triad interactions combine with the climatological linear operator interactions to simultaneously produce both strong nonlinear dissipation and correlated additive and multiplicative cam stochastic noise for a single low frequency variable the dyad interactions and climatological linear operator alone produce a normal form with cam noise from advection of the large scales by the small scales and simultaneously strong cubic damping these normal forms should prove useful for developing systematic strategies for the estimation of stochastic models from climate data as an illustrative example the one dimensional normal form is applied below to low frequency patterns such as the north atlantic oscillation nao in a climate model the results here also illustrate the short comings of a recent linear scalar cam noise model proposed elsewhere for low frequency variability

stochastic mode reduction for particle based simulation methods for complex microfluid systems we illustrate the stochastic mode reduction procedure as formulated recently by majda timofeyev and vanden eijnden comm pure appl math 54 2001 pp 891 974 mtv on the equations of motion underlying various particle based simulation approaches such as stokesian dynamics and brownian dynamics and the conceptually distinct dissipative particle dynamics dpd simulation approaches for complex microfluid systems the resulting coarse grained dynamics are compared and contrasted with each other we show that the stochastic mode reduction procedure provides a way to recover the smoluchowski dynamics for a standard model of multiple interacting particles in a fluid the dpd however has some subtle aspects which obstruct the application of the stochastic mode reduction procedure we discuss the mathematical and physical properties of the dpd method that underlie thisdifficulty

white noise limits for discrete dynamical systems driven by fast deterministic dynamics abstract we study a class of singularly perturbed dynamical systems that have fast and slow components e 1 being the fast to slow timescale ratio the fast components are governed by a strongly mixing discrete map which is iterated at time intervals e the slow components are governed by a first order finite difference equation that uses a time step e as e tends to zero the fast components may be eliminated giving rise to sdes for the slow components the emerging stochastic calculus is in the general case of neither ito nor stratonovich type but depends on the correlation time of the mixing process

adaptive sub sampling for parametric estimation of gaussian diffusions we consider a gaussian diffusion xt ornstein uhlenbeck process with drift coefficient and diffusion coefficient 2 and an approximating process y varepsilon t converging to xt in l2 as e 0 we study estimators hat gamma varepsilon hat sigma 2 varepsilon which are asymptotically equivalent to the maximum likelihood estimators of and 2 respectively we assume that the estimators are based on the available n n e observations extracted by sub sampling only from the approximating process y varepsilon t with time step e we characterize all such adaptive sub sampling schemes for which hat gamma varepsilon hat sigma 2 varepsilon are consistent and asymptotically efficient estimators of and 2 as e 0 the favorable adaptive sub sampling schemes are identified by the conditions e 0 0 e and n which implies that we sample from the process y varepsilon t with a vanishing but coarse time step e e this study highlights the necessity to sub sample at adequate rates when the observations are not generated by the underlying stochastic model whose parameters are being estimated the adequate sub sampling rates we identify seem to retain their validity in much wider contexts such as the additive triad application we briefly outline

the role of additive and multiplicative noise in filtering complex dynamical systems covariance inflation is an ad hoc treatment that is widely used in practical real time data assimilation algorithms to mitigate covariance underestimation owing to model errors nonlinearity or an

efficient statistically accurate algorithms for the fokker planck equation in large dimensions abstract solving the fokker planck equation for high dimensional complex turbulent dynamical systems is an important and practical issue however most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions pdfs of complex systems in turbulence neuroscience and excitable media in this article efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of fokker planck equations associated with high dimensional nonlinear turbulent dynamical systems with conditional gaussian structures the algorithms involve a hybrid strategy that requires only a small number of ensembles here a conditional gaussian mixture in a high dimensional subspace via an extremely efficient parametric method is combined with a judicious non parametric gaussian kernel density estimation in the remaining low dimensional subspace particularly the parametric method provides closed analytical formulae for determining the conditional gaussian distributions in the high dimensional subspace and is therefore computationally efficient and accurate the full non gaussian pdf of the system is then given by a gaussian mixture different from traditional particle methods each conditional gaussian distribution here covers a significant portion of the high dimensional pdf therefore a small number of ensembles is sufficient to recover the full pdf which overcomes the curse of dimensionality notably the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high dimensional non gaussian pdfs and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems it is shown in a stringent set of test problems that the method only requires an order of o 100 ensembles to successfully recover the highly non gaussian transient pdfs in up to 6 dimensions with only small errors

on the foundations of the stochastic immersed boundary method we explore the theoretical foundations for the inclusion of thermal fluctuations in the immersed boundary method for simulating microscale fluid systems with immersed flexible structures as in cellular and subcellular biology we investigate in particular the physical validity of the thermal forcing scheme with respect to the coupling of fluid and immersed structural degrees of freedom and non equilibrium conditions we discuss also the shortcomings of a natural alternative scheme in which the thermal fluctuations are applied directly to the structural degrees of freedom through langevin type dynamics

predictability of extreme events in a nonlinear stochastic dynamical model the objective of this work is to evaluate the potential of reduced order models to reproduce the extreme event and predictability characteristics of higher dimensional dynamical systems a nonlinear toy model is used which contains key features of comprehensive climate models first we demonstrate that the systematic stochastic mode reduction strategy leads to a reduced order model with the same extreme value characteristics as the full dynamical models for a wide range of time scale separations second we find that extreme events in this model follow a generalized pareto distribution with a negative shape parameter thus extreme events are bounded in this model third we show that a precursor approach has good forecast skill for extreme events we then find that the reduced stochastic models capture the predictive skill of extreme events of the full dynamical models well consistent with previous studies we also find that the larger the extreme events the better predictable they are our results suggest that systematically derived reduced order models have the potential to be used for the modeling and statistical prediction of weather and climate related extreme events and possibly in other areas of science and engineering too

numerical experience with a superfast real toeplitz solver abstract we briefly describe the generalized schur algorithm for the superfast solution of positive definite toeplitz systems of equations and its relationship with schur s algorithm and the szego recursions we then present some experimental results obtained with our fortran implementation of this superfast toeplitz solver the algorithm displays favorable behavior in that the growth rates of the resulting residuals and errors are comparable with those of the szego recursions

fast direct solution methods for symmetric banded toeplitz systems based on the sine transform we present new fast direct methods for solving a large symmetric banded toeplitz system of order n with bandwidth p we make use of structured matrices which can be diagonalized by the discrete sine transform matrix sometimes called matrices a first method writes the toeplitz matrix as the sum of a matrix and a low rank matrix a second method embeds the toeplitz matrix in a larger matrix of order m the methods are similar to jain ieee trans acoust speech signal process 26 1978 121 and linzer linear algebra appl 170 1992 1 who worked with circulant matrices both algorithms consist in solving two systems and two smaller systems a system of order n can be solved in o n log n by using a discrete sine transform if n 1 has small prime factors therefore the second algorithm is preferable since we can choose m such that m 1 has small prime factors on the other hand in the second method the smaller systems can become large when m differs too much from n w hile in the first method the order is always p 1 in both methods the small systems have low displacement rank so we can use fast methods to solve them 2002 elsevier science inc all rights reserved

exact multimagnon states in one dimensional ferromagnetic spin chains with a short range interaction the authors propose the procedure of constructing the exact wavefunctions of m magnons on an infinite one dimensional lattice in the one parameter model with an exponentially decreasing interaction between spins for m or 4 the problem is solved explicitly the bethe wavefunctions in the heisenberg xxx chain appear in the limit of nearest neighbour interaction

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

parametric estimation from approximate data non gaussian diffusions we study the problem of parameters estimation in indirect observability contexts where x t in r r is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process y varepsilon t which is close to x t in l 4 norm we construct consistent parameter estimators which are smooth functions of the sub sampled empirical mean and empirical lagged covariance matrices computed from the observable data we derive explicit optimal sub sampling schemes specifying the best paired choices of sub sampling time step and number of observations we show that these choices ensure that our parameter estimators reach optimized asymptotic l 2 convergence rates which are constant multiples of the l 4 norm y varepsilon t x t

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

variance reduction for particle filters of systems with time scale separation we present a particle filter construction for a system that exhibits time scale separation the separation of time scales allows two simplifications that we exploit 1 the use of the averaging principle for the dimensional reduction of the dynamics for each particle during the prediction step and 2 the factorization of the transition probability for the rao blackwellization of the update step the resulting particle filter is faster and has smaller variance than the particle filter based on the original system the method is tested on a multiscale stochastic differential equation and on a multiscale pure jump diffusion motivated by chemical reactions

numerical methods for stochastic partial differential equations with multiple scales a new method for solving numerically stochastic partial differential equations spdes with multiple scales is presented the method combines a spectral method with the heterogeneous multiscale method hmm presented in w e d liu e vanden eijnden analysis of multiscale methods for stochastic differential equations commun pure appl math 58 11 2005 1544 1585 the class of problems that we consider are spdes with quadratic nonlinearities that were studied in d blomker m hairer g a pavliotis multiscale analysis for stochastic partial differential equations with quadratic nonlinearities nonlinearity 20 7 2007 1721 1744 for such spdes an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales our method based on micro and macro solvers allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem numerical experiments illustrate the behavior of the proposed method

reduced models of atmospheric low frequency variability parameter estimation and comparative performance abstract low frequency variability lfv of the atmosphere refers to its behavior on time scales of 10 100 days longer than the life cycle of a mid latitude cyclone but shorter than a season this behavior is still poorly understood and hard to predict the present study compares various model reduction strategies that help in deriving simplified models of lfv three distinct strategies are applied here to reduce a fairly realistic high dimensional quasi geostrophic 3 level qg3 atmospheric model to lower dimensions i an empirical dynamical method which retains only a few components in the projection of the full qg3 model equations onto a specified basis and finds the linear deterministic and the stochastic corrections empirically as in selten 1995 5 ii a purely dynamics based technique employing the stochastic mode reduction strategy of majda et al 2001 62 and iii a purely empirical multi level regression procedure which specifies the functional form of the reduced model and finds the model coefficients by multiple polynomial regression as in kravtsov et al 2005 3 the empirical dynamical and dynamical reduced models were further improved by sequential parameter estimation and benchmarked against multi level regression models the extended kalman filter was used for the parameter estimation overall the reduced models perform better when more statistical information is used in the model construction thus the purely empirical stochastic models with quadratic nonlinearity and additive noise reproduce very well the linear properties of the full qg3 model s lfv i e its autocorrelations and spectra as well as the nonlinear properties i e the persistent flow regimes that induce non gaussian features in the model s probability density function the empirical dynamical models capture the basic statistical properties of the full model s lfv such as the variance and integral correlation time scales of the leading lfv modes as well as some of the regime behavior features but fail to reproduce the detailed structure of autocorrelations and distort the statistics of the regimes dynamical models that use data assimilation corrections do capture the linear statistics to a degree comparable with that of empirical dynamical models but do much less well on the full qg3 model s nonlinear dynamics these results are discussed in terms of their implications for a better understanding and prediction of lfv

stochastic suspensions of heavy particles turbulent suspensions of heavy particles in incompressible flows have gained much attention in recent years a large amount of work focused on the impact that the inertia and the dissipative dynamics of the particles have on their dynamic and statistical properties substantial progress followed from the study of suspensions in model flows which although much simpler reproduce most of the important mechanisms observed in real turbulence this paper presents recent developments made on the relative motion of a pair of particles suspended in time uncorrelated and spatially self similar gaussian flows this review is complemented by new results by introducing a time dependent stokes number it is demonstrated that inertial particle relative dispersion recovers asymptotically richardson s diffusion associated to simple tracers a perturbative homogeneization technique is used in the small stokes number asymptotics and leads to interpreting first order corrections to tracer dynamics in terms of an effective drift this expansion implies that the correlation dimension deficit behaves linearly as a function of the stokes number the validity and the accuracy of this prediction is confirmed by numerical simulations

low dimensional chaotic dynamics versus intrinsic stochastic noise a paradigm model several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables these models are built by coupling a four dimensional ode with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables a systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four dimensional reduced stochastic equations for the slow variables due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system furthermore it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath the detailed structure of the stochastic terms derived through the mode elimination procedure is essential for reproducing the statistical behavior of the slow dynamics

an ensemble kalman filter for statistical estimation of physics constrained nonlinear regression models a central issue in contemporary science is the development of nonlinear data driven statistical dynamical models for time series of noisy partial observations from nature or a complex model it has been established recently that ad hoc quadratic multi level regression models can have finite time blow up of statistical solutions and or pathological behavior of their invariant measure recently a new class of physics constrained nonlinear regression models were developed to ameliorate this pathological behavior here a new finite ensemble kalman filtering algorithm is developed for estimating the state the linear and nonlinear model coefficients the model and the observation noise covariances from available partial noisy observations of the state several stringent tests and applications of the method are developed here in the most complex application the perfect model has 57 degrees of freedom involving a zonal east west jet two topographic rossby waves and 54 nonlinearly interacting rossby waves the perfect model has significant non gaussian statistics in the zonal jet with blocked and unblocked regimes and a non gaussian skewed distribution due to interaction with the other 56 modes we only observe the zonal jet contaminated by noise and apply the ensemble filter algorithm for estimation numerically we find that a three dimensional nonlinear stochastic model with one level of memory mimics the statistical effect of the other 56 modes on the zonal jet in an accurate fashion including the skew non gaussian distribution and autocorrelation decay on the other hand a similar stochastic model with zero memory levels fails to capture the crucial non gaussian behavior of the zonal jet from the perfect 57 mode model