additive units of product systems we introduce the notion of additive units or addits of a pointed arveson system and demonstrate their usefulness through several applications by a pointed arveson system we mean a spatial arveson system with a fixed normalised reference unit we show that the addits form a hilbert space whose codimension one subspace of roots is isomorphic to the index space of the arveson system and that the addits generate the type i part of the arveson system consequently the isomorphism class of the hilbert space of addits is independent of the reference unit the addits of a pointed inclusion system are shown to be in natural correspondence with the addits of the generated pointed product system the theory of amalgamated products is developed using addits and roots and an explicit formula for the amalgamation of pointed arveson systems is given providing a new proof of its independence of the particular reference units this independence justifies the terminology spatial product of spatial arveson systems finally a cluster construction for inclusion subsystems of an arveson system is introduced and we demonstrate its correspondence with the action of the cantor bendixson derivative in the context of the random closed set approach to product systems due to tsirelson and liebscher

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

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

linear stability analysis of parallel shear flows for an inviscid generalized two dimensional fluid system the linear stability of parallel shear flows for an inviscid generalized two dimensional 2d fluid system the so called turbulence system is studied this system is characterized by the relation q 2 between the advected scalar q and the stream function here is a real number not exceeding 3 and q is referred to as the generalized vorticity in this study a sufficient condition for linear stability of parallel shear flows is derived using the conservation of wave activity a stability analysis is then performed for a sheet vortex that violates the stability condition the instability of a sheet vortex in the 2d euler system 2 is referred to as a kelvin helmholtz kh instability such an instability for the generalized 2d fluid system is investigated for 0 3 the sheet vortex is unstable in the sense that a sinusoidal perturbation applied to it grows exponentially with time the growth rate is finite and depends on the wavenumber of the perturbation as k3 for 1 3 where k is the wavenumber of the perturbation in contrast for 0 1 the growth rate is infinite in other words a transition of the growth rate of the perturbation occurs at 1 a physical model for kh instability in the generalized 2d fluid system which can explain the transition of the growth rate of the perturbation at 1 is proposed

effective degrees of nonlinearity in a family of generalized models of two dimensional turbulence we study the small scale behavior of generalized two dimensional turbulence governed by a family of model equations in which the active scalar theta delta alpha 2 psi is advected by the incompressible flow u psi y psi x here psi is the stream function delta is the laplace operator and alpha is a positive number the dynamics of this family are characterized by the material conservation of theta whose variance theta2 is preferentially transferred to high wave numbers direct transfer as this transfer proceeds to ever smaller scales the gradient nublatheta grows without bound this growth is due to the stretching term nablatheta nabla u whose effective degree of nonlinearity differs from one member of the family to another this degree depends on the relation between the advecting flow u and the active scalar theta i e on alpha and is wide ranging from approximately linear to highly superlinear linear dynamics are realized when nablau is a quantity of no smaller scales than theta so that it is insensitive to the direct transfer of the variance of theta which is nearly passively advected this case corresponds to alpha or 2 for which the growth of nablatheta is approximately exponential in time and nonaccelerated for alpha 2 superlinear dynamics are realized as the direct transfer of theta2 entails a growth in nablau thereby enhancing the production of nablatheta this superlinearity reaches the familiar quadratic nonlinearity of three dimensional turbulence at alpha 1 and surpasses that for alpha 1 the usual vorticity equation alpha 2 is the border line where nablau and theta are of the same scale separating the linear and nonlinear regimes of the small scale dynamics we discuss these regimes in detail with an emphasis on the locality of the direct transfer

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

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

an index theory for quantum dynamical semigroups w arveson showed a way of associating continuous tensor product systems of hilbert spaces with endomorphism semigroups of type i factors we do the same for general quantum dynamical semigroups through a dilation procedure the product system so obtained is the index and its dimension is a numerical invariant for the original semigroup

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

discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics many physical systems are described by nonlinear differential equations that are too complicated to solve in full a natural way to proceed is to divide the variables into those that are of direct interest and those that are not formulate solvable approximate equations for the variables of greater interest and use data and statistical methods to account for the impact of the other variables in the present paper we consider time dependent problems and introduce a fully discrete solution method which simplifies both the analysis of the data and the numerical algorithms the resulting time series are identified by a narmax nonlinear autoregression moving average with exogenous input representation familiar from engineering practice the connections with the mori zwanzig formalism of statistical physics are discussed as well as an application to the lorenz 96 system

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

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

green s function for a generalized two dimensional fluid a green s function for a generalized two dimensional 2d fluid in an unbounded domain the so called turbulence system is discussed the generalized 2d fluid is characterized by a relationship between an advected quantity q and the stream function namely q 2 here is a real number and q is referred to as the vorticity in this study the green s function refers to the stream function produced by a delta functional distribution of q i e a point vortex with unit strength the green s function has the form g r r 2 except when is an even number where r is the distance from the point vortex this functional form is known as the riesz potential when is a positive even number the logarithmic correction to the riesz potential has the form g r r 2 in r in contrast when is a negative even number g is given by the higher order laplacian of the delta function the transition of the small scale behavior of q at 2 a well known property of forced and dissipative turbulence is explained in terms of the green s function moreover the azimuthal velocity around the point vortex is derived from the green s function the functional form of the azimuthal velocity indicates that physically realizable systems for the generalized 2d fluid exist only when 3 the green s function and physically realizable systems for an anisotropic generalized 2d fluid are presented as an application of the present study

type i product systems of hilbert modules abstract christensen and evans showed that in the language of hilbert modules a bounded derivation on a von neumann algebra with values in a two sided von neumann module i e a sufficiently closed two sided hilbert module are inner then they use this result to show that the generator of a normal uniformly continuous completely positive cp semigroup on a von neumann algebra decomposes into a suitably normalized cp part and a derivation like part the backwards implication is left open in these notes we show that both statements are equivalent among themselves and equivalent to a third one namely that any type i tensor product system of von neumann modules has a unital central unit from existence of a central unit we deduce that each such product system is isomorphic to a product system of time ordered fock modules we thus find the analogue of arveson s result that type i product systems of hilbert spaces are symmetric fock spaces on the way to our results we have to develop a number of tools interesting on their own right inspired by a very similar notion due to accardi and kozyrev we introduce the notion of semigroups of completely positive definite kernels cpd semigroups being a generalization of both cp semigroups and schur semigroups of positive definite c valued kernels the structure of a type i system is determined completely by its associated cpd semigroup and the generator of the cpd semigroup replaces arveson s covariance function as another tool we give a complete characterization of morphisms among product systems of time ordered fock modules in particular the concrete form of the projection endomorphisms allows us to show that subsystems of time ordered systems are again time ordered systems and to find a necessary and sufficient criterion when a given set of units generates the whole system as a byproduct we find a couple of characterizations of other subclasses of morphisms we show that the set of contractive positive endomorphisms are order isomorphic to the set of cpd semigroups dominated by the cpd semigroup associated with type i system

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

on the product system of a completely positive semigroup abstract given a w continuous semigroup of unital normal completely positive maps of b h we introduce its continuous tensor product system e if is a minimal dilation e 0 semigroup of with arveson product system f then e is canonically isomorphic to f we apply this construction to a class of semigroups of b l 2 r arising from a modified weyl moyal quantization of convolution semigroups of borel probability measures on r 2 this class includes the heat flow on the ccr algebra studied recently by arveson we prove that the minimal dilations of all such semigroups are completely spatial and additionally we prove that the minimal dilation of the heat flow is cocyle conjugate to the car ccr flow of index two

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

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

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

an efficient algorithm for estimating noise covariances in distributed systems we present an efficient computational algorithm for estimating the noise covariance matrices of large linear discrete stochasticdynamic systems such systems arise typically by discretizing distributed parameter systems and their size renders computational efficiency a major consideration our adaptive filtering algorithm is based on the ideas of belanger and is algebraically equivalent to his algorithm the earlier algorithm however has computational complexity proportional to p6 where p is the number of observations of the system state while the new algorithm has complexity proportional to only p3 furthermore our formulation of noise covariance estimation as a secondary filter analogous to state estimation as a primary filter suggests several generalizations of the earlier algorithm the performance of the proposed algorithm is demonstrated for a distributed system arising in numerical weather prediction

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

growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids the growth rates of scalar gradients are studied numerically and analytically in a family of two dimensional 2d incompressible fluid equations related to the surface quasigeostrophic sqg equation the active scalar is related to the stream function by 2 0 2 a notable difference is observed in a comparison of the instantaneous growth rates in l p and in l norms depending on the stage of the time evolution the crux is the phase shift effect of singular integral operators which displaces the peak location of the scalar gradient from that of the strain rate on this basis a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow up moreover it is found in the long time evolution that a solution of the sqg equation whose regularity is not known is less singular than that of the 2d euler equations known to be regular on the time interval covered by this computation this consistently expands an earlier observation by majda and tabak physica d 98 515 1996 in some detail a 1d model problem is discussed to illustrate the present method and extensions to the 3d case are also are briefly discussed

the spatial product of arveson systems is intrinsic abstract we prove that the spatial product of two spatial arveson systems is independent of the choice of the reference units this also answers the same question for the minimal dilation of the powers sum of two spatial cp semigroups it is independent up to cocycle conjugacy

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