poincare bendixson theory for leaves of codimension one the level of a local minimal set of a c2 codimension one foliation of a compact manifold is a nonnegative integer defined inductively level zero corresponding to the minimal sets in the usual sense each leaf of a local minimal set at level k is at level k the authors develop a theory of local minimal sets level and how leaves at level k asymptotically approach leaves at lower level this last generalizes the classical poincare bendixson theorem and provides information relating growth topological type and level e g if l is a totally proper leaf at level k then l has exactly polynomial growth of degree k and topological type k

state estimation with partially observed inputs a unified kalman filtering approach for linear stochastic time varying state space models with gaussian noises this paper investigates state estimation for the scenario where the input variables of the state equation are not fully observed but rather the input data are available only at an aggregate level unlike the existing filters for unknown inputs that are based on the approach of minimum variance unbiased estimation this paper does not impose the unbiasedness condition for state estimation instead it incorporates a bayesian approach to derive a modified kalman filter by pooling the prior knowledge about the state vector at the aggregate level with the measurements on the output variables at the original level of interest the estimated state vector is shown to be a minimum mean square error estimator the developed filter provides a unified approach to state estimation it includes the existing filters obtained under two extreme scenarios as its special cases i e the classical kalman filter where all the inputs are observed and the filter for unknown inputs

best approximation in c algebras on developpe des methodes permettant d obtenir les meilleures approximations des ideaux de sous algebres et de sous espaces des c algebres

failures of sequential bayesian filters and the successes of shadowing filters in tracking of nonlinear deterministic and stochastic systems sequential bayesian filters such as particle filters are often presented as an ideal means of tracking the state of nonlinear systems here shadowing filters are demonstrated to perform better than sequential filters at tracking under specific circumstances the success of shadowing filters is attributed to avoiding both well known deficiencies of particle filters and some newly identified problems

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

semi crossed products of c algebras abstract given a c algebra u and endomorphim there is an associated nonselfadjoint operator algebra z x u called the semi crossed product of u with if is an automorphim z x u can be identified with a subalgebra of the c crossed product z x u if u is commutative and is an automorphim satisfying certain conditions z x u is an operator algebra of the type studied by arveson and josephson suppose s is a locally compact hausdorff space s s is a continuous and proper map and is the endomorphim of u c0 s given by necessary and sufficient conditions on the map are given to insure that the semi crossed product z x c0 s is i semiprime ii semisimple ii strongly semisimple

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

a stochastic version of center manifold theory random dynamical systems arise naturally if the influence of white or real noise on the parameters of a nonlinear determinstic dynamical system is studied in this situation lyapunov exponents attached to the linearized flow replace the real parts of the eigenvalues and describe the stability behavior of the linear system if at least one of them vanishes then it is possible to prove the existence of a stochastic analogue of the deterministic center manifold the asymptotic behavior of the entire system can then be derived from the lower dimensional system restricted to this stochastic center manifold a dynamical characterization of the stochastic center manifold is given and approximation results are proved

an interlaced extended kalman filter an estimation algorithm for a class of discrete time nonlinear systems is proposed the system structure we deal with is partitionable into in subsystems each affine w r t the corresponding part of the state vector the algorithm consists of a bank of m interlaced kalman filters and each of them estimates a part of the state considering the remaining parts as known time varying parameters whose values are evaluated by the other filters at the previous step the procedure neglects the subsystem coupling terms in the covariance matrix of the state estimation error and counteracts the errors so introduced by suitably increasing the noise covariance matrices comparisons through numerical simulations with the extended kalman filter and its modified versions proposed in the literature illustrate the good trade off provided by the algorithm between the reduction of the computational load and the estimation accuracy

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

annuli in generalized heegaard splittings and degeneration of tunnel number abstract we analyze how a family of essential annuli in a compact 3 manifold will induce from a strongly irreducible generalized heegaard splitting of the ambient manifold generalized heegaard splittings of the complementary components there are specific applications to the subadditivity of tunnel number of knots improving somewhat bounds of kowng kw for example in the absence of 2 bridge summands the tunnel number of the sum of n knots is no less than frac 2 5 the sum of the tunnel numbers

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

state and parameter estimation in stochastic dynamical models abstract this paper derives generalized maximum likelihood estimates of state and model parameters of a stochastic dynamical model in contrast to previous studies the change in background distribution due to changes in model parameters is taken into account an ensemble approach to solving the maximum likelihood estimates is proposed an exact solution for the ensemble update based on a square root kalman filter is derived this solution involves a two step procedure in which an ensemble is first produced by a standard ensemble kalman filter and then corrected to account for parameter estimation thereby allowing a user to take advantage of an existing ensemble filter the solution is illustrated with simple low dimensional stochastic dynamical models and shown to work well and outperform augmentation methods for estimating stochastic parameters

general two stage extended kalman filters a general two stage extended kalman filter gtsekf which extends the linear general two stage kalman filter to nonlinear systems is further proposed a new nonlinear two stage transformation is introduced to facilitate achieving this extension as in the linear one the gtsekf is derived mainly by applying the nonlinear two stage transformation to the well known extended kalman filter ekf and is shown to be equivalent to the ekf with a decoupled computing structure a nonlinear filter for estimating constant parameters in dynamic systems is presented to illustrate one application of the proposed gtsekf a literature example is also given to demonstrate the correctness and usefulness of the proposed results

semicrossed products of operator algebras and their c envelopes abstract let a be a unital operator algebra and let be an automorphism of a that extends to a automorphism of its c envelope c env a we introduce the isometric semicrossed product a is z and we show that c env a is z c env a z in contrast the c envelope of the familiar contractive semicrossed product a z may not equal c env a z our main tool for calculating c envelopes for semicrossed products is a new concept the relative semicrossed product of an operator algebra by an endomorphism as an application of our theory we show that if t x is the tensor algebra of a c correspondence x a and is a extendible automorphism of t x that fixes the diagonal elementwise then the contractive semicrossed product satisfies c env t x z o x z where o x denotes the cuntz pimsner algebra of x a this extends the main result of davidson and katsoulis 2010 6

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

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

adaptive error covariances estimation methods for ensemble kalman filters this paper presents a computationally fast algorithm for estimating both the system and observation noise covariances of nonlinear dynamics that can be used in an ensemble kalman filtering framework the new method is a modification of belanger s recursive method to avoid an expensive computational cost in inverting error covariance matrices of product of innovation processes of different lags when the number of observations becomes large when we use only product of innovation processes up to one lag the computational cost is indeed comparable to a recently proposed method by berry sauer s however our method is more flexible since it allows for using information from product of innovation processes of more than one lag extensive numerical comparisons between the proposed method and both the original belanger s and berry sauer s schemes are shown in various examples ranging from low dimensional linear and nonlinear systems of sdes and 40 dimensional stochastically forced lorenz 96 model our numerical results suggest that the proposed scheme is as accurate as the original belanger s scheme on low dimensional problems and has a wider range of more accurate estimates compared to berry sauer s method on l 96 example

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

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

strong tracking filtering of nonlinear time varying stochastic systems with coloured noise application to parameter estimation and empirical robustness analysis the strong tracking filter stf proposed by zhou et al in 1992 which was developed for nonlinear systems with white noise is extended to a class of nonlinear time varying stochastic systems with coloured noise a new concept of softening factor is introduced to make the state estimator much smoother its value can be preselected by computer simulations via a heuristic searching scheme the stf is then used to estimate the parameters of a class of nonlinear time varying stochastic systems in the presence of coloured noise the robustness against model uncertainty of the stf is thoroughly studied via monte carlo simulations the results show that the stf has strong robustness against model plant parameter mismatches in the statistics of the initial conditions the statistics of the process noise and the measurement noise the system parameters and the parameters in the measurement noise model to a great extent the stf can give bias free parameter estimations where the parameters may be randomly time

optimal solution of the two stage kalman estimator the two stage kalman estimator was originally proposed to reduce the computational complexity of the augmented state kalman filter it was also applied to the tracking of maneuvering targets by treating the target acceleration as a bias term except in certain restrictive conditions the conventional two stage estimators are suboptimal in the sense that they are not equivalent to the augmented state filter in this paper the authors propose a new two stage kalman estimator i e new structure which is an extension of friedland s estimator and is optimal in general conditions in addition we provide some analytic results to demonstrate the computational advantages of two stage estimators over augmented ones

speed and rotor flux estimation of induction machines using a two stage extended kalman filter this paper presents an effective implementation of an extended kalman filter used for the estimation of both rotor flux and rotor velocity of an induction motor an algorithm proposed by hsieh and chen in hsieh c s chen f c 1999 optimal solution of the two stage kalman estimator ieee transactions on automatic control 44 1 194 199 for linear parameter estimation is extended to non linear estimation where parameters such as the velocity of an induction machine are present in the transition matrix and in the augmented state space compared to a straightforward implementation of an extended kalman filter our modified optimal two stage kalman estimator reduces the number of arithmetic operations by 25 allowing higher sampling rate or the use of a cheaper microcontroller

semisimple triangular af algebras abstract we give several necessary and sufficient conditions for a triangular af algebra to be semisimple in particular a triangular af algebra which can be written using the standard embedding infinitely often is semisimple we also give a semisimple triangular af algebra which does not have a presentation of this form if two triangular af algebras have the same peters poon wagner diagonal invariant then either both are semisimple or both are not however we give two algebras with the same diagonal invariant where one has the jacobson radical equal to the strong radical and the other does not semisimplicity can be characterized in terms of power s fundamental relation we give generalizations of these results to triangular subalgebras of groupoid c algebras it is shown that there is a unique maximal bimodule over the diagonal which does not intersect the radical using this we show that the wedderburn principal theorem does not hold for all triangular af algebras a necessary condition for the theorem to hold is that the above mentioned unique bimodule is a subalgebra

extension of friedland s separate bias estimation to randomly time varying bias of nonlinear systems by extending b friedland s 1969 separate bias estimation algorithm for linear systems to nonlinear systems and combining the result with the suboptimal fading extended kalman filter proposed by d h zhou 1990 and by d h zhou et al a pseudo separate bias estimation algorithm for randomly time varying bias of a class of nonlinear time varying stochastic systems is obtained a simulation example is presented to illustrate the effectiveness of the algorithm

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

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

estimating sudden changes of biases in linear dynamic systems a procedure is presented for estimating sudden changes in the bias vector in a linear dynamic system this procedure is the result of combining the recently developed method 1 2 of tracking a piecewise constant vector observed directly through additive noise with the method 3 for separating the estimation of the bias in a linear system from the estimation of the dynamic state variables the application of the procedure is illustrated by a second order dynamic system with two biases subject to transition its performance is demonstrated by a simulation

an alternate derivation and extension of friendland s two stage kalman estimator an alternate simplified derivation of friedland s two stage kalman estimator is given for a somewhat more general class of problems than considered by friedland friedland s result is also extended to encompass two variations on the basic idea which are of practical interest

probabilistic description of extreme events in intermittently unstable dynamical systems excited by correlated stochastic processes massachusetts institute of technology naval engineering education center grant 3002883706

on the optimality of two stage state estimation in the presence of random bias sufficient conditions for the optimality of a two stage state estimator in the presence of random bias are derived under an algebraic constraint on the correlation between the state and bias process noises the optimal estimate of the system state can be obtained as a linear combination of the output of the first stage a bias free filter and the second stage a bias filter because the algebraic constraint is restrictive in practice the results indirectly indicate that for most practical systems the proposed solution to the two stage estimation problem will be suboptimal

c operator systems and crossed products abstract the purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces in particular we shall put emphasis to reveal the full power of the universal properties of the universal crossed products it turns out to make things consistent one has to perform the constructions in some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non discrete groups in the case of operator systems this larger category is what we call a c operator system i e a selfadjoint subspace x of some b h which contains a c algebra a such that a x x x a in the case of operator spaces the larger category is given by what we call c operator bimodules we introduce the notion of crossed products in these categories and show that the classical imai takai and katayama duality theorems for crossed products by group co actions on c algebras extend to our notion of crossed products by c operator systems and c operator bimodules

perturbation theory for groups and lattices abstract a generalization of a theorem of dan voiculescu on perturbations of separable c algebras is proved this is applied to solve two problems relating to the perturbation theory of unitary group representations and of commutative subspace lattices the latter generalizes a theorem of niels toft andersen on compact perturbations of nests

introduction to stochastic processes preface to second edition preface to first edition preliminaries introduction linear differential equations linear difference equations exercises finite markov chains definitions and examples large time behavior and invariant probability classification of states return times transient states examples exercises countable markov chains introduction recurrence and transience positive recurrence and null recurrence branching process exercises continuous time markov chains poisson process finite state space birth and death processes general case exercises optimal stopping optimal stopping of markov chains optimal stopping with cost optimal stopping with discounting exercises martingales conditional expectation definition and examples optional sampling theorem uniform integrability martingale convergence theorem maximal inequalities exercises renewal processes introduction renewal equation discrete renewal processes m g 1 and g m 1 queues exercises reversible markov chains reversible processes convergence to equilibrium markov chain algorithms a criterion for recurrence exercises brownian motion introduction markov property zero set of brownian motion brownian motion in several dimensions recurrence and transience fractal nature of brownian motion scaling rules brownian motion with drift exercises stochastic integration integration with respect to random walk integration with respect to brownian motion ito s formula extensions if ito s formula continuous martingales girsanov transformation feynman kac formula black scholes formula simulation exercises suggestions for further reading index

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

manifold learning with contracting observers for data driven time series analysis analyzing signals arising from dynamical systems typically requires many modeling assumptions in high dimensions this modeling is particularly difficult due to the curse of dimensionality in this paper we propose a method for building an intrinsic representation of such signals in a purely data driven manner first we apply a manifold learning technique diffusion maps to learn the intrinsic model of the latent variables of the dynamical system solely from the measurements second we use concepts and tools from control theory and build a linear contracting observer to estimate the latent variables in a sequential manner from new incoming measurements the effectiveness of the presented framework is demonstrated by applying it to a toy problem and to a music analysis application in these examples we show that our method reveals the intrinsic variables of the analyzed dynamical systems

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

knot floer homology detects genus one fibred knots ozsv ath and szab o conjectured that knot floer homology detects fibred knots we propose a strategy to approach this conjecture based on contact topology and gabai s theory of sutured manifold decomposition we implement this strategy for genus one knots obtaining as a corollary that if rational surgery on a knot k gives the poincar e homology sphere sigma 2 3 5 then k is the left handed trefoil knot

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

crossed products of operator algebras applications of takai duality abstract let g be a partially ordered abelian group with haar measure let a g be a dynamical system and let a be the associated semicrossed product using takai duality we establish a stable isomorphism a s a k g ad g where k g denotes the compact operators in the csl algebra alg l g and denotes the right regular representation of g we also show that there exists a complete lattice isomorphism between the invariant ideals of a and the ad invariant ideals of a k g using takai duality we also continue our study of the radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra we also show that the permanence of semisimplicity fails for crossed products by r a final section of the paper is devoted to the study of radically tight dynamical systems i e dynamical systems a g for which the identity rad a g rad a g persists a broad class of such dynamical systems is identified

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

the ideal structure of certain nonselfadjoint operator algebras soit x un systeme dynamique localement compact et z x co x la sous algebre close en norme du produit croise zx co x engendre par les puissances non negatives de dans le cas homeomorphisme on determine la structure ideale de ces algebres dans le cas ou agit librement

isomorphisms between topological conjugacy algebras a family of algebras which we call topological conjugacy algebras is associated with each proper continuous map on a locally compact hausdorff space assume that i xi xi is a continuous proper map on a locally compact hausdorff space xi for i 1 2 we show that the dynamical systems x1 1 and x2 2 are conjugate if and only if some topological conjugacy algebra of x1 1 is isomorphic as an algebra to some topological conjugacy algebra of x2 2 this implies as a corollary the complete classification of the semicrossed products c0 x z which was previously considered by arveson and josephson 3 peters 19 hadwin and hoover 8 and power 21 we also obtain a complete classification of all semicrossed products of the form a d z where a d denotes the disc algebra and d d a continuous map which is analytic on the interior in this case a surprising dichotomy appears in the classification scheme which depends on the fixed point set of we also classify more general semicrossed products of uniform algebras

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

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

wiener filter design using polynomial equations a simplified way of deriving realizable and explicit wiener filters is presented discrete time problems are discussed in a polynomial equation framework optimal filters predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations a tool for obtaining these equations for a given problem structure is described it is based on the evaluation of orthogonality in the frequency domain by means of canceling stable poles with zeros comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical wiener solution the simplicity of the proposed derivation method is particularly evident in multistage filtering problems to illustrate two examples are discussed a filtering and a generalized deconvolution problem a new solvability condition for linear polynomial equation appearing in scalar problems is also presented

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

the hyperrigidity of tensor algebras of c correspondences abstract given a c correspondence x we give necessary and sufficient conditions for the tensor algebra t x to be hyperrigid in the case where x is coming from a topological graph we obtain a complete characterization

the jacobson radical for analytic crossed products we characterise the jacobson radical of an analytic crossed product c0 x z answering a question first raised by arveson and josephson in 1969 in fact we characterise the jacobson radical of analytic crossed products c0 x zd this consists of all elements whose fourier coefficients vanish on the recurrent points of the dynamical system and the first one is zero the multidimensional version requires a variation of the notion of recurrence taking into account the various degrees of freedom

3 manifolds with planar presentations and the width of satellite knots we consider compact 3 manifolds m having a submersion h to r in which each generic point inverse is a planar surface the standard height function on a submanifold of s 3 is a motivating example to m h we associate a connectivity graph for m s 3 is a tree if and only if there is a fox reimbedding of m which carries horizontal circles to a complete collection of complementary meridian circles on the other hand if the connectivity graph of s 3 m is a tree then there is a level preserving reimbedding of m so that s 3 m is a connected sum of handlebodies corollary the width of a satellite knot is no less than the width of its pattern knot and so w k 1 k 2 max w k 1 w k 2

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

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

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

nonparametric forecasting of low dimensional dynamical systems this paper presents a nonparametric modeling approach for forecasting stochastic dynamical systems on low dimensional manifolds the key idea is to represent the discrete shift maps on a smooth basis which can be obtained by the diffusion maps algorithm in the limit of large data this approach converges to a galerkin projection of the semigroup solution to the underlying dynamics on a basis adapted to the invariant measure this approach allows one to quantify uncertainties in fact evolve the probability distribution for nontrivial dynamical systems with equation free modeling we verify our approach on various examples ranging from an inhomogeneous anisotropic stochastic differential equation on a torus the chaotic lorenz three dimensional model and the ni no 3 4 data set which is used as a proxy of the el ni no southern oscillation

unbiased minimum variance estimation for systems with unknown exogenous inputs a new method is developed for the state estimation of linear discrete time stochastic systems in the presence of an unknown disturbance the filter obtained is optimal in the unbiased minimum variance sense the necessary and sufficient conditions for the existence and the stability of the filter are given

a strong tracking extended kalman observer for nonlinear discrete time systems the authors show how the extended kalman filter used as an observer for nonlinear discrete time systems or extended kalman observer eko becomes a useful state estimator when the arbitrary matrices namely r sub k and q sub k are adequately chosen as a first step we use the linearization technique given by boutayed et al 1997 which consists of introducing unknown diagonal matrices to take the approximation errors into account it is shown that the decreasing lyapunov function condition leads to a linear matrix inequality lmi problem which points out the connection between a good convergence behavior of the eko and the instrumental matrices r sub k and q sub k in order to satisfy the obtained lmi a particular design of q sub k is given high performances of the proposed technique are shown through numerical examples under the worst conditions

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

semicrossed products of simple c algebras let mathcal a alpha and mathcal b beta be c dynamical systems and assume that mathcal a is a separable simple c algebra and that and are automorphisms then the semicrossed products mathcal a times alpha mathbb z and mathcal b times beta mathbb z are isometrically isomorphic if and only if the dynamical systems mathcal a alpha and mathcal b beta are outer conjugate

a polynomial approach to bias aware fixed lag smoothing problem in the present technical note a solution to the problem of fixed lag smoothing of siso systems in presence of a dynamical bias is presented in a polynomial framework the bias aware smoothing problem is solved in three steps namely the design of the general structure of the smoother the estimation of the dynamical bias by means of a deconvolution technique and then the combination of the previous results to obtain the bias aware fixed lag smoother applied to an example this approach shows its efficiency

parameter estimation for multiscale diffusions we study the problem of parameter estimation for time series possessing two widely separated characteristic time scales the aim is to understand situations where it is desirable to fit a homogenized single scale model to such multiscale data we demonstrate numerically and analytically that if the data is sampled too finely then the parameter fit will fail in that the correct parameters in the homogenized model are not identified we also show numerically and analytically that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly the ideas are studied in the context of thermally activated motion in a two scale potential however the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data

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

filtering a statistically exactly solvable test model for turbulent tracers from partial observations a statistically exactly solvable model for passive tracers is introduced as a test model for the authors nonlinear extended kalman filter nekf as well as other filtering algorithms the model involves a gaussian velocity field and a passive tracer governed by the advection diffusion equation with an imposed mean gradient the model has direct relevance to engineering problems such as the spread of pollutants in the air or contaminants in the water as well as climate change problems concerning the transport of greenhouse gases such as carbon dioxide with strongly intermittent probability distributions consistent with the actual observations of the atmosphere one of the attractive properties of the model is the existence of the exact statistical solution in particular this unique feature of the model provides an opportunity to design and test fast and efficient algorithms for real time data assimilation based on rigorous mathematical theory for a turbulence model problem with many active spatiotemporal scales here we extensively study the performance of the nekf which uses the exact first and second order nonlinear statistics without any approximations due to linearization the role of partial and sparse observations the frequency of observations and the observation noise strength in recovering the true signal its spectrum and fat tail probability distribution are the central issues discussed here the results of our study provide useful guidelines for filtering realistic turbulent systems with passive tracers through partial observations

a separate bias u d factorization filter two forms of friedland s separate bias estimation algorithm with u d factorization of the covariance matrices are provided each is suited to implementation in a particular computing environment we consider matlab and compiled computer languages we reduce the computation time substantially primarily at the time propagation stage by using a separated bias formulation while retaining the desirable numerical properties of the u d factorization the perecentage reduction typically increases with ratio of bias state dimension to dynamic state dimension a numerical evaluation is given for the matlab algorithm

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