measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

nonlinear analysis of river flow time sequences within the field of chaos theory several methods for the analysis of complex dynamical systems have recently been proposed in light of these ideas we study the dynamics which control the behavior over time of river flow investigating the existence of a low dimension deterministic component the present article follows the research undertaken in the work of porporato and ridolfi 1996a in which some clues as to the existence of chaos were collected particular emphasis is given here to the problem of noise and to nonlinear prediction with regard to the latter the benefits obtainable by means of the interpolation of the available time series are reported and the remarkable predictive results attained with this nonlinear method are shown

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

estimating dimension from small samples abstract details of a new algorithm to estimate the dimension of a measure from point samples are described the new algorithm emphasizes that dimension need not be the same at all scales that is there need not be a scaling region the algorithm is verified with gaussian measures and is found to be reliable and have less demanding data requirements than conventional estimators using certain convergence properties of gaussian measures it is shown that good estimates can be made from very small samples

noise robust estimates of correlation dimension and k2 entropy using gaussian kernels to define the correlation sum we derive simple formulas that correct the noise bias in estimates of the correlation dimension and k2 entropy of chaotic time series the corrections are only based on the difference of correlation dimensions for adjacent embedding dimensions and hence preserve the full functional dependencies on both the scale parameter and embedding dimension it is shown theoretically that the estimates which are derived for additive white gaussian noise are also robust for moderately colored noise simulations underline the usefulness of the proposed correction schemes it is demonstrated that the method gives satisfactory results also for non gaussian and dynamical noise

the efficient rotational pressure correction schemes for the coupling stokes darcy problem abstract in this paper the rotational pressure correction methods for the stokes darcy system are developed and analyzed the central advantage of these methods is a time dependent version of domain decomposition these methods have first order second order accuracy without the incompressibility constraint of the stokes darcy system their main feature is the implementation efficiency in that we only solve one vector valued elliptic equation and one scalar valued poisson equation for the stokes equations per time step the unconditional stability and long time stability are established and numerical experiments are also presented to show their performance

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

on perceptual distortion measures and parametric modeling over the past two decades there has been much interest in incorporating human sound perception in signal processing algorithms often in the form of perceptual distortion measure or an approximation thereof an example of this is mdct based audio coding where a good quality can be achieved at very low bit rates by taking masking effects into account more recently the same principles have been applied to parametric modeling and coding of audio signals we discuss the inherent tradeoffs in choosing a perceptual distortion measure and a parametric model and the pros and cons of various ways of implementing such perceptual distortion measures is discussed an important question that we seek to answer is whether perception should be taken into account in the estimation of model parameters or this should be done in a separate step

subgrid stabilized projection method for 2d unsteady flows at high reynolds numbers a subgrid stabilization technique is developed for solving the two dimensional incompressible navier stokes equations at high reynolds numbers the time marching algorithm is based on a well established fractional step pressure correction projection method the advection diffusion step is enriched by an implicit subgrid stabilizing term and by an explicit dissipative shock capturing term the former is calculated by means of a hierarchical finite element setting the latter is included to avoid gibbs phenomenon in the boundary layer convergence tests on prototypical two dimensional examples are reported and the method is used to simulate the viscous incompressible flows around the airfoil naca0012 at zero incidence and reynolds numbers ranging from 105 to 106

mixed stabilized finite element methods based on backward difference adams bashforth scheme for the time dependent variable density incompressible flows in this paper we consider a second order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows the originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal order finite element pairs a second order backward difference bdf for the temporal term and a second order adams bashforth ab for the explicit treatment of the nonlinear term lead to the presented second order bdf ab scheme the stability of the method was proved and the performance of the method is numerically illustrated finally comparison with some established methods a series of numerical experiments are given to show that this method has better stability and accuracy

unconditionally stable gauge uzawa finite element schemes for incompressible natural convection problems with variable density abstract we construct in this paper two gauge uzawa schemes one in conserved form and the other in convective form for solving natural convection problems with variable density and prove that the first order versions of both schemes are unconditionally stable we also show that a full discretized version of the conserved scheme with finite elements is also unconditionally stable these schemes lead to a sequence of decoupled elliptic equations to solve at each step hence they are very efficient and easy to implement we present several numerical tests to validate the analysis and demonstrate the effectiveness of these schemes for simulating natural convection problems with large density differences

efficient implementation of the gaussian kernel algorithm in estimating invariants and noise level from noisy time series data we describe an efficient algorithm which computes the gaussian kernel correlation integral from noisy time series this is subsequently used to estimate the underlying correlation dimension and noise level in the noisy data the algorithm first decomposes the integral core into two separate calculations reducing computing time from o n 2 ifmmode times else texttimes fi n b to o n 2 n b 2 with other further improvements this algorithm can speed up the calculation of the gaussian kernel correlation integral by a factor of ensuremath gamma ensuremath sim 2 10 n b we use typical examples to demonstrate the use of the improved gaussian kernel algorithm

a non subjective approach to the gp algorithm for analysing noisy time series we present an adaptation of the standard grassberger proccacia gp algorithm for estimating the correlation dimension of a time series in a non subjective manner the validity and accuracy of this approach are tested using different types of time series such as those from standard chaotic systems pure white and colored noise and chaotic systems with added noise the effectiveness of the scheme in analysing noisy time series particularly those involving colored noise is investigated one interesting result we have obtained is that for the same percentage of noise addition data with colored noise is more distinguishable from the corresponding surrogates than data with white noise as examples of real life applications analyses of data from an astrophysical x ray object and a human brain eeg are presented

low bias local intrinsic dimension estimation from expected simplex skewness in exploratory high dimensional data analysis local intrinsic dimension estimation can sometimes be used in order to discriminate between data sets sampled from different low dimensional structures global intrinsic dimension estimators can in many cases be adapted to local estimation but this leads to problems with high negative bias or high variance we introduce a method that exploits the curse blessing of dimensionality and produces local intrinsic dimension estimators that have very low bias even in cases where the intrinsic dimension is higher than the number of data points in combination with relatively low variance we show that our estimators have a very good ability to classify local data sets by their dimension compared to other local intrinsic dimension estimators furthermore we provide examples showing the usefulness of local intrinsic dimension estimation in general and our method in particular for stratification of real data sets

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

convergence of the discontinuous galerkin method for discontinuous solutions we consider linear first order scalar equations of the form rho t rm div rho v a rho f with appropriate initial and boundary conditions it is shown that approximate solutions computed using the discontinuous galerkin method will converge in ltwoltwo when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions in particular v need not be lipschitz so characteristics of the equation may not be defined and the solutions being approximated my not have bounded variation

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

test your surrogate data before you test for nonlinearity the schemes for the generation of surrogate data in order to test the null hypothesis of linear stochastic process undergoing nonlinear static transform are investigated as to their consistency in representing the null hypothesis in particular we pinpoint some important caveats of the prominent algorithm of amplitude adjusted fourier transform surrogates aaft and compare it to the iterated aaft which is more consistent in representing the null hypothesis it turns out that in many applications with real data the inferences of nonlinearity after marginal rejection of the null hypothesis were premature and have to be reinvestigated taking into account the inaccuracies in the aaft algorithm mainly concerning the mismatching of the linear correlations in order to deal with such inaccuracies we propose the use of linear together with nonlinear polynomials as discriminating statistics the application of this setup to some well known real data sets cautions against the use of the aaft algorithm

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

gauge uzawa methods for incompressible flows with variable density two new gauge uzawa schemes are constructed for incompressible flows with variable density one is in the conserved form while the other is in the convective form it is shown that the first order versions of both schemes in their semi discretized form are unconditionally stable numerical experiments indicate that the first order resp second order versions of the two schemes lead to first order resp second order convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

fully iterative ilu preconditioning of the unsteady navier stokes equations for gpgpu abstract in this work we investigate the numerical difficulties that arise in optimizing the efficiency of newtonian fluids simulations on a massively parallel computing hardware like a gpu in particular we will concentrate on the resulting algebraic problem we will present an approximate fully iterative ilu preconditioner and we will show that this solution is very efficient on a gpu if compared with an intrinsic massively parallel preconditioner like the diagonal preconditioner which indeed goes faster than more robust techniques like ilu despite their strong decrease in the number of iterations we refer to gmres as the iterative scheme used to solve the linear system in particular we will deal with the solution of incompressible flows with variable density and we will investigate the interplay between reynolds and atwood numbers we will show that the numerical simulation at medium high reynolds numbers produces linear systems whose matrices can be reasonably preconditioned with the diagonal preconditioner while at low reynolds numbers the higher viscosity of the fluid makes the diagonal preconditioner ineffective in the solution time requested from gmres and decreasing the reynolds number unable to let gmres converge at all in this situation we will show how an adequate iterative approach to the parallel solution of the triangular systems that result from the ilu preconditioning turns out to be robust and efficient we will show numerical results for variable density fluids discretized with the scheme described in calgaro et al 2008 in classical benchmarks and in particular in the well known rayleigh taylor instability

uniform framework for the recurrence network analysis of chaotic time series we propose a general method for the construction and analysis of unweighted e recurrence networks from chaotic time series the selection of the critical threshold e c in our scheme is done empirically and we show that its value is closely linked to the embedding dimension m in fact we are able to identify a small critical range e numerically that is approximately the same for the random and several standard chaotic time series for a fixed m this provides us a uniform framework for the nonsubjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors we explicitly show that the degree distribution of the recurrence network constructed by our scheme is characteristic to the structure of the attractor and display statistical scale invariance with respect to increase in the number of nodes n we also present two practical applications of the scheme detection of transition between two dynamical regimes in a time delayed system and identification of the dimensionality of the underlying system from real world data with a limited number of points through recurrence network measures the merits limitations and the potential applications of the proposed method are also highlighted

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

convergence analysis of a class of massively parallel direction splitting algorithms for the navier stokes equations in simple domains we provide a convergence analysis for a new fractional timestepping technique for the incompressible navier stokes equations based on direction splitting this new technique is of linear complexity unconditionally stable and convergent and suitable for massive parallelization

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a new fractional time stepping method for variable density incompressible flows this paper describes a new method for solving variable density incompressible viscous flows we have dealt with the momentum equation and the divergence free constraint in a new manner by rewriting the original equations the originality of the proposed approach is that we have used different numerical methods to evaluate the evolution of the velocity and pressure compared with some established methods the proposed approach is parameter free more flexible and can avoid the difficulties caused by the original equations the stability analysis of the method is performed to show that our method is stable finally numerical experiments are given to show the accuracy efficiency and validness of this method for variable density incompressible flows

parameter estimation of superimposed signals using the em algorithm a computationally efficient algorithm for parameter estimation of superimposed signals based on the two step iterative em estimate and maximize with an e step and an m step algorithm is developed the idea is to decompose the observed data into their signal components and then to estimate the parameters of each signal component separately the algorithm iterates back and forth using the current parameter estimates to decompose the observed data better and thus increase the likelihood of the next parameter estimates the application of the algorithm to the multipath time delay and multiple source location estimation problems is considered

a fast algorithm for direct simulation of particulate flows using conforming grids this paper presents a development of the direction splitting algorithm for flows in complex geometries proposed in 2 to the case of flows containing rigid particles the main novelty of this method is that the grid is fit to the time dependent domain at each time step which allows for an optimal spatial approximation of the navier stokes equations in general this is a very hard task for multidimensional problems however it is significantly simplified if the momentum equations are discretized by a direction splitting scheme such a scheme requires only solving a set of one dimensional problems and the grid can be easily fit to the boundary in the direction of each of these problems here we use a mac discretization stencil but the same idea can be applied to other discretizations the equations of motion of each particle are discretized explicitly and the computed particle velocities are imposed as dirichlet boundary conditions for the navier stokes equations on the adapted grid the pressure is extended within the particles in a fictitious domain fashion but the divergence stencil is fit to the particle boundaries in this paper the direction splitting algorithm with boundary fitting is compared to pure fictitious domain methods

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

constrained surrogate time series with preservation of the mean and variance structure a method is presented for generating surrogates that are constrained realizations of a time series but which preserve the local mean and variance of the original signal the method is based on the popular iterated amplitude adjusted fourier transform method but makes use of a wavelet transform to constrain behavior in the time domain using this method it is possible to test for local changes in the nonlinear properties of the signal we present an example for a change in hurst exponent in a time series produced by fractional brownian motion

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

error estimate of gauge uzawa methods for incompressible flows with variable density abstract in this paper we construct a positivity preserving gauge uzawa method for the semi discrete in time scheme of incompressible viscous flows with variable density and establish its stability and error estimates we also construct and implement a fully discrete scheme with finite elements in space and derive its positivity preserving and stability result

estimating invariants of noisy attractors we propose a method for estimating the correlation dimension and correlation entropy of a time series it is based on a generalization of the correlation integral that is specifically useful when the time series is corrupted with gaussian measurement noise from computational experiments we conclude that reasonable estimates for the noise level correlation dimension and correlation entropy can be found for time series with up to 20 noise the method appears to be fairly robust with respect to the noise distribution

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of the proposed method

effective scaling regime for compution the correlating dimension from chaotic time series abstract in the analysis of chaotic time series a standard technique is to reconstruct an image of the original dynamical system using delay coordinates if the original dynamical system has an attractor then the correlation dimension d2 of its image in the reconstruction can be estimated using the grassberger procaccia algorithm the quality of the reconstruction can be probed by measuring the length of the linear scaling region used in this estimation in this paper we show that the quality is constrained by both the embedding dimension m and more importantly by the delay time for a given embedding dimension and a finite time series there exists a maximum allowed delay time beyond which the size of the scaling region is no longer reliably discernible we derive an upper bound for this maximum delay time numerical experiments on several model chaotic time series support the theoretical argument they also clearly indicate the different roles played by the embedding dimension and the delay time in the reconstruction as the embedding dimension is increased it is necessary to reduce the delay time substantially to guarantee a reliable estimate of d2 our results imply that it is the delay time itself rather than the total observation time m 1 which plays the most critical role in the determination of the correlation dimension

determination of the noise level of chaotic time series we propose a method to determine the amount of measurement noise present in a chaotic time series if the data are embedded in a space of higher dimension than that strictly required to reconstruct the dynamics the extra dimensions are dominated by the noise which results in a certain shape of the correlation integral for the case in which only gaussian noise is present this shape can be calculated analytically as a function of the noise level thus the noise level can be obtained from a simple function fit the analytical result also shows that a noise level of more than 2 will obscure any possible scaling of the correlation integral and thus makes it impossible to estimate the correlation dimension

high resolution source coding for non difference distortion measures multidimensional companding entropy coded vector quantization is studied using high resolution multidimensional companding over a class of nondifference distortion measures for distortion measures which are locally quadratic a rigorous derivation of the asymptotic distortion and entropy coded rate of multidimensional companders is given along with conditions for the optimal choice of the compressor function this optimum compressor when it exists depends on the distortion measure but not on the source distribution the rate distortion performance of the companding scheme is studied using an asymptotic expression for the rate distortion function which parallels the shannon lower bound for difference distortion measures it is proved that the high resolution performance of the scheme is arbitrarily close to the rate distortion limit for large quantizer dimensions if the compressor function and the lattice quantizer used in the companding scheme are optimal extending an analogous statement for entropy coded lattice quantization and mse distortion the companding approach is applied to obtain a high resolution quantizing scheme for noisy sources

convergence of numerical approximations of the incompressible navier stokes equations with variable density and viscosity we consider numerical approximations of incompressible newtonian fluids having variable possibly discontinuous density and viscosity since solutions of the equations with variable density and viscosity may not be unique numerical schemes may not converge if the solution is unique then approximate solutions computed using the discontinuous galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution if the solution is not unique a subsequence of these approximate solutions will converge to a solution

a fundamental bias in calculating dimensions from finite data sets abstract one bias inherent in calculating dimension for limited time series data is investigated the bias is derived from the fluctuation of the distribution of measures in the phase space and distorts the scaling with respect to each reference point to be concave up or down these distortions are pronounced for the experimental data whose number of points is not sufficient and whose scaling region is restricted to a relatively small interval it is possible that the grassberger procaccia algorithm and all its modified ones are affected by the bias we evaluate the distortion quantitatively and show the procedure required for the correction of the bias taking the case of an electroencephalogram eeg

estimating the dimension of high dimensional attractors a comparison between two algorithms abstract we compare two algorithms for the numerical estimation of the correlation dimension from a finite set of vectors the classical algorithm of grassberger and procaccia gpa and the recently proposed algorithm of judd ja data set size requirements and their relations to systematic and statistical errors of the estimates are investigated it is demonstrated that correlation dimensions of the order of 6 can correctly be resolved on the basis of the about 100000 data points in the case of a continuous trajectory on a strange attractor the minimum data set size is however noticeably dependent on the geometrical structure of the system from which the vectors were sampled

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

an improved estimator of dimension and some comments on providing confidence intervals abstract an estimator of dimension is described that has many advantages over the conventional implementation of the grassberger procaccia method it can in some circumstances provide a confidence interval for the dimension estimate for high dimension sets greater than about 3 5 the dimension estimate has a bias which depends on the structure of the set if one has suitable knowledge of the structure of the set it is possible to make a bias adjustment some numerical calculations are presented which indicate the largest error in the dimension estimate that can be expected the problem of providing confidence intervals for dimension estimates in general is not solved completely but we do take a step away from a meaningless number

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

parametric characterisation of a chaotic attractor using the two scale cantor measure abstract a chaotic attractor is usually characterised by its multifractal spectrum which gives a geometric measure of its complexity here we present a characterisation using a minimal set of independent parameters which is uniquely determined by the underlying process that generates the attractor the method maps the f spectrum of a chaotic attractor on to that of a general two scale cantor measure we show that the mapping can be done in practice with reasonable accuracy for many of the standard chaotic attractors in order to implement this procedure we also propose a generalisation of the standard equations for the two scale cantor set in one dimension to that in higher dimensions another interesting result we have obtained both theoretically and numerically is that the f characterisation gives information only up to two scales even when the underlying process generating the multifractal involves more than two scales

stability analysis of pressure correction schemes for the navier stokes equations with traction boundary conditions abstract we present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions first we provide a stability analysis for a rotational version of the grad div stabilized scheme of bonito et al 2015 this scheme turns out to be unconditionally stable provided the stabilization parameter is suitably chosen we also establish a conditional stability result for the boundary correction scheme presented in bansch 2014 these results are shown by employing the equivalence between stabilized gauge uzawa methods and rotational pressure correction schemes with traction boundary conditions

comparisons of new nonlinear modeling techniques with applications to infant respiration abstract this paper concerns the application of new nonlinear time series modeling methods to recordings of infant respiratory patterns the techniques used combine the concept of minimum description length modeling with radial basis models our first application of the methods produced results that were not entirely satisfactory particularly with respect to accurately modeling long term quantitative and qualitative features of respiration patterns this paper describes a number of modifications of the original methods and makes a comparison of the improvements the various modifications gave the modifications made were increasing the class of basis function broadening the range of possible embedding strategies improving the optimization of the likelihood of the model parameters and calculating a closer approximation to description length the criteria used in the comparisons were description length root mean square prediction error model size free run behavior and amplitude size and variation we use surrogate data analysis to confirm the hypothesis that the recorded data are consistent with the nonlinear models we construct and not consistent with simpler models we also investigate the free run dynamics of the model systems to see if the models exhibit features consistent with physiological characteristics observed independently in the respiratory recording this involved modeling breathing patterns prior to a sigh and onset of a phenomenon called periodic breathing and comparing the period of cyclic amplitude modulation of the free run dynamics of the models and the period of the subsequent periodic breathing observed in respiration recording the periods were consistent in six out of seven recordings

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed