optimal construction of koopman eigenfunctions for prediction and control this article presents a novel data driven framework for constructing eigenfunctions of the koopman operator geared toward prediction and control the method leverages the richness of the spectrum of the koopman operator away from attractors to construct a set of eigenfunctions such that the state or any other observable quantity of interest is in the span of these eigenfunctions and hence predictable in a linear fashion the eigenfunction construction is optimization based with no dictionary selection required once a predictor for the uncontrolled part of the system is obtained in this way the incorporation of control is done through a multistep prediction error minimization carried out by a simple linear least squares regression the predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the koopman model predictive control mpc framework of m korda and i mezic 2018 to control nonlinear dynamical systems using linear mpc tools the method is entirely data driven and based predominantly on convex optimization the novel eigenfunction construction method is also analyzed theoretically proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions in addition the method is extended to construct generalized eigenfunctions that also give rise koopman invariant subspaces and hence can be used for linear prediction detailed numerical examples demonstrate the approach both for prediction and feedback control code for the numerical examples is available from https homepages laas fr mkorda eigfuns zip

comparison of systems with complex behavior abstract we present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the koopman operator we first compare invariant measures and discuss this in terms of a statistical takens theorem proved here we also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic theoretic treatment of a class of spectral functionals that allow for this the formalism is extended for a class of stochastic systems discrete random dynamical systems the ideas introduced in this paper can be used for parameter identification and model validation of driven nonlinear models with complicated behavior as an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of a class of models that have the form of a random dynamical system

explicit inverse of a generalized vandermonde matrix in this paper the author gives an explicit closed form expression for the nxn inverse matrix v g k n 1 of the generalized nxn vandermonde matrix v g k n by using the elementary symmetric functions symbolic and numerical results are presented

semiparametric forecasting and filtering correcting low dimensional model error in parametric models semiparametric forecasting and filtering are introduced as a method of addressing model errors arising from unresolved physical phenomena while traditional parametric models are able to learn high dimensional systems from small data sets their rigid parametric structure makes them vulnerable to model error on the other hand nonparametric models have a very flexible structure but they suffer from the curse of dimensionality and are not practical for high dimensional systems the semiparametric approach loosens the structure of a parametric model by fitting a data driven nonparametric model for the parameters given a parametric dynamical model and a noisy data set of historical observations an adaptive kalman filter is used to extract a time series of the parameter values a nonparametric forecasting model for the parameters is built by projecting the discrete shift map onto a data driven basis of smooth functions existing techniques for filtering and forecasting algorithms extend naturally to the semiparametric model which can effectively compensate for model error with forecasting skill approaching that of the perfect model semiparametric forecasting and filtering are a generalization of statistical semiparametric models to time dependent distributions evolving under dynamical systems

precursors of extreme increments we investigate precursors and the predictability of extreme increments in a time series the events we are focusing on consist in large increments within successive time steps we are especially interested in understanding how the quality of the predictions depends on the strategy to choose precursors on the size of the event and on the correlation strength we study the prediction of extreme increments analytically in an autoregressive process of order 1 and numerically in wind speed recordings and long range correlated autoregressive moving average processes data we evaluate the success of predictions via receiver operator characteristics roc curves furthermore we observe an increase of the quality of predictions with increasing event size and with decreasing correlation in all examples both effects can be understood by using the likelihood ratio as a summary index for smooth roc curves

optimal filtering of complex turbulent systems with memory depth through consistency constraints in this article we develop a linear theory for optimal filtering of complex turbulent signals with model errors through linear autoregressive models we will show that when the autoregressive model parameters are chosen such that they satisfy absolute stability and consistency conditions of at least order 2 of the classical multistep method for solving initial value problems the filtered solutions with autoregressive models of order p 2 are optimal in the sense that they are comparable to the estimates obtained from the true filter with perfect model this result is reminiscent of the lax equivalence fundamental theorem in the analysis of finite difference discretization scheme for the numerical solutions of partial differential equations we will apply this linear theory to filter two nonlinear problems the slowest mode of the truncated burgers hopf and the lorenz 96 model on these nonlinear problems we will show that whenever these linear conditions are satisfied the filtered solutions accuracy is significantly improved finally we will also apply the recently developed offline test criteria to understand the robustness of the multistep filter on various turbulent nature including the stochastically forced linear advection diffusion equation and a toy model for barotropic turbulent rossby waves

operator theoretic framework for forecasting nonlinear time series with kernel analog techniques abstract kernel analog forecasting kaf alternatively known as kernel principal component regression is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data this paper synthesizes descriptions of kernel methods and koopman operator theory in order to provide a single consistent account of kaf the framework presented here illuminates the property of the kaf method that under measure preserving and ergodic dynamics it consistently approximates the conditional expectation of observables that are acted upon by the koopman operator of the dynamical system and are conditioned on the observed data at forecast initialization more precisely kaf yields optimal predictions in the sense of minimal root mean square error with respect to the invariant measure in the asymptotic limit of large data the presented framework facilitates moreover the analysis of generalization error and quantification of uncertainty extensions of kaf to the construction of conditional variance and conditional probability functions as well as to non symmetric kernels are also shown illustrations of various aspects of kaf are provided with applications to simple examples namely a periodic flow on the circle and the chaotic lorenz 63 system

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

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

influence of the event magnitude on the predictability of an extreme event we investigate the predictability of extreme events in time series the focus of this work is to understand under which circumstances large events are better predictable than smaller events therefore we use a simple prediction algorithm based on precursory structures which are identified using the maximum likelihood principle using the receiver operator characteristic curve as a measure for the quality of predictions we find that the dependence on the event size is closely linked to the probability distribution function of the underlying stochastic process we evaluate this dependence on the probability distribution function analytically and numerically if we assume that the optimal precursory structures are used to make the predictions we find that large increments are better predictable if the underlying stochastic process has a gaussian probability distribution function whereas larger increments are harder to predict if the underlying probability distribution function has a power law tail in the case of an exponential distribution function we find no significant dependence on the event size furthermore we compare these results with predictions of increments in correlated data namely velocity increments of a free jet flow the velocity increments in the free jet flow are in dependence on the time scale either asymptotically gaussian or asymptotically exponential distributed the numerical results for predictions within free jet data are in good agreement with the previous analytical considerations for random numbers

principal dynamical components a procedure is proposed for a dimension reduction in time series similarly to principal components the procedure seeks a low dimensional manifold that minimizes information loss unlike principal components however the procedure involves dynamical considerations through the proposal of a predictive dynamical model in the reduced manifold hence the minimization of the uncertainty is not only over the choice of a reduced manifold as in principal components but also over the parameters of the dynamical model as in autoregressive analysis and principal interaction patterns further generalizations are provided to nonautonomous and non markovian scenarios which are then applied to historical sea surface temperature data 2012 wiley periodicals inc

linear predictors for nonlinear dynamical systems koopman operator meets model predictive control this paper presents a class of linear predictors for nonlinear controlled dynamical systems the basic idea is to lift or embed the nonlinear dynamics into a higher dimensional space where its evolution is approximately linear in an uncontrolled setting this procedure amounts to numerical approximations of the koopman operator associated to the nonlinear dynamics in this work we extend the koopman operator to controlled dynamical systems and apply the extended dynamic mode decomposition edmd to compute a finite dimensional approximation of the operator in such a way that this approximation has the form of a linear controlled dynamical system in numerical examples the linear predictors obtained in this way exhibit a performance superior to existing linear predictors such as those based on local linearization or the so called carleman linearization importantly the procedure to construct these linear predictors is completely data driven and extremely simple it boils down to a nonlinear transformation of the data the lifting and a linear least squares problem in the lifted space that can be readily solved for large data sets these linear predictors can be readily used to design controllers for the nonlinear dynamical system using linear controller design methodologies we focus in particular on model predictive control mpc and show that mpc controllers designed in this way enjoy computational complexity of the underlying optimization problem comparable to that of mpc for a linear dynamical system with the same number of control inputs and the same dimension of the state space importantly linear inequality constraints on the state and control inputs as well as nonlinear constraints on the state can be imposed in a linear fashion in the proposed mpc scheme similarly cost functions nonlinear in the state variable can be handled in a linear fashion we treat both the full state measurement case and the input output case as well as systems with disturbances noise numerical examples demonstrate the approach

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

traveling salesperson problems for the dubins vehicle in this paper we study minimum time motion planning and routing problems for the dubins vehicle i e a nonholonomic vehicle that is constrained to move along planar paths of bounded curvature without reversing direction motivated by autonomous aerial vehicle applications we consider the traveling salesperson problem for the dubins vehicle dtsp given n points on a plane what is the shortest dubins tour through these points and what is its length first we show that the worst case length of such a tour grows linearly with n and we propose a novel algorithm with performance within a constant factor of the optimum for the worst case point sets in doing this we also obtain an upper bound on the optimal length in the classical point to point problem second we study a stochastic version of the dtsp where the n targets are randomly and independently sampled from a uniform distribution we show that the expected length of such a tour is of order at least n 2 3 and we propose a novel algorithm yielding a solution with length of order n 2 3 with probability one third and finally we study a dynamic version of the dtsp given a stochastic process that generates target points is there a policy that guarantees that the number of unvisited points does not diverge over time if such stable policies exist what is the minimum expected time that a newly generated target waits before being visited by the vehicle we propose a novel stabilizing algorithm such that the expected wait time is provably within a constant factor from the optimum

uncertainty propagation in dynamical systems dynamical propagation of parametric and initial condition uncertainty is studied the notion of input measure of an observable is defined and its propagation to output measure of the observable is studied by means of transfer operators uncertainty of these measures is defined in terms of their cumulative probability distributions comparison with alternative uncertainty metrics such as variance and entropy is pursued the developed formalism is illustrated through an analysis of the effect of pitchfork bifurcation on uncertainty finally the implications of these concepts in the design of nonlinear systems are discussed

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

isostables isochrons and koopman spectrum for the action angle representation of stable fixed point dynamics abstract for asymptotically periodic systems a powerful phase reduction of the dynamics is obtained by computing the so called isochrons i e the sets of points that converge toward the same trajectory on the limit cycle motivated by the analysis of excitable systems a similar reduction has been attempted for non periodic systems admitting a stable fixed point in this case the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories instead the sets of interest that we call isostables are defined in the literature as the sets of points that converge toward the same trajectory on a stable slow manifold of the fixed point however it turns out that this definition of the isostables holds only for systems with slow fast dynamics also efficient methods for computing the isostables are missing the present paper provides a general framework for the definition and the computation of the isostables of stable fixed points which is based on the spectral properties of the so called koopman operator more precisely the isostables are defined as the level sets of a particular eigenfunction of the koopman operator through this approach the isostables are unique and well defined objects related to the asymptotic properties of the system also the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action angle coordinates for the dynamics in addition an efficient algorithm for computing the isostables is obtained which relies on the evaluation of laplace averages along the trajectories the method is illustrated with the excitable fitzhugh nagumo model and with the lorenz model finally we discuss how these methods based on the koopman operator framework relate to the global linearization of the system and to the derivation of special lyapunov functions

assimilating irregularly spaced sparsely observed turbulent signals with hierarchical bayesian reduced stochastic filters in this paper we consider a practical filtering approach for assimilating irregularly spaced sparsely observed turbulent signals through a hierarchical bayesian reduced stochastic filtering framework the proposed hierarchical bayesian approach consists of two steps blending a data driven interpolation scheme and the mean stochastic model msm filter we examine the potential of using the deterministic piecewise linear interpolation scheme and the ordinary kriging scheme in interpolating irregularly spaced raw data to regularly spaced processed data and the importance of dynamical constraint through msm in filtering the processed data on a numerically stiff state estimation problem in particular we test this approach on a two layer quasi geostrophic model in a two dimensional domain with a small radius of deformation to mimic ocean turbulence our numerical results suggest that the dynamical constraint becomes important when the observation noise variance is large second we find that the filtered estimates with ordinary kriging are superior to those with linear interpolation when observation networks are not too sparse such robust results are found from numerical simulations with many randomly simulated irregularly spaced observation networks various observation time intervals and observation error variances third when the observation network is very sparse we find that both the kriging and linear interpolations are comparable

clustering of extreme and recurrent events in deterministic chaotic systems we study the nontrivial clustering properties of extreme or recurrent events in the context of deterministic chaotic systems we find that correlations between return times of such events can depend nonmonotonically on the threshold used to define the events which leads to counterintuitive behavior in particular the distribution of the conditional return intervals can indicate clustering as well as repelling of extreme events for the same condition but different thresholds in sharp contrast to what has been observed for stochastic processes with long range correlations as well as for independent and identically distributed random variables this has important implications for the time dependent hazard assessment of extreme events indicating that possible threshold dependencies should always be taken into account

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

improving filtering and prediction of spatially extended turbulent systems with model errors through stochastic parameter estimation the filtering and predictive skill for turbulent signals is often limited by the lack of information about the true dynamics of the system and by our inability to resolve the assumed dynamics with sufficiently high resolution using the current computing power the standard approach is to use a simple yet rich family of constant parameters to account for model errors through parameterization this approach can have significant skill by fitting the parameters to some statistical feature of the true signal however in the context of real time prediction such a strategy performs poorly when intermittent transitions to instability occur alternatively we need a set of dynamic parameters one strategy for estimating parameters on the fly is a stochastic parameter estimation through partial observations of the true signal in this paper we extend our newly developed stochastic parameter estimation strategy the stochastic parameterization extended kalman filter spekf to filtering sparsely observed spatially extended turbulent systems which exhibit abrupt stability transition from time to time despite a stable average behavior for our primary numerical example we consider a turbulent system of externally forced barotropic rossby waves with instability introduced through intermittent negative damping we find high filtering skill of spekf applied to this toy model even in the case of very sparse observations with only 15 out of the 105 grid points observed and with unspecified external forcing and damping additive and multiplicative bias corrections are used to learn the unknown features of the true dynamics from observations we also present a comprehensive study of predictive skill in the one mode context including the robustness toward variation of stochastic parameters imperfect initial conditions and finite ensemble effect furthermore the proposed stochastic parameter estimation scheme applied to the same spatially extended rossby wave system demonstrates high predictive skill comparable with the skill of the perfect model for a duration of many eddy turnover times especially in the unstable regime

mathematical strategies for filtering complex systems regularly spaced sparse observations real time filtering of noisy turbulent signals through sparse observations on a regularly spaced mesh is a notoriously difficult and important prototype filtering problem simpler off line test criteria are proposed here as guidelines for filter performance for these stiff multi scale filtering problems in the context of linear stochastic partial differential equations with turbulent solutions filtering turbulent solutions of the stochastically forced dissipative advection equation through sparse observations is developed as a stringent test bed for filter performance with sparse regular observations the standard ensemble transform kalman filter etkf has poor skill on the test bed and even suffers from filter divergence surprisingly at observable times with resonant mean forcing and a decaying energy spectrum in the partially observed signal systematic alternative filtering strategies are developed here including the fourier domain kalman filter fdkf and various reduced filters called strongly damped approximate filter sdaf variance strongly damped approximate filter vsdaf and reduced fourier domain kalman filter rfdkf which operate only on the primary fourier modes associated with the sparse observation mesh while nevertheless incorporating into the approximate filter various features of the interaction with the remaining modes it is shown below that these much cheaper alternative filters have significant skill on the test bed of turbulent solutions which exceeds etkf and in various regimes often exceeds fdkf provided that the approximate filters are guided by the off line test criteria the skill of the various approximate filters 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 precise fashion elucidated here

measuring the distance between time series abstract to evaluate models of dynamical systems researchers have traditionally used quantitative measures of short term prediction errors however for chaotic or stochastic systems comparison of long term qualitative behaviors may be more relevant let x x0 xn be a sequence of real numbers generated by sampling a dynamical system or stochastic process and suppose y y0 yn is another sequence generated by a mathematical model of the process which generated x in this paper we consider several ways of assigning a distance d x y which measures the difference in long term behavior

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

statistical properties of controlled fluid flows with applications to control of mixing abstract in this paper we study statistical properties of fluid flows that are actively controlled statistical properties such as lagrangian and eulerian time averages are important flow quantities in fluid flows particularly during mixing processes due to the assumption of incompressibility the transformations in the state space can be described by a sequence of measure preserving transformations on a measure space the classical birkhoff s pointwise ergodic theorem does not necessarily apply in the context of sequences of transformations we call b regular a sequence for which this theorem holds motivated by mixing control concepts we define three notions of asymptotic equivalence for sequences of transformations we show an example in which birkhoff s pointwise ergodic theorem does not hold even when a strong asymptotic equivalence to a b regular sequence is assumed under a very strong asymptotic equivalence condition we prove b regularity in the context of optimize then stabilize strategy for mixing control we also prove that very strong asymptotic equivalence to a mixing sequence implies mixing the mean ergodic theorem and the poincare recurrence theorem are also proven for sequences of transformations under suitable asymptotic equivalence assumptions

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

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

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 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

predictability of threshold exceedances in dynamical systems in a low order model of the general circulation of the atmosphere we examine the predictability of threshold exceedance events of certain observables the likelihood of such binary events the cornerstone also for the categoric as opposed to probabilistic prediction of threshold exceedances is established from long time series of one or more observables of the same system the prediction skill is measured by a summary index of the roc curve that relates the hit and false alarm rates our results for the examined systems suggest that exceedances of higher thresholds are more predictable or in other words rare large magnitude i e extreme events are more predictable than frequent typical events we find this to hold provided that the bin size for binning time series data is optimized but not necessarily otherwise this can be viewed as a confirmation of a counterintuitive and seemingly contrafactual statement that was previously formulated for more simple autoregressive stochastic processes however we argue that for dynamical systems in general it may be typical only but not universally true we argue that when there is a sufficient amount of data depending on the precision of observation the skill of a class of data driven categoric predictions of threshold exceedances approximates the skill of the analogous model driven prediction assuming strictly no model errors therefore stronger extremes in terms of higher threshold levels are more predictable both in case of data and model driven prediction furthermore we show that a quantity commonly regarded as a measure of predictability the finite time maximal lyapunov exponent does not correspond directly to the roc based measure of prediction skill when they are viewed as functions of the prediction lead time and the threshold level this points to the fact that even if the lyapunov exponent as an intrinsic property of the system measuring the instability of trajectories determines predictability it does that in a nontrivial manner

filtering nonlinear spatio temporal chaos with autoregressive linear stochastic models abstract fundamental barriers in practical filtering of nonlinear spatio temporal chaotic systems are model errors attributed to the stiffness in resolving multiscale features recently reduced stochastic filters based on linear stochastic models have been introduced to overcome such stiffness one of them is the mean stochastic model msm based on a diagonal ornstein uhlenbeck process in fourier space despite model errors the msm shows very encouraging filtering skill especially when the hidden signal of interest is strongly chaotic in this regime the dynamical system statistical properties resemble to those of the energy conserving equilibrium statistical mechanics with gaussian invariant measure therefore the ornstein uhlenbeck process with appropriate parameters is sufficient to produce reasonable statistical estimates for the filter model in this paper we consider a generalization of the msm with a diagonal autoregressive linear stochastic model in fourier space as a filter model for chaotic signals with long memory depth with this generalization the filter prior model becomes slightly more expensive than the msm but it is still less expensive relative to integrating the perfect model which is typically unknown in real problems furthermore the associated kalman filter on each fourier mode is computationally as cheap as inverting a matrix of size d where d is the number of observed variables on each fourier mode in our numerical example d 1 using the lorenz 96 l 96 model as a testbed we show that the non markovian nature of this autoregressive model is an important feature in capturing the highly oscillatory modes with long memory depth second we show that the filtering skill with autoregressive models supersedes that with msm in weakly chaotic regime where the memory depth is longer in strongly chaotic regime the performance of the ar p filter is still better or at least comparable to that of the msm most importantly we find that this reduced filtering strategy is not as sensitive as standard ensemble filtering strategies to additional intrinsic model errors that are often encountered when model parameters are incorrectly specified

polynomial time approximation schemes for euclidean traveling salesman and other geometric problems we present a polynomial time approximation scheme for euclidean tsp in fixed dimensions for every fixed c 1 and given any n nodes in r 2 a randomized version of the scheme finds a 1 1 c approximation to the optimum traveling salesman tour in o n log n o c time when the nodes are in r d the running time increases to o n log n o d c d 1 for every fixed c d the running time is n poly log n that is nearly linear in n the algorithmm can be derandomized but this increases the running time by a factor o n d the previous best approximation algorithm for the problem due to christofides achieves a 3 2 aproximation in polynomial time we also give similar approximation schemes for some other np hard euclidean problems minimum steiner tree k tsp and k mst the running times of the algorithm for k tsp and k mst involve an additional multiplicative factor k the previous best approximation algorithms for all these problems achieved a constant factor approximation we also give efficient approximation schemes for euclidean min cost matching a problem that can be solved exactly in polynomial time all our algorithms also work with almost no modification when distance is measured using any geometric norm such as e p for p 1 or other minkowski norms they also have simple parallel i e nc implementations

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

approximation algorithms for curvature constrained shortest paths let b be a point robot in the plane whose path is constrained to have curvature of at most 1 and let omega be a set of polygonal obstacles with n vertices we study the collision free optimal path planning problem for b given a parameter varepsilon we present an o n 2 eps 4 log n time algorithm for computing a collision free curvature constrained path between two given positions whose length is at most 1 varepsilon times the length of an optimal path provided it is robust roughly speaking a path is robust if it remains collision free even if certain positions on the path are perturbed our algorithm thus runs significantly faster than the previously best known algorithm by jacobs and canny whose running time is o frac n l eps 2 2 n 2 frac n l eps 2 log n where l is the total edge length of the obstacles more importantly the running time of our algorithm does not depend on the size of obstacles the path returned by this algorithm is not necessarily robust we present an o n 2 5 eps 4 log n time algorithm that returns a robust path whose length is at most 1 varepsilon times the length of an optimal path provided it is robust we also give a stronger characterization of curvature constrained shortest paths which apart from being crucial for our algorithm is interesting in its own right roughly speaking we prove that except in some special cases a shortest path touches obstacles at points that have a visible vertex nearby

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

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

extreme phase sensitivity in systems with fractal isochrons abstract sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors however even systems with quasi periodic dynamics can exhibit it in this context we report on the fractal properties of the isochrons of some continuous time asymptotically periodic systems we define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons similar results are also obtained with discrete time systems as an illustration of the framework we compute the phase sensitivity coefficient for popular models of bursting neurons suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive unreliable phase response

numerical aspects for approximating governing equations using data abstract we present effective numerical algorithms for approximating unknown governing differential equations from measurement data we employ a set of standard basis functions e g polynomials to approximate the governing equation with high accuracy upon recasting the problem into a function approximation problem we discuss several important aspects for accurate approximation most notably we discuss the importance of using a large number of short bursts of trajectory data rather than using data from a single long trajectory several options for the numerical algorithms to perform accurate approximation are then presented along with an error estimate of the final equation approximation we then present an extensive set of numerical examples of both linear and nonlinear systems to demonstrate the properties and effectiveness of our equation approximation algorithms

linearization in the large of nonlinear systems and koopman operator spectrum abstract according to the hartman grobman theorem a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point here we extend this linearization around stable unstable equilibria or periodic orbits to the whole basin of attraction for both discrete diffeomorphisms and flows we discuss the connection of the linearizing transformation to the spectrum of koopman operator

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

interpolating irregularly spaced observations for filtering turbulent complex systems we present a numerically fast reduced filtering strategy the fourier domain kalman filter with appropriate interpolations to account for irregularly spaced observations of complex turbulent signals the design of such a reduced filter involves i interpolating irregularly spaced observations to the model regularly spaced grid points ii understanding under which situation the small scale oscillatory artifact from such interpolation won t degrade the filtered solutions iii understanding when the interpolated covariance structure can be approximated by its diagonal terms when observations are corrupted by independent gaussian noise and iv applying a scalar kalman filter formula on each fourier component independently with an approximate diagonal interpolated covariance matrix from the practical point of view there is an emerging need to understand the effect of i toward the filtered solutions for example in utilizing the data produced from various interpolation techniques that merge multiple satellite measurements of atmospheric and ocean dynamical quantities to understand point iii above and to see how many of nondiagonal terms are effectively ignored in iv we compute a ratio lambda between the largest nondiagonal components and the smallest diagonal components of the interpolated covariance matrix we find that for piecewise linear interpolation this ratio lambda is always smaller than that of the alternative interpolation schemes such as trigonometric and cubic spline for any irregularly spaced observation networks when observations are not so sparse with small noise we find that the small scale oscillatory artifact in ii above is negligible when piecewise linear interpolation is used whereas for the other schemes such as the nearest neighbor trigonometric and cubic spline interpolation the oscillatory artifact degrades the filtered solutions significantly finally we also find that the reduced filtering strategy with piecewise linear interpolation produces more accurate filtered solutions than conventional approaches when observations are extremely irregularly spaced such that the ratio lambda is not so small and very sparse

generalized langevin equation and the linear regression model with memory because the existence of cheap emulators can give rise to potentially drastic computational savings as compared to direct numerical simulations of complex turbulence models the study of reduced models has good practical relevance and the guidelines regarding their construction from the true dynamical system model are much in demand in accordance with this contemporary trend in science and engineering we provide a new approach for the rigorous derivation of the linear reduced model with memory from the one dimensional majda mclaughlin tabak mmt prototypical wave turbulence in thermal equilibrium the basic idea in obtaining the physics constrained autoregressive model is to perform the discretization in time of the generalized langevin equation gle governing a single wave profile of the true dynamical system model the gle formalized in the mori zwanzig mz projection theory is the exact reduced order equation and the closed form rearrangement of the canonical equation of motion for the hamiltonian system we study the performance of the resulting linear non markovian model in addressing two different manners of uncertainty quantification uq problems prediction and filtering in doing so we perform a comparison analysis against the linear and nonlinear markovian models describing the mmt system which are also built upon the gle under the mz framework finally we discuss an optimal selection of the statistical model in applying the reduced model approach to the uq of the turbulent signal

delay coordinate maps and the spectra of koopman operators the koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system including attractor reconstruction and forecasting koopman eigenfunctions represent the non mixing component of the dynamics they factor the dynamics which can be chaotic into quasiperiodic rotations on tori here we describe a method through which these eigenfunctions can be obtained from a kernel integral operator which also annihilates the continuous spectrum we show that incorporating a large number of delay coordinates in constructing the kernel of that operator results in the limit of infinitely many delays in the creation of a map into the point spectrum subspace of the koopman operator this enables efficient approximation of koopman eigenfunctions in systems with pure point or mixed spectra we illustrate our results with applications to product dynamical systems with mixed spectra

an algebraic method for constructing stable and consistent autoregressive filters in this paper we introduce an algebraic method to construct stable and consistent univariate autoregressive ar models of low order for filtering and predicting nonlinear turbulent signals with memory depth by stable we refer to the classical stability condition for the ar model by consistent we refer to the classical consistency constraints of adams bashforth methods of order two one attractive feature of this algebraic method is that the model parameters can be obtained without directly knowing any training data set as opposed to many standard regression based parameterization methods it takes only long time average statistics as inputs the proposed method provides a discretization time step interval which guarantees the existence of stable and consistent ar model and simultaneously produces the parameters for the ar models in our numerical examples with two chaotic time series with different characteristics of decaying time scales we find that the proposed ar models produce significantly more accurate short term predictive skill and comparable filtering skill relative to the linear regression based ar models these encouraging results are robust across wide ranges of discretization times observation times and observation noise variances finally we also find that the proposed model produces an improved short time prediction relative to the linear regression based ar models in forecasting a data set that characterizes the variability of the madden julian oscillation a dominant tropical atmospheric wave pattern

numerical strategies for filtering partially observed stiff stochastic differential equations in this paper we present a fast numerical strategy for filtering stochastic differential equations with multiscale features this method is designed such that it does not violate the practical linear observability condition and more importantly it does not require the computationally expensive cross correlation statistics between multiscale variables that are typically needed in standard filtering approach the proposed filtering algorithm comprises of a macro filter that borrows ideas from the heterogeneous multiscale methods and a micro filter that reinitializes the fast microscopic variables to statistically reflect the unbiased slow macroscopic estimate obtained from the macro filter and macroscopic observations at asynchronous times we will show that the proposed micro filter is equivalent to solving an inverse problem for parameterizing differential equations numerically we will show that this microscopic reinitialization is an important novel feature for accurate filtered solutions especially when the microscopic dynamics is not mixing at all

c convexity of norm unit balls abstract we characterize the norms on b h whose unit balls are c convex we call such norms m norms and investigate their dual norms named l norms we show that the class of l norms greater than a given norm enjoys a minimum element and the class of m norms less than a given norm enjoys a maximum element these minimum and maximum elements will be determined in some cases finally we give a constructive result to obtain m norms and l norms on b h

using dynamic mode decomposition to extract cyclic behavior in the stock market abstract the presence of cyclic expansions and contractions in the economy has been known for over a century the work reported here searches for similar cyclic behavior in stock valuations the variations are subtle and can only be extracted through analysis of price variations of a large number of stocks koopman mode analysis is a natural approach to establish such collective oscillatory behavior the difficulty is that even non cyclic and stochastic constituents of a finite data set may be interpreted as a sum of periodic motions however deconvolution of these irregular dynamical facets may be expected to be non robust i e to depend on specific data set we propose an approach to differentiate robust and non robust features in a time series it is based on identifying robust features with reproducible koopman modes i e those that persist between distinct sub groupings of the data our analysis of stock data discovered four reproducible modes one of which has period close to the number of trading days year to the best of our knowledge these cycles were not reported previously it is particularly interesting that the cyclic behaviors persisted through the great recession even though phase relationships between stocks within the modes evolved in the intervening period

stability of multiobjective predictive control we propose a utopia tracking strategy to handle multiple conflicting objectives in model predictive control the controller minimizes the distance of its vector of objectives to that of the compromise solution the point along the steady state pareto front closest to the utopia point where all the objectives are independently minimized we establish conditions for asymptotic stability and propose numerical implementation variants one of the key advantages of the approach is that it avoids the computation of pareto fronts in real time environments in addition the approach can handle general objectives of different nature such as economic and regularization

test models for improving filtering with model errors through stochastic parameter estimation the filtering skill for turbulent signals from nature is often limited by model errors created by utilizing an imperfect model for filtering updating the parameters in the imperfect model through stochastic parameter estimation is one way to increase filtering skill and model performance here a suite of stringent test models for filtering with stochastic parameter estimation is developed based on the stochastic parameterization extended kalman filter spekf these new spekf algorithms systematically correct both multiplicative and additive biases and involve exact formulas for propagating the mean and covariance including the parameters in the test model a comprehensive study is presented of robust parameter regimes for increasing filtering skill through stochastic parameter estimation for turbulent signals as the observation time and observation noise are varied and even when the forcing is incorrectly specified the results here provide useful guidelines for filtering turbulent signals in more complex systems with significant model errors

memory effects in recurrent and extreme events a dynamical approach to recurrent and extreme events is developed focusing on the role of correlations and memory in the structure of the probability distributions and their low order moments the procedure is illustrated on homogeneous first and second order markov chains non markovian and nonhomogeneous processes and deterministic dynamical systems substantial differences with classical statistical theory as applied to independent identically distributed random variables are identified

implicit sampling for particle filters we present a particle based nonlinear filtering scheme related to recent work on chainless monte carlo designed to focus particle paths sharply so that fewer particles are required the main features of the scheme are a representation of each new probability density function by means of a set of functions of gaussian variables a distinct function for each particle and step and a resampling based on normalization factors and jacobians the construction is demonstrated on a standard ill conditioned test problem

statistical energy conservation principle for inhomogeneous turbulent dynamical systems understanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science with important societal impact in such inhomogeneous turbulent dynamical systems there is a large dimensional phase space with a large dimension of unstable directions where a large scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other these complex features strongly impact practical prediction and uncertainty quantification a systematic energy conservation principle is developed here in a theorem that precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations this statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence this result applies to general inhomogeneous turbulent dynamical systems including the above applications the theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self contained treatment is presented below corollary 1 and corollary 2 illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds provided that the negative symmetric dissipation matrix is diagonal in a suitable basis implications of the energy principle for low order closure modeling and automatic estimates for the single point variance are discussed below

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

filtering skill for turbulent signals for a suite of nonlinear and linear extended kalman filters the filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model in particular real time filtering and prediction when very limited or no a posteriori analysis is possible e g spread of pollutants storm surges tsunami detection etc introduces a number of additional challenges to the problem here a suite of filters implementing stochastic parameter estimation for mitigating model error through additive and multiplicative bias correction is examined on a nonlinear exactly solvable stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent transient instabilities associated with positive finite time lyapunov exponents to laminar behavior stochastic parameterization extended kalman filter spekf used as a benchmark here involves exact formulas for propagating the mean and covariance of the augmented forecast model including the unresolved parameters the remaining filters use the same nonlinear forecast model but they introduce model error through different moment closure approximations and or linear tangent approximation used for computing the second order statistics of the augmented stochastic forecast model a comprehensive study of filter performance is carried out in the presence of various moment closure errors which are enhanced by additional model errors due to incorrect parameters inducing additive and multiplicative stochastic biases the estimation skill of the unresolved stochastic parameters is also discussed and it is shown that the linear tangent filter despite its popularity is completely unreliable in many turbulent regimes for both parameter estimation and filtering moreover regimes of filter divergence for the linear tangent filter are identified the results presented here provide useful guidelines for filtering turbulent high dimensional spatially extended systems with more general model errors as well as for designing more skillful methods for superparameterization of unresolved intermittent processes in complex multi scale models they also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended kalman filters using incorrect statistics on an exactly solvable test bed with rich and realistic dynamics

lyapunov measure for almost everywhere stability this paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system based upon a stochastic representation of deterministic dynamics a lyapunov measure is proposed for these purposes this measure is shown to be a stochastic counterpart of stability transience just as an invariant measure is a counterpart of the attractor recurrence it is a dual of the lyapunov function and is useful for the study of more general weaker and set wise notions of stability in addition to the theoretical framework constructive methods for computing approximations to the lyapunov measures are presented these methods are based upon set oriented numerical approaches several equivalent descriptions including a series formula and a system of linear inequalities are provided for computational purposes these descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets finally in certain cases the exact relationship between lyapunov functions and lyapunov measures is also given

forecasting turbulent modes with nonparametric diffusion models learning from noisy data abstract in this paper we apply a recently developed nonparametric modeling approach the diffusion forecast to predict the time evolution of fourier modes of turbulent dynamical systems while the diffusion forecasting method assumes the availability of a noise free training data set observing the full state space of the dynamics in real applications we often have only partial observations which are corrupted by noise to alleviate these practical issues following the theory of embedology the diffusion model is built using the delay embedding coordinates of the data we show that this delay embedding biases the geometry of the data in a way which extracts the most stable component of the dynamics and reduces the influence of independent additive observation noise the resulting diffusion forecast model approximates the semigroup solutions of the generator of the underlying dynamics in the limit of large data and when the observation noise vanishes as in any standard forecasting problem the forecasting skill depends crucially on the accuracy of the initial conditions we introduce a novel bayesian method for filtering the discrete time noisy observations which works with the diffusion forecast to determine the forecast initial densities numerically we compare this nonparametric approach with standard stochastic parametric models on a wide range of well studied turbulent modes including the lorenz 96 model in weakly chaotic to fully turbulent regimes and the barotropic modes of a quasi geostrophic model with baroclinic instabilities we show that when the only available data is the low dimensional set of noisy modes that are being modeled the diffusion forecast is indeed competitive to the perfect model

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

numerical approximation of some linear stochastic partial differential equations driven by special additive noises this paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises a special representation of the noise is considered and it is compared with general representations of noises in the infinite dimensional setting convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods the effects of the noises on the accuracy of the approximations are illustrated results of the numerical experiments are provided

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

an effective implementation of the lin kernighan traveling salesman heuristic abstract this paper describes an implementation of the lin kernighan heuristic one of the most successful methods for generating optimal or near optimal solutions for the symmetric traveling salesman problem tsp computational tests show that the implementation is highly effective it has found optimal solutions for all solved problem instances we have been able to obtain including a 13 509 city problem the largest non trivial problem instance solved to optimality today

duality and normal parts of operator modules for an operator bimodule x over von neumann algebras a b h and b b k the space of all completely bounded a b bimodule maps from x into b k h is the bimodule dual of x basic duality theory is developed with a particular attention to the haagerup tensor product over von neumann algebras to x a normal operator bimodule xn is associated so that completely bounded a b bimodule maps from x into normal operator bimodules factorize uniquely through xn a construction of xn in terms of biduals of x a and b is presented various operator bimodule structures are considered on a banach bimodule admitting a normal such structure

controllability for a class of area preserving twist maps abstract in this paper we study controllability of two dimensional integrable twist maps with bounded area preserving time dependent control perturbations in contrast to the time independent perturbation case of the kolmogorov arnold moser theorem there are no invariant sets other than the whole phase space if the perturbation is made a function of time we give necessary and sufficient conditions for global controllability of these maps

a non equilibrium analysis and control framework for active queue management we present a non equilibrium analysis and control approach for the active queue management aqm problem in communication networks using simplified fluid models we carry out a bifurcation study of the complex dynamic queue behavior to show that non equilibrium methods are essential for analysis and optimization in the aqm problem we investigate an ergodic theoretic framework for stochastic modeling of the non equilibrium behavior in deterministic models and use it to identify parameters of a fluid model from packet level simulations for computational tractability we use set oriented numerical methods to construct finite dimensional markov models including control markov chains and hidden markov models subsequently we develop and analyze an example aqm algorithm using a markov decision process mdp based control framework the control scheme developed is optimal with respect to a reward function defined over the queue size and aggregate flow rate we implement and simulate our illustrative aqm algorithm in the ns 2 network simulator the results obtained confirm the theoretical analysis and exhibit promising performance when compared with well known alternative schemes under persistent non equilibrium queue behavior

shortest dubins paths through three points the 3 point dubins problem 3pdp consists of steering a dubins vehicle through three consecutive points with prescribed heading orientations at initial and final points so that the resulting path is the shortest characterizing the path of the 3pdp is important because 1 it provides insightful views on the solution paths of the dubins traveling salesman problem dtsp and the curvature constrained shortest path problem ccspp as they are natural extensions of the 3pdp and 2 some algorithms in the literature for solving the dtsp and the ccspp require efficient methods for solving the 3pdp in this paper pontryagin s maximum principle is used to show that the path of 3pdp must lie in a sufficient family of 18 types moreover a formula in terms of the parameters of the 3pdp for all the 18 types is established and this formula reveals the relationship between the unknown orientation angle at mid point and known parameters by observing that the formula can be converted into some polynomials the 3pdp can therefore be efficiently solved by finding zeros of those polynomials finally numerical simulations illustrate the developments by comparing with the straightforward discretization based method

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

mathematical test criteria for filtering complex systems plentiful observations an important emerging scientific issue is the real time filtering through observations of noisy turbulent signals for complex systems as well as the statistical accuracy of spatio temporal discretizations for such systems these issues are addressed here in detail for the setting with plentiful observations for a scalar field through explicit mathematical test criteria utilizing a recent theory a j majda m j grote explicit off line criteria for stable accurate time filtering of strongly unstable spatially extended systems proceedings of the national academy of sciences 104 4 2007 1124 1129 for plentiful observations the number of observations equals the number of mesh points these test criteria involve much simpler decoupled complex scalar filtering test problems with explicit formulas and elementary numerical experiments which are developed here as guidelines for filter performance the theory includes information criteria to avoid filter divergence with large model errors asymptotic kalman gain filter stability and accurate filtering with small ensemble size as well as rigorous results delineating the role of various turbulent spectra for filtering under mesh refinement these guidelines are also applied to discrete approximations for filtering the stochastically forced dissipative advection equation with very turbulent and noisy signals with either an equipartition of energy or 5 3 turbulent spectrum with infrequent observations as severe test problems the theory and companion simulations demonstrate accurate statistical filtering in this context with implicit schemes with large time step with very small ensemble sizes and even with unstable explicit schemes under appropriate circumstances provided the filtering strategies are guided by the off line theoretical criteria the surprising failure of other strongly stable filtering strategies is also explained through these off line criteria

a factorization problem for normal completely bounded mappings abstract given an operator space x and a von neumann algebra a we consider a contractive mapping q a eh x eh a ncb x a formally defined by q aj xjk bk xjk ajbk from the extended haagerup tensor product a eh x eh a into the space of w continuous completely bounded maps from x into a we characterize elements of the range space im q by a factorization property involving decomposable operators and investigate various properties of that space in the case when x b is the predual of a von neumann algebra b im q is included in the space dec b a of decomposable operators from b into a regarding q as having values in that space we show that q is a quotient map onto its range then we prove that dec b a is a normal dual operator a bimodule and that im q dec b a is a strong operator a bimodule

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

quantifying predictability through information theory small sample estimation in a non gaussian framework many situations in complex systems require quantitative estimates of the lack of information in one probability distribution relative to another in short term climate and weather prediction examples of these issues might involve the lack of information in the historical climate record compared with an ensemble prediction or the lack of information in a particular gaussian ensemble prediction strategy involving the first and second moments compared with the non gaussian ensemble itself the relative entropy is a natural way to quantify the predictive utility in this information and recently a systematic computationally feasible hierarchical framework has been developed in practical systems with many degrees of freedom computational overhead limits ensemble predictions to relatively small sample sizes here the notion of predictive utility in a relative entropy framework is extended to small random samples by the definition of a sample utility a measure of the unlikeliness that a random sample was produced by a given prediction strategy the sample utility is the minimum predictability with a statistical level of confidence which is implied by the data two practical algorithms for measuring such a sample utility are developed here the first technique is based on the statistical method of null hypothesis testing while the second is based upon a central limit theorem for the relative entropy of moment based probability densities these techniques are tested on known probability densities with parameterized bimodality and skewness and then applied to the lorenz 96 model a recently developed toy climate model with chaotic dynamics mimicking the atmosphere the results show a detection of non gaussian tendencies of prediction densities at small ensemble sizes with between 50 and 100 members with a 95 confidence level

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

on the dubins traveling salesman problem we study the traveling salesman problem for a dubins vehicle we prove that this problem is np hard and provide lower bounds on the approximation ratio achievable by some recently proposed heuristics we also describe new algorithms for this problem based on heading discretization and evaluate their performance numerically

reconstruction of normal forms by learning informed observation geometries from data abstract the discovery of physical laws consistent with empirical observations is at the heart of applied science and engineering these laws typically take the form of nonlinear differential equations depending on parameters dynamical systems theory provides through the appropriate normal forms an intrinsic prototypical characterization of the types of dynamical regimes accessible to a given model using an implementation of data informed geometry learning we directly reconstruct the relevant normal forms a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics interestingly the state variables and the parameters of these realizations are inferred from the empirical observations without prior knowledge or understanding they parametrize the dynamics intrinsically without explicit reference to fundamental physical quantities