on the gaussian measure of the intersection the gaussian correlation conjecture states that for any two symmetric convex sets in n dimensional space and for any centered gaussian measure on that space the measure of the intersection is greater than or equal to the product of the measures in this paper we obtain several results which substantiate this conjecture for example in the standard gaussian case we show there is a positive constant c such that the conjecture is true if the two sets are in the euclidean ball of radius c n further we show that if for every n the conjecture is true when the sets are in the euclidean ball of radius n then it is true in general our most concrete result is that the conjecture is true if the two sets are arbitrary centered ellipsoids

numerical analysis of the leray reduced order model abstract standard roms generally yield spurious numerical oscillations in the simulation of convection dominated flows regularized roms use explicit rom spatial filtering to decrease these spurious numerical oscillations the leray rom is a recently introduced regularized rom that utilizes explicit rom spatial filtering of the convective term in the navier stokes equations this paper presents the numerical analysis of the finite element discretization of the leray rom error estimates for the rom differential filter which is the explicit rom spatial filter used in the leray rom are proved these rom filtering error estimates are then used to prove error estimates for the leray rom finally both the rom filtering error estimates and the leray rom error estimates are numerically investigated in the simulation of the two dimensional navier stokes equations with an analytic solution

small deviations of riemann liouville processes in l_ lowercase q spaces with respect to fractal measures we investigate riemann liouville processes r h with h 0 and fractional brownian motions b h for 0 h 1 and study their small deviation properties in the spaces l q 0 1 mu of special interest here are thin fractal measures mu that is those that are singular with respect to the lebesgue measure we describe the behavior of small deviation probabilities by numerical quantities of mu called mixed entropy numbers characterizing size and regularity of the underlying measure for the particularly interesting case of self similar measures the asymptotic behavior of the mixed entropy is evaluated explicitly we also provide two sided estimates for this quantity in the case of random measures generated by subordinators while the upper asymptotic bound for the small deviation probability is proved by purely probabilistic methods the lower bound is verified by analytic tools concerning entropy and kolmogorov numbers of riemann liouville operators

on the stability and convergence of a galerkin reduced order model rom of compressible flow with solid wall and far field boundary treatment a reduced order model rom based on the proper orthogonal decomposition pod galerkin projection method is proposed as an alternative discretization of the linearized compressible euler equations it is shown that the numerical stability of the rom is intimately tied to the choice of inner product used to define the galerkin projection for the linearized compressible euler equations a symmetry transformation motivates the construction of a weighted l 2 inner product that guarantees certain stability bounds satisfied by the rom sufficient conditions for well posedness and stability of the present galerkin projection method applied to a general linear hyperbolic initial boundary value problem ibvp are stated and proven well posed and stable far field and solid wall boundary conditions are formulated for the linearized compressible euler rom using these more general results a convergence analysis employing a stable penalty like formulation of the boundary conditions reveals that the rom solution converges to the exact solution with refinement of both the numerical solution used to generate the rom and of the pod basis an a priori error estimate for the computed rom solution is derived and examined using a numerical test case published in 2010 by john wiley sons ltd

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

limits to the experimental detection of nonlinear synchrony chaos synchronization is often characterized by the existence of a continuous function between the states of the components however in coupled systems without inherent symmetries the synchronization set can be extremely complicated we describe and illustrate three typical complications that can arise and we discuss how existing methods for detecting synchronization will be hampered by the presence of these features

reduced order controllers for spatially distributed systems via proper orthogonal decomposition a method for reducing controllers for systems described by partial differential equations pdes is presented this approach differs from an often used method of reducing the model and then designing the controller the controller reduction is accomplished by projection of a large scale finite element approximation of the pde controller onto low order bases that are computed using the proper orthogonal decomposition pod two methods for constructing input collections for pod and hence low order bases are discussed and computational results are included the first uses the method of snapshots found in pod literature the second is a new idea that uses an integral representation of the feedback control law specifically the kernels or functional gains are used as data for pod a low order controller derived by applying the pod process to functional gains avoids subjective criteria associated with implementing a time snapshot approach and performs favorably

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

efficient non linear model reduction via a least squares petrov galerkin projection and compressive tensor approximations the first author acknowledges the partial support by a national science foundation graduate fellowship and the partial support by a national defense science and engineering graduate fellowship the second and third authors acknowledge the partial support by the motor sports division of the toyota motor corporation under agreement number 48737 and the partial support by a research grant from the academic excellence alliance program between king abdullah university of science and technology kaust and stanford university all authors also acknowledge the constructive comments received during the review process

model reduction of dynamical systems by proper orthogonal decomposition error bounds and comparison of methods using snapshots from the solution and the time derivatives abstract we consider two proper orthogonal decomposition pod methods for dimension reduction of dynamical systems the first method m1 uses only time snapshots of the solution while the second method m2 augments the snapshot set with time derivative snapshots the goal of the paper is to analyze and compare the approximation errors resulting from the two methods by using error bounds we derive several new bounds of the error from pod model reduction by each of the two methods the new error bounds involve a multiplicative factor depending on the time steps between the snapshots for method m1 the factor depends on the second power of the time step while for method 2 the dependence is on the fourth power of the time step suggesting that method m2 can be more accurate for small between snapshot intervals however three other factors also affect the size of the error bounds these include i the norm of the second for m1 and fourth derivatives m2 ii the first neglected singular value and iii the spectral properties of the projection of the system s jacobian in the reduced space because of the interplay of these factors neither method is more accurate than the other in all cases finally we present numerical examples demonstrating that when the number of collected snapshots is small and the first neglected singular value has a value of zero method m2 results in a better approximation

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

supg reduced order models for convection dominated convection diffusion reaction equations abstract this paper presents a streamline upwind petrov galerkin supg reduced order model rom based on proper orthogonal decomposition pod this rom is investigated theoretically and numerically for convection dominated convection diffusion reaction problems the supg finite element method was used on realistic meshes for computing the snapshots leading to some noise in the pod data numerical analysis is used to propose the scaling of the stabilization parameter for the supg rom two approaches are used one based on the underlying finite element discretization and the other one based on the pod truncation the resulting supg roms and the standard galerkin rom g rom are studied numerically for many settings the results obtained with the supg roms are more accurate finally one of the choices for the stabilization parameter is recommended

optimal periodic orbits of chaotic systems invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis which indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor

model reduction for large scale systems with high dimensional parametric input space a model constrained adaptive sampling methodology is proposed for the reduction of large scale systems with high dimensional parametric input spaces our model reduction method uses a reduced basis approach which requires the computation of high fidelity solutions at a number of sample points throughout the parametric input space a key challenge that must be addressed in the optimization control and probabilistic settings is the need for the reduced models to capture variation over this parametric input space which for many applications will be of high dimension we pose the task of determining appropriate sample points as a pde constrained optimization problem which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters the methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21 which describe thermal analysis and design of a heat conduction fin and compared with statistically based sampling methods for these examples the model constrained adaptive sampling leads to reduced models that for a given basis size have error several orders of magnitude smaller than that obtained using the other methods

robust synchronization of chaotic systems the question of robustness of synchronization with respect to small arbitrary perturbations of the underlying dynamical systems is addressed we present examples of chaos synchronization demonstrating that normal hyperbolicity is a necessary and sufficient condition for the synchronization manifold to be smooth and persistent under small perturbations the same examples however show that in real applications normal hyperbolicity is not sufficient to give quantitative bounds for deformations of the synchronization manifold i e even in the case of normal hyperbolicity two almost identical systems may cause large synchronization errors

differentiable generalized synchronization of chaos we consider simple lyapunov exponent based conditions under which the response of a system to a chaotic drive is a ital smooth function of the drive state we call this ital differentiable generalized synchronization dgs when dgs does not hold we quantify the degree of nondifferentiability using the h umlt o lder exponent we also discuss the consequences of dgs and give an illustrative numerical example copyright ital 1997 ital the american physical society

generalized synchronization of chaos in noninvertible maps the properties of functional relation between a noninvertible chaotic drive and a response map in the regime of generalized synchronization of chaos are studied it is shown that despite a very fuzzy image of the relation between the current states of the maps the functional relation becomes apparent when a sufficient interval of driving trajectory is taken into account this paper develops a theoretical framework of such functional relation and illustrates the main theoretical conclusions using numerical simulations

detectability of nondifferentiable generalized synchrony generalized synchronization of chaos is a type of cooperative behavior in directionally coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator in many practical cases this function is nondifferentiable and has a very complex shape the generalized synchrony in such cases seems to be undetectable and only the cases in which a differentiable synchronization function exists are considered to make sense in practice we show that this viewpoint is not always correct and the nondifferentiable generalized synchrony can be revealed in many practical cases conditions for detection of generalized synchrony are derived analytically and illustrated numerically with a simple example of nondifferentiable generalized synchronization

a state space error estimate for pod deim nonlinear model reduction this paper derives state space error bounds for the solutions of reduced systems constructed using proper orthogonal decomposition pod together with the discrete empirical interpolation method deim recently developed for nonlinear dynamical systems siam j sci comput 32 2010 pp 2737 2764 the resulting error estimates are shown to be proportional to the sums of the singular values corresponding to neglected pod basis vectors both in galerkin projection of the reduced system and in the deim approximation of the nonlinear term the analysis is particularly relevant to ode systems arising from spatial discretizations of parabolic pdes the derivation clearly identifies where the parabolicity is crucial it also explains how the deim approximation error involving the nonlinear term comes into play

on a characterization of operators from lq into a banach space of type p with some applications to eigenvalue problems abstract let s be an operator admitting a factorization where d is a diagonal operator and t an arbitrary operator with the image in a banach space of type p we shall characterize these operators by entropy numbers using a well known inequality between eigenvalues and entropy numbers we establish some results about distributions of eigenvalues

pod and cvt based reduced order modeling of navier stokes flows abstract a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the construction and application of reduced order bases reviews of the pod proper orthogonal decomposition and cvt centroidal voronoi tessellation approaches to reduced order modeling are provided including descriptions of pod and cvt reduced order bases their construction from snapshot sets and their application to the low cost simulation of the navier stokes system some concrete incompressible flow examples are used to illustrate the construction and use of pod and cvt reduced order bases and to compare and contrast the two approaches to reduced order modeling

energy balance and mass conservation in reduced order models of fluid flows abstract in this paper we investigate theoretically and computationally the conservation properties of reduced order models roms for fluid flows specifically we investigate whether the roms satisfy the same or similar energy balance and mass conservation as those satisfied by the navier stokes equations all of our theoretical findings are illustrated and tested in numerical simulations of a 2d flow past a circular cylinder at a reynolds number r e 100 first we investigate the rom energy balance we show that using the snapshot average for the centering trajectory which is a popular treatment of nonhomogeneous boundary conditions in roms yields an incorrect energy balance then we propose a new approach in which we replace the snapshot average with the stokes extension theoretically the stokes extension produces an accurate energy balance numerically the stokes extension yields more accurate results than the standard snapshot average especially for longer time intervals our second contribution centers around rom mass conservation we consider roms created using two types of finite elements the standard taylor hood th element which satisfies the mass conservation weakly and the scott vogelius sv element which satisfies the mass conservation pointwise theoretically the error estimates for the sv rom are sharper than those for the th rom numerically the sv rom yields significantly more accurate results especially for coarser meshes and longer time intervals

detecting generalized synchrony an improved approach we examine some of the difficulties involved in detecting generalized synchrony gs in systems that exhibit noninvertibility and or wrinkling these latter features severely hinder identification of gs by conventional techniques it is shown that it is possible to greatly improve detection by reducing the pseudofalse neighbors effects here we propose the ensuremath delta p neighbor method to overcome the noninvertibility effect and the ensuremath delta p q method to detect gs in systems with wrinkled structures

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

error estimation for reduced order models of dynamical systems the use of reduced order models to describe a dynamical system is pervasive in science and engineering often these models are used without an estimate of their error or range of validity in this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition we show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method most important the proposed approach allows the assessment of regions of validity for reduced models i e ranges of perturbations in the original system over which the reduced model is still appropriate numerical examples validate our approach the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude

approximation metric entropy and small ball estimates for gaussian measures a precise link proved by kuelbs and li relates the small ball behavior of a gaussian measure on a banach space e with the metric entropy behavior of k the unit ball of the reproducing kernel hilbert space of in e we remove the main regularity assumption imposed on the unknown function in the link this enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete gaussian processes moreover we show that the small ball behavior of a gaussian process is also tightly connected with the speed of approximation by finite rank processes

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

proper orthogonal decomposition closure models for turbulent flows a numerical comparison this paper puts forth two new closure models for the proper orthogonal decomposition reduced order modeling of structurally dominated turbulent flows the dynamic subgrid scale model and the variational multiscale model these models which are considered state of the art in large eddy simulation together with the mixing length and the smagorinsky closure models are tested in the numerical simulation of 3d turbulent flow past a circular cylinder at re 1000 five criteria are used to judge the performance of the proper orthogonal decomposition reduced order models the kinetic energy spectrum the mean velocity the reynolds stresses the root mean square values of the velocity fluctuations and the time evolution of the pod coefficients all the numerical results are benchmarked against a direct numerical simulation based on these numerical results we conclude that the dynamic subgrid scale and the variational multiscale models are the most accurate

model reduction for compressible flows using pod and galerkin projection abstract we present a framework for applying the method of proper orthogonal decomposition pod and galerkin projection to compressible fluids for incompressible flows only the kinematic variables are important and the techniques are well known in a compressible flow both the kinematic and thermodynamic variables are dynamically important and must be included in the configuration space we introduce an energy based inner product which may be used to obtain pod modes for this configuration space we then obtain an approximate version of the navier stokes equations valid for cold flows at moderate mach number and project these equations onto a pod basis the resulting equations of motion are quadratic and are much simpler than projections of the full compressible navier stokes equations

small deviations of weighted fractional processes and average non linear approximation we investigate the small deviation problem for weighted fractional brownian motions in l q norm 1 q let b h be a fractional brownian motion with hurst index 0 h 1 if 1 r h 1 q then our main result asserts formula math provided the weight function p satisfies a condition slightly stronger than the r integrability thus we extend earlier results for brownian motion i e h 1 2 to the fractional case our basic tools are entropy estimates for fractional integration operators a non linear approximation technique for gaussian processes as well as sharp entropy estimates for l q sums of linear operators defined on a hilbert space

metric entropy and the small ball problem for gaussian measures abstract we establish a precise link between the small ball problem for a gaussian measure on a separable banach space and the metric entropy of the unit ball of the hubert space h generating this link allows us to compute small ball probabilities from metric entropy results and vice versa

isogeometric analysis and proper orthogonal decomposition for parabolic problems we investigate the combination of isogeometric analysis iga and proper orthogonal decomposition pod based on the galerkin method for model order reduction of linear parabolic partial differential equations for the proposed fully discrete scheme the associated numerical error features three components due to spatial discretization by iga time discretization with the theta scheme and eigenvalue truncation by pod first we prove a priori error estimates of the spatial iga semi discrete scheme then we show stability and prove a priori error estimates of the space time discrete scheme and the fully discrete iga theta pod galerkin scheme numerical tests are provided to show efficiency and accuracy of nurbs based iga for model order reduction in comparison with standard finite element based pod techniques

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

are the snapshot difference quotients needed in the proper orthogonal decomposition this paper presents a theoretical and numerical investigation of the following practical question should the time difference quotients dqs of the snapshots be used to generate the proper orthogonal decomposition pod basis functions the answer to this question is important since some published numerical studies use the time dqs whereas other numerical studies do not the criterion used in this paper to answer this question is the optimality of the convergence rate of the error of the reduced order model with respect to the number of pod basis functions since to the best of our knowledge a definition of the optimality of the convergence rate is not available we propose one in definition 3 1 in this paper two cases are considered the no dq case in which the snapshot dqs are not used and the dq case in which the snapshot dqs are used for each case two types of pod bases are used the l 2 pod basis in which the basis is generated in the l 2 norm and the h 1 pod basis in which the basi

small values of gaussian processes and functional laws of the iterated logarithm we estimate small ball probabilities for locally nondeterministic gaussian processes with stationary increments a class of processes that includes the fractional brownian motions these estimates are used to prove chung type laws of the iterated logarithm

new pod error expressions error bounds and asymptotic results for reduced order models of parabolic pdes the derivations of existing error bounds for reduced order models of time varying partial differential equations pdes constructed using proper orthogonal decomposition pod have relied on bounding the error between the pod data and various pod projections of that data furthermore the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the pod data approximation error bounds we consider time varying data taking values in two different hilbert spaces h and v with v subset h and prove exact expressions for the pod data approximation errors considering four different pod projections and the two different hilbert space error norms furthermore the exact error expressions can be computed using only the pod eigenvalues and modes and we prove the errors converge to zero as the number of pod modes increases we consider the pod error estimation approaches of kunisch and volkwein siam j numer anal 40 2002 pp 492 515 and chapelle gariah an