an analytic approach to practical state space reconstruction we study the three standard methods for reconstructing a state space from a time series delays derivatives and principal components we derive a closed form solution to principal component analysis in the limit of small window widths this solution explains the relationship between delays derivatives and principal components it shows how the singular spectrum scales with dimension and delay time and it explains why the eigenvectors resemble the legendre polynomials most importantly the solution allows us to derive a guideline for choosing a good window width unlike previous suggestions this guideline is based on first principles and simple quantities we argue that discrete legendre polynomials provide a quick and not so dirty substitute for principal component analysis and that they are a good practical method for state space reconstruction

dimensional hyper reduction of nonlinear finite element models via empirical cubature we present a general framework for the dimensional reduction in terms of number of degrees of freedom as well as number of integration points hyper reduction of nonlinear parameterized finite element fe models the reduction process is divided into two sequential stages the first stage consists in a common galerkin projection onto a reduced order space as well as in the condensation of boundary conditions and external forces for the second stage reduction in number of integration points we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized the distinguishing features of the proposed method are 1 the minimization problem is posed in terms of orthogonal basis vector obtained via a partitioned singular value decomposition rather that in terms of snapshots of the integrand 2 the volume of the domain is exactly integrated 3 the selection algorithm need not solve in all iterations a nonnegative least squares problem to force the positiveness of the weights furthermore we show that the proposed method converges to the absolute minimum zero integration error when the number of selected points is equal to the number of internal force modes included in the objective function we illustrate this model reduction methodology by two nonlinear structural examples quasi static bending and resonant vibration of elastoplastic composite plates in both examples the number of integration points is reduced three order of magnitudes with respect to fe analyses without significantly sacrificing accuracy

stable galerkin reduced order models for linearized compressible flow the galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations the numerical stability of galerkin models is shown to depend on the choice of inner product for the projection for the linearized euler equations a symmetry transformation leads to a stable formulation for the inner product boundary conditions for compressible flow that preserve stability of the reduced order model are constructed preservation of stability for the discrete implementation of the galerkin projection is made possible using a piecewise smooth finite element basis stability of the reduced order model using this approach is demonstrated on several model problems where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation

on the stability of the reduced basis method for stokes equations in parametrized domains we present an application of reduced basis method for stokes equations in domains with affine parametric dependence the essential components of the method are i the rapid convergence of global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n selected points in parameter space ii the off line on line computational procedures decoupling the generation and projection stages of the approximation process the operation count for the on line stage in which given a new parameter value we calculate an output of interest depends only on n typically very small and the parametric complexity of the problem the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation design optimization and real time control particular attention is given i to the pressure treatment of incompressible stokes problem ii to find an equivalent inf sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem iii to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions iv to reduce computational costs in order to allow real time solution of parametrized problem

structure preserving stability and accuracy properties of the energy conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models summary the computational efficiency of a typical projection based nonlinear model reduction method hinges on the efficient approximation for explicit computations of the scalar projections onto a subspace of a residual vector for implicit computations it also hinges on the additional efficient approximation of similar projections of the jacobian of this residual with respect to the solution the computation of both approximations is often referred to in the literature as hyper reduction to this effect this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced order models of solids and structures of the recently developed energy conserving mesh sampling and weighting ecsw hyper reduction method unlike most alternative approaches this method approximates the scalar projections of residuals and or jacobians directly instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest in this paper it is shown that ecsw distinguishes itself furthermore from other hyper reduction methods through its preservation of the lagrangian structure associated with hamilton s principle for second order dynamical systems this enables it to also preserve the numerical stability properties of the discrete system to which it is applied it is also shown that for a fixed set of parameter values the approximation error committed online by ecsw is bounded by its counterpart error committed off line during the training of this method therefore this error can be estimated in this case a priori and controlled the performance of ecsw is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems for both of them ecsw is shown to preserve numerical stability and deliver the desired level of accuracy whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability the potential of ecsw for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme for this simulation ecsw is reported to enable the reduction of the cpu time required by the high dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude while achieving a very good level of accuracy copyright 2015 john wiley sons ltd

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

a new selection operator for the discrete empirical interpolation method improved a priori error bound and extensions this paper introduces a new framework for constructing the discrete empirical interpolation method sf deim projection operator the interpolation node selection procedure is formulated using the qr factorization with column pivoting and it enjoys a sharper error bound for the sf deim projection error furthermore for a subspace mathcal u given as the range of an orthonormal mathsf u the sf deim projection does not change if mathsf u is replaced by mathsf u omega with arbitrary unitary matrix omega in a large scale setting the new approach allows modifications that use only randomly sampled rows of mathsf u but with the potential of producing good approximations with corresponding probabilistic error bounds another salient feature of the new framework is that robust and efficient software implementation is easily developed based on readily available high performance linear algebra packages

a generalized framework for nodal first derivative summation by parts operators a generalized framework is presented that extends the classical theory of finite difference summation by parts sbp operators to include a wide range of operators where the main extensions are i non repeating interior point operators ii nonuniform nodal distribution in the computational domain iii operators that do not include one or both boundary nodes necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the sbp property it is proven that the positive definite norm matrix of each sbp operator must be associated with a quadrature rule moreover given a quadrature rule there exists a corresponding sbp operator where for diagonal norm sbp operators the weights of the quadrature rule must be positive the generalized framework gives a straightforward means of posing many known approximations to the first derivative as sbp operators several are surveyed such as discontinuous galerkin discretizations based on the legendre gauss quadrature points and shown to be sbp operators moreover the new framework provides a method for constructing sbp operators by starting from quadrature rules this is illustrated by constructing novel sbp operators from known quadrature rules to demonstrate the utility of the generalization the legendre gauss and legendre gauss radau quadrature points are used to construct sbp operators that do not include one or both boundary nodes

minimal subspace rotation on the stiefel manifold for stabilization and enhancement of projection based reduced order models for the compressible navier stokes equations for a projection based reduced order model rom of a fluid flow to be stable and accurate the dynamics of the truncated subspace must be taken into account this paper proposes an approach for stabilizing and enhancing projection based fluid roms in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace attention is focused on the full non linear compressible navier stokes equations in specific volume form as a step toward a more general formulation for problems with generic non linearities unlike traditional approaches no empirical turbulence modeling terms are required and consistency between the rom and the navier stokes equation from which the rom is derived is maintained mathematically the approach is formulated as a trace minimization problem on the stiefel manifold the reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems including a problem involving laminar flow over an airfoil with a high angle of attack and a channel driven cavity flow problem

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

preserving lagrangian structure in nonlinear model reduction with application to structural dynamics this work proposes a model reduction methodology that preserves lagrangian structure and achieves computational efficiency in the presence of high order nonlinearities and arbitrary parameter dependence as such the resulting reduced order model retains key properties such as energy conservation and symplectic time evolution maps we focus on parameterized simple mechanical systems subjected to rayleigh damping and external forces and consider an application to nonlinear structural dynamics to preserve structure the method first approximates the system s lagrangian ingredients the riemannian metric the potential energy function the dissipation function and the external force and subsequently derives reduced order equations of motion by applying the forced euler lagrange equation with these quantities from the algebraic perspective key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness matri

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

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

polynomial search and global modeling two algorithms for modeling chaos global modeling aims to build mathematical models of concise description polynomial model search pomos and global modeling glomo are two complementary algorithms freely downloadable at the following address http www cesbio ups tlse fr us pomos et glomo html designed for the modeling of observed dynamical systems based on a small set of time series models considered in these algorithms are based on ordinary differential equations built on a polynomial formulation more specifically pomos aims at finding polynomial formulations from a given set of 1 to n time series whereas glomo is designed for single time series and aims to identify the parameters for a selected structure glomo also provides basic features to visualize integrated trajectories and to characterize their structure when it is simple enough one allows for drawing the first return map for a chosen poincare section in the reconstructed space another one computes the lyapunov exponent along the trajectory in the present paper global modeling from single time series is considered a description of the algorithms is given and three examples are provided the first example is based on the three variables of the rossler attractor the second one comes from an experimental analysis of the copper electrodissolution in phosphoric acid for which a less parsimonious global model was obtained in a previous study the third example is an exploratory case and concerns the cycle of rainfed wheat under semiarid climatic conditions as observed through a vegetation index derived from a spatial sensor

reliable reduced order models for time dependent linearized euler equations development of optimal reduced order models for linearized euler equations is investigated recent methods based on proper orthogonal decomposition pod applicable for high order systems are presented and compared particular attention is paid to the link between the choice of the projection and the efficiency of the reduced model a stabilizing projection is introduced to induce a stable reduced order model at finite time even if the energy of the physical model is growing the proposed method is particularly well adapted for time dependent hyperbolic systems and intrinsically skew symmetric models this paper also provides a common methodology to reliably reduce very large nonsymmetric physical problems

state space reconstruction parameters in the analysis of chaotic time series the role of the time window length abstract the most common state space reconstruction method in the analysis of chaotic time series is the method of delays mod many techniques have been suggested to estimate the parameters of mod i e the time delay and the embedding dimension m we discuss the applicability of these techniques with a critical view as to their validity and point out the necessity of determining the overall time window length w for successful embedding emphasis is put on the relation between w and the dynamics of the underlying chaotic system and we suggest to set w p the mean orbital period p is approximated from the oscillations of the time series the procedure is assessed using the correlation dimension for both synthetic and real data for clean synthetic data values of w larger than p always give good results given enough data and thus p can be considered as a lower limit w p for noisy synthetic data and real data an upper limit is reached for w which approaches p for increasing noise amplitude

aerodynamic data reconstruction and inverse design using proper orthogonal decomposition the application of proper orthogonal decomposition for incomplete gappy data for compressible external aerodynamic problems has been demonstrated successfully in this paper for the first time using this approach it is possible to construct entire aerodynamic flowfields from the knowledge of computed aerodynamic flow data or measured flow data specified on the aerodynamic surface thereby demonstrating a means to effectively combine experimental and computational data the sensitivity of flow reconstruction results to available measurements and to experimental error is analyzed another new extension of this approach allows one to cast the problem of inverse airfoil design as a gappy data problem the gappy methodology demonstrates a great simplification for the inverse airfoil design problem and is found to work well on a range of examples including both subsonic and transonic cases

structure preserving model reduction of parametric hamiltonian systems while reduced order models roms have been popular for efficiently solving large systems of differential equations the stability of reduced models over long time integration presents challenges we present a greedy approach for a rom generation of parametric hamiltonian systems that captures the symplectic structure of hamiltonian systems to ensure stability of the reduced model through the greedy selection of basis vectors two new vectors are added at each iteration to the linear vector space to increase the accuracy of the reduced basis we use the error in the hamiltonian due to model reduction as an error indicator to search the parameter space and identify the next best basis vectors under natural assumptions on the set of all solutions of the hamiltonian system under variation of the parameters we show that the greedy algorithm converges at an exponential rate moreover we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplecti

global vector field reconstruction by using a multivariate polynomial l 2 approximation on nets a multivariate polynomial mathit l 2 approximation on nets is designed for global vector field reconstructions of time continuous dynamical systems the technique is tested by investigating standard forms of the r ossler band the lorenz mask and a chaotic attractor produced by a simple model of thermal lens oscillations

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

phase space reconstruction for symmetric dynamical systems abstract since the pioneering work of packard et al and takens it has become customary to reconstruct the topology of attractors in phase space from a time series of one dimensional experimental observations by using delay coordinates many practical refinements of the original methods have been developed many experimental systems possess symmetry and bifurcations can cause changes in the symmetry of observed states these changes are quite subtle when the dynamics is chaotic it is therefore important to reconstruct not just the topology of the attractor but its symmetry we indicate how this can be done by extending the packard takens approach to a single equivariant observation taking values not in the real numbers r but in a linear representation v of the symmetry group g in effect a single set of symmetrically related observations is required our central point is that not all plausible choices for such a set can generate embeddings in order for the method to produce an embedding it is necessary that v should be sufficiently complicated more precisely the phase space m must be subordinate to v in a sense introduced by wassermann this concept is technical but unavoidable in this context and it greatly clarifies the issue of embeddability using it we state a symmetric version of the takens embedding theorem and sketch the proof we also discuss the issue of setwise versus pointwise symmetry of an attractor and relate this to the transition from spatial order to spatial disorder in temporally chaotic systems

on selecting models for nonlinear time series abstract constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models or to choose between competing classes this paper discusses a method of building nonlinear models of possibly chaotic systems from data while maintaining good robustness against noise the models that are built are close to the simplest possible according to a description length criterion the method will deliver a linear model if that has shorter description length than a nonlinear model we show how our models can be used for prediction smoothing and interpolation in the usual way we also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series

entropy stable high order discontinuous galerkin methods with suitable quadrature rules for hyperbolic conservation laws abstract it is well known that semi discrete high order discontinuous galerkin dg methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws jiang and shu 1994 39 and symmetric hyperbolic systems hou and liu 2007 36 in any space dimension and for any triangulations however this property holds only for the square entropy and the integrations in the dg methods must be exact it is significantly more difficult to design dg methods to satisfy entropy inequalities for a non square convex entropy and or when the integration is approximated by a numerical quadrature in this paper we develop a unified framework for designing high order dg methods which will satisfy entropy inequalities for any given single convex entropy through suitable numerical quadrature which is specific to this given entropy our framework applies from one dimensional scalar cases all the way to multi dimensional systems of conservation laws for the one dimensional case our numerical quadrature is based on the methodology established in carpenter et al 2014 5 and gassner 2013 19 the main ingredients are summation by parts sbp operators derived from legendre gauss lobatto quadrature the entropy conservative flux within elements and the entropy stable flux at element interfaces we then generalize the scheme to two dimensional triangular meshes by constructing sbp operators on triangles based on a special quadrature rule a local discontinuous galerkin ldg type treatment is also incorporated to achieve the generalization to convection diffusion equations extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable dg methods

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

stabilization of projection based reduced order models summary a rigorous method for stabilizing projection based linear reduced order models without significantly affecting their accuracy is proposed unlike alternative approaches this method is computationally efficient it requires primarily the solution of a small scale convex optimization problem furthermore it is nonintrusive in the sense that it operates directly on readily available reduced order operators these can be precomputed using any data compression technique including balanced truncation balanced proper orthogonal decomposition proper orthogonal decomposition or moment matching the proposed method is illustrated with three applications the stabilization of the reduction of the computational fluid dynamics based model of a linearized unsteady supersonic flow the reduction of a computational structural dynamics system and the stabilization of the reduction of a coupled computational fluid dynamics computational structural dynamics model of a linearized aeroelastic system in the transonic flow regime copyright 2012 john wiley sons ltd

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts framework are used to derive entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations a semi discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator the data at the boundary are weakly imposed using a penalty flux approach and a simultaneous approximation term penalty technique although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy the new boundary conditions are compatible with any diagonal norm summation by parts spatial operator including finite element finite difference finite volume discontinuous galerkin and flux reconstruction correction procedure via reconstruction schemes the proposed boundary treatment is tested for three dimensional subsonic and supersonic flows the numerical computations corroborate the non linear stability entropy stability and accuracy of the boundary conditions

an lp empirical quadrature procedure for reduced basis treatment of parametrized nonlinear pdes abstract we present a model reduction formulation for parametrized nonlinear partial differential equations pdes our approach builds on two ingredients reduced basis rb spaces which provide rapidly convergent approximations to the parametric manifold sparse empirical quadrature rules which provide rapid evaluation of the nonlinear residual and output forms associated with the rb spaces we identify both the rb spaces and the sparse quadrature rules in the offline stage through a greedy training procedure over the parameter domain the procedure requires the dual norm of the finite element fe residual at many training points in the parameter domain but only very few fe solutions the snapshots retained in the rb space the quadrature rules are identified by a linear program lp empirical quadrature procedure eqp which i admits efficient solution by a simplex method and ii directly controls the solution error induced by the approximate quadrature we demonstrate the formulation for a parametrized neo hookean beam the dimension of the approximation space and the number of quadrature points are both reduced by two orders of magnitude relative to fe treatment with commensurate savings in computational cost

dynamical effects of overparametrization in nonlinear models abstract this paper is concemed with dynamical reconstruction for nonlinear systems the effects of the driving function and of the complexity of a given representation on the bifurcation patter are investigated it is shown that the use of different driving functions to excite the system may yield models with different bifurcation patterns the complexity of the reconstructions considered is quantified by the embedding dimension and the number of estimated parameters in this respect it appears that models which reproduce the original bifurcation behaviour are of limited complexity and that excessively complex models tend to induce ghost bifurcations and spurious dynamical regimes moreover some results suggest that the effects of overparametrization on the global dynamical behaviour of a nonlinear model may be more deleterious than the presence of moderate noise levels in order to precisely quantify the complexity of the reconstructions global polynomials are used although the results are believed to apply to a much wider class of representations including neural networks

structure preserving tangential interpolation for model reduction of port hamiltonian systems port hamiltonian systems result from port based network modeling of physical systems and are an important example of passive state space systems in this paper we develop a framework for model reduction of large scale multi input multi output port hamiltonian systems via tangential rational interpolation the resulting reduced model is a rational tangential interpolant that retains the port hamiltonian structure hence it remains passive we introduce an h 2 inspired algorithm for effective choice of interpolation points and tangent directions and present several numerical examples illustrating its effectiveness

split form nodal discontinuous galerkin schemes with summation by parts property for the compressible euler equations fisher and carpenter 2013 12 found a remarkable equivalence of general diagonal norm high order summation by parts operators to a subcell based high order finite volume formulation this equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux we show that besides the construction of entropy stable high order schemes a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms thus by changing the subcell finite volume flux to a specific choice we are able to generate in a systematic way all common split forms of the compressible euler advection terms such as the ducros splitting and the kennedy and gruber splitting although these split forms are not entropy stable we present a systematic way to prove which of those split forms are at least kinetic energy preserving with this we construct a unified high order split form dg framework we investigate with three dimensional numerical simulations of the inviscid taylorgreen vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows in fact we show that for certain test cases the novel split form discontinuous galerkin schemes are more robust than the discontinuous galerkin scheme with over integration

on discretely entropy conservative and entropy stable discontinuous galerkin methods abstract high order methods based on diagonal norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic pdes 1 2 these methods can also be interpreted as nodal discontinuous galerkin methods with diagonal mass matrices 3 4 5 6 in this work we describe how use flux differencing quadrature based projections and sbp like operators to construct discretely entropy conservative schemes for dg methods under more arbitrary choices of volume and surface quadrature rules the resulting methods are semi discretely entropy conservative or entropy stable with respect to the volume quadrature rule used numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible euler equations in one and two dimensions

local false nearest neighbors and dynamical dimensions from observed chaotic data the time delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension the minimum necessary global embedding dimension mathit d mathit e may still be larger than the actual dimension of the underlying dynamics mathit d mathit l the embedding theorem only guarantees that the attractor of the system is fully unfolded using mathit d mathit e greater than 2 mathit d mathit a with mathit d mathit a the fractal attractor dimension using the idea of local false nearest neighbors we discuss methods for determining the integer valued mathit d mathit l

efficient entropy stable gauss collocation methods the construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of gauss legendre lobatto collocation points and their equivalence

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

discretely conservative finite difference formulations for nonlinear conservation laws in split form theory and boundary conditions the lax wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities the discrete operator however need not be derived from the divergence form of the continuous equations indeed conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal norm skew symmetric summation by parts sbp spatial operator yield discrete operators that are conservative furthermore split form discretely conservation operators can be derived for periodic or finite domain sbp spatial operators of any order examples are presented of a fourth order sbp finite difference operator with second order boundary closures sixth and eighth order constructions are derived and are supplied in an accompanying text file

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

lower bounds for algebraic computation trees with integer inputs let w subseteq r n be scale invariant and rationally dispersed i e tilde x in w implies lambda tilde x in w for all real lambda 0 and the rationale are dense in both w and r n w it is shown that in the algebraic computation tree model the problem of deciding whether an input tilde x in r n with integer coordinates is a member of w has complexity omega log 2 hat beta w 2n where hat beta w is the number of connected components of w that are not of measure 0 this theorem can be used to prove tight lower bounds for the integral constrained form of many basic problems such as element distinctness set disjointness finding convex hull etc through further transformations it leads to lower bounds for problems such as integer max gap and closest pair of a simple polygon the proof involves a nontrivial extension of the milnor thorn techniques for finding upper bounds on the betti numbers of algebraic varieties

observability of multivariate differential embeddings the present paper extends some results recently developed for the analysis of observability in nonlinear dynamical systems the aim of the paper is to address the problem of embedding an attractor using more than one observable a multivariate nonlinear observability matrix is proposed which includes the monovariable nonlinear and linear observability matrices as particular cases using the developed framework and a number of worked examples it is shown that the choice of embedding coordinates is critical moreover in some cases to reconstruct the dynamics using more than one observable could be worse than to reconstruct using a scalar measurement finally using the developed framework it is shown that increasing the embedding dimension observability problems diminish and can even be eliminated this seems to be a physically meaningful interpretation of the takens embedding theorem

galerkin v least squares petrov galerkin projection in nonlinear model reduction least squares petrov galerkin lspg model reduction techniques such as the gauss newton with approximated tensors gnat method have shown promise as they have generated stable accurate solutions for large scale turbulent compressible flow problems where standard galerkin techniques have failed however there has been limited comparative analysis of the two approaches this is due in part to difficulties arising from the fact that galerkin techniques perform optimal projection associated with residual minimization at the time continuous level while lspg techniques do so at the time discrete level this work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators linear multistep schemes and runge kutta schemes we present a number of new findings including conditions under which the lspg rom has a time continuous representation conditions under which the two techniques are equivalent and time discrete error bounds for the two approaches perhaps most surprisingly we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the lspg rom instead the time step should be matched to the spectral content of the reduced basis in numerical experiments carried out on a turbulent compressible flow problem with over one million unknowns we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the lspg reduced order model by an order of magnitude

optimal reconstruction of dynamical systems a noise amplification approach in this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements these statistics can be evaluated on any reconstructed attractor thereby allowing a direct comparison among different approaches uniform or nonuniform delay vectors pca legendre coordinates etc it can also be used to select the most appropriate parameters of a reconstruction strategy in the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function its definition is based on theoretical arguments on noise amplification the complexity of the reconstructed attractor and a direct measure of local stretch which constitutes an irrelevance measure the proposed method is demonstrated on synthetic and experimental time series

stabilization of projection based reduced order models for linear time invariant systems via optimization based eigenvalue reassignment abstract a new approach for stabilizing unstable reduced order models roms for linear time invariant lti systems through an a posteriori post processing step applied to the algebraic rom system is developed the key idea is to modify the unstable eigenvalues of the rom system by moving these eigenvalues into the stable half of the complex plane it is demonstrated that this modification to the rom system eigenvalues can be accomplished using full state feedback a k a pole placement algorithms from control theory this approach ensures that the modified rom is stable provided the system s unstable poles are controllable and observable however the accuracy of the stabilized rom is not guaranteed to remedy this difficulty and guarantee an accurate stabilized rom a constrained nonlinear least squares optimization problem for the stabilized rom eigenvalues in which the error in the rom output is minimized is formulated this optimization problem is small and therefore computationally inexpensive to solve performance of the proposed algorithms is evaluated on two test cases for which roms constructed via the proper orthogonal decomposition pod galerkin method suffer from instabilities

entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible navier stokes equations staggered grid entropy stable discontinuous spectral collocation operators of any order are developed for the compressible euler and navier stokes equations on unstructured hexahedral elements this generalization of previous entropy stable spectral collocation work m h carpenter t c fisher e j nielsen and s h frankel siam j sci comput 36 2014 pp b835 b867 m parsani m h carpenter and e j nielsen j comput phys 292 2015 pp 88 113 extends the applicable set of points from tensor product legendre gauss lobatto lgl to a combination of tensor product legendre gauss lg and lgl points the new semidiscrete operators discretely conserve mass momentum energy and satisfy a mathematical entropy inequality for the compressible navier stokes equations in three spatial dimensions they are valid for smooth as well as discontinuous flows the staggered lg and conventional lgl point formulations are compared on several challenging test problems the staggered lg ope

entropy stable reduced order modeling of nonlinear conservation laws abstract reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model we introduce projection based hyper reduced models of nonlinear conservation laws which are globally conservative and inherit a semi discrete entropy inequality independently of the choice of basis and choice of parameters

speed scaling to manage energy and temperature speed scaling is a power management technique that involves dynamically changing the speed of a processor we study policies for setting the speed of the processor for both of the goals of minimizing the energy used and the maximum temperature attained the theoretical study of speed scaling policies to manage energy was initiated in a seminal paper by yao et al 1995 and we adopt their setting we assume that the power required to run at speed s is p s e s for some constant 1 we assume a collection of tasks each with a release time a deadline and an arbitrary amount of work that must be done between the release time and the deadline yao et al 1995 gave an offline greedy algorithm yds to compute the minimum energy schedule they further proposed two online algorithms average rate avr and optimal available oa and showed that avr is 2 1 competitive with respect to energy we provide a tight bound on the competitive ratio of oa with respect to energy we initiate the study of speed scaling to manage temperature we assume that the environment has a fixed ambient temperature and that the device cools according to newton s law of cooling we observe that the maximum temperature can be approximated within a factor of two by the maximum energy used over any interval of length 1 b where b is the cooling parameter of the device we define a speed scaling policy to be cooling oblivious if it is simultaneously constant competitive with respect to temperature for all cooling parameters we then observe that cooling oblivious algorithms are also constant competitive with respect to energy maximum speed and maximum power we show that yds is a cooling oblivious algorithm in contrast we show that the online algorithms oa and avr are not cooling oblivious we then propose a new online algorithm that we call bkp we show that bkp is cooling oblivious we further show that bkp is e competitive with respect to the maximum speed and that no deterministic online algorithm can have a better competitive ratio bkp also has a lower competitive ratio for energy than oa for 5 finally we show that the optimal temperature schedule can be computed offline in polynomial time using the ellipsoid algorithm

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

construction of phenomenological models from numerical scalar time series abstract the problem of reconstruction of ordinary differential equations from numerical scalar time series is discussed techniques are exemplified for rossler and lorenz chaotic attractors with emphasis on improved algorithms with respect to those previously published the steps which are still required in order to investigate experimental noisy data are discussed

structure preserving galerkin pod reduced order modeling of hamiltonian systems abstract the proper orthogonal decomposition reduced order model pod rom has been widely used as a computationally efficient surrogate model in large scale numerical simulations of complex systems however when it is applied to a hamiltonian system a naive application of the pod method can destroy the hamiltonian structure in the reduced order model in this paper we develop a new reduced order modeling approach for hamiltonian systems which modifies the galerkin projection based pod rom so that the appropriate hamiltonian structure is preserved since the pod truncation can degrade the approximation of the hamiltonian function we propose to use a pod basis from shifted snapshots to improve the approximation to the hamiltonian function we further derive a rigorous a priori error estimate for the structure preserving rom and demonstrate its effectiveness in several numerical examples this approach can be readily extended to dissipative hamiltonian systems port hamiltonian systems etc

symbolic observability coefficients for univariate and multivariate analysis in practical problems the observability of a system not only depends on the choice of observable s but also on the space which is reconstructed in fact starting from a given set of observables the reconstructed space is not unique since the dimension can be varied and in the case of multivariate measurement functions there are various ways to combine the measured observables using a graphical approach recently introduced we analytically compute symbolic observability coefficients which allow to choose from the system equations the best observable in the case of scalar reconstructions and the best way to combine the observables in the case of multivariate reconstructions it is shown how the proposed coefficients are also helpful for analysis in higher dimension

symplectic model reduction of hamiltonian systems in this paper a symplectic model reduction technique proper symplectic decomposition psd with symplectic galerkin projection is proposed to save the computational cost for the simplification of large scale hamiltonian systems while preserving the symplectic structure as an analogy to the classical proper orthogonal decomposition pod galerkin approach psd is designed to build a symplectic subspace to fit empirical data while the symplectic galerkin projection constructs a reduced hamiltonian system on the symplectic subspace for practical use we introduce three algorithms for psd which are based upon the cotangent lift complex singular value decomposition and nonlinear programming the proposed technique has been proven to preserve system energy and stability moreover psd can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear hamiltonian systems owing to these properties the proposed technique is better suited than the classical po

reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries roles of the inf sup stability constants in this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady stokes flows in affinely parametrized geometries focusing on the role played by the brezzi s and babuska s stability constants the crucial ingredients of the methodology are a galerkin projection onto a low dimensional space of basis functions properly selected an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables the combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction ideally suited for real time simulation and many query contexts e g optimization control or parameter identification in particular in this work we focus on i the stability of the reduced basis approximation based on the brezzi s saddle point theory and the introduction of a supremizer operator on the pressure terms ii a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the babuska s inf sup constant including residuals calculations iii the computation of a lower bound of the stability constant and iv different options for the reduced basis spaces construction we present some illustrative results for both interior and external steady stokes flows in parametrized geometries representing two parametrized classical poiseuille and couette flows a channel contraction and a simple flow control problem around a curved obstacle

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

structure preserving model reduction for mechanical systems this paper focuses on methods of constructing of reduced order models of mechanical systems which preserve the lagrangian structure of the original system these methods may be used in combination with standard spatial decomposition methods such as the karhunen loeve expansion balancing and wavelet decompositions the model reduction procedure is implemented for three dimensional finite element models of elasticity and we show that using the standard newmark implicit integrator significant savings are obtained in the computational costs of simulation in particular simulation of the reduced model scales linearly in the number of degrees of freedom and parallelizes well

the shifted proper orthogonal decomposition a mode decomposition for multiple transport phenomena transport dominated phenomena provide a challenge for common mode based model reduction approaches we present a model reduction method which is suited for these kinds of systems it extends the proper orthogonal decomposition pod by introducing time dependent shifts of the snapshot matrix the approach called shifted proper orthogonal decomposition spod features a determination of the multiple transport velocities and a separation of these one and two dimensional test examples reveal the good performance of the spod for transport dominated phenomena and its superiority in comparison to the pod

invariant domains and first order continuous finite element approximation for hyperbolic systems we propose a numerical method for solving general hyperbolic systems in any space dimension using forward euler time stepping and continuous finite elements on nonuniform grids the properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant set containing the initial data is an invariant domain for the method the invariant domain property is proved for any hyperbolic system provided a cfl condition holds the solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system the method is formally first order accurate in space and can be made high order in time by using strong stability preserving algorithms this technique extends to continuous finite elements the work of d hoff math comp 33 1979 pp 1171 1193 d hoff trans amer math soc 289 1985 pp 591 610 and h frid arch ration mech anal 160 2001 pp 245 269

construction of energy stable projection based reduced order models an approach for building energy stable galerkin reduced order models roms for linear hyperbolic or incompletely parabolic systems of partial differential equations pdes using continuous projection is developed this method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow the key idea is to apply to the pdes a transformation induced by the lyapunov function for the system and to build the rom in the transformed variables for linear problems the desired transformation is induced by a special inner product termed the symmetry inner product which is derived herein for several systems of physical interest connections are established between the proposed approach and other stability preserving model reduction methods giving the paper a review flavor more specifically it is shown that a discrete counterpart of this inner product is a weighted l 2 inner product obtained by solving a lyapunov equation first proposed by rowley et al and termed herein the lyapunov inner product comparisons between the symmetry inner product and the lyapunov inner product are made and the performance of roms constructed using these inner products is evaluated on several benchmark test cases

an efficient algorithm for computing optimal discrete voltage schedules we consider the problem of job scheduling on a variable voltage processor with d discrete voltage speed levels we give an algorithm which constructs a minimum energy schedule for n jobs in o d n log n time previous approaches solve this problem by first computing the optimal continuous solution in o n 3 time and then adjusting the speed to discrete levels in our approach the optimal discrete solution is characterized and computed directly from the inputs we also show that o n log n time is required hence the algorithm is optimal for fixed d

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

hyper reduction of mechanical models involving internal variables we propose to improve the efficiency of the computation of the reduced state variables related to a given reduced basis this basis is supposed to be built by using the snapshot proper orthogonal decomposition pod model reduction method in the framework of non linear mechanical problems involving internal variables the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model this drawback is due to the fact that using a galerkin formulation the size of the reduced basis has no effect on the complexity of the constitutive equations in this paper we show how a reduced basis approximation and a petrov galerkin formulation enable to reduce the computational effort related to the internal variables the key concept is a reduced integration domain where the integration of the constitutive equations is performed the local computations being not made over the entire domain we extrapolate the computed internal variable over the full domain using pod vectors related to the internal variables this paper shows the improvement of the computational saving obtained by the hyper reduction of the elasto plastic model of a simple structure

structure preserving model reduction for nonlinear port hamiltonian systems this paper presents a structure preserving model reduction approach applicable to large scale nonlinear port hamiltonian systems structure preservation in the reduction step ensures the retention of port hamiltonian structure which in turn ensures the stability and passivity of the reduced model our analysis provides a priori error bounds for both state variables and outputs three techniques are considered for constructing bases needed for the reduction one that utilizes proper orthogonal decompositions one that utilizes mathcal h 2 mathcal h infty derived optimized bases and one that is a mixture of the two the complexity of evaluating the reduced nonlinear term is managed efficiently using a modification of the discrete empirical interpolation method that also preserves port hamiltonian structure the efficiency and accuracy of this model reduction framework are illustrated with two examples a nonlinear ladder network and a tethered toda lattice

supremizer stabilization of pod galerkin approximation of parametrized steady incompressible navier stokes equations in this work we present a stable proper orthogonal decomposition pod galerkin approximation for parametrized steady incompressible navier stokes equations with low reynolds number supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced order system considering in particular the fulfillment of an inf sup condition the stability analysis is first carried out from a theoretical standpoint then confirmed by numerical tests performed on a parametrized two dimensional backward facing step flow

energy optimal schedules for jobs with multiple active intervals in this paper we study the scheduling problem of jobs with multiple active intervals each job in the problem instance has n n 1 disjoint active time intervals where it can be executed and a workload characterized by the required number of cpu cycles previously people studied multiple interval job scheduling problem where each job must be assigned enough cpu cycles in one of its active intervals we study a different practical version where the partial work done by the end of an interval remains valid and each job is considered finished if total cpu cycles assigned to it in all its active intervals reach the requirement the goal is to find a feasible schedule that minimizes energy consumption by adapting the algorithm for single interval jobs proposed in yao demers and shenker 1995 1 one can still obtain an optimal schedule however the two phases in that algorithm critical interval finding and scheduling the critical interval can no longer be carried out directly we present polynomial time algorithms to solve the two phases for jobs with multiple active intervals and therefore can still compute the optimal schedule in polynomial time

modeling and synchronizing chaotic systems from time series data the problem of obtaining ordinary differential equations ode s from chaotic time series data is addressed the vector fields for the ode s are polynomials constructed from a basis set that is orthonormal on the data the method for constructing the model is similar to the integration of ode s using adams predictor corrector integration the method is compared to the usual euler model and is shown to be accurate for much larger sampling intervals in addition the method used to construct the model is capable of determining the optimal polynomial vector field for the given data finally we demonstrate that it is possible to synchronize in the sense of fujisaka and yamada prog theor phys 69 32 1983 as well as pecora and carroll phys rev lett 64 821 1990 phys rev a 44 2374 1991 the model to a time series synchronization is used as a nontrivial test to determine how close the model vector field is to the true vector field implications and possible applications of synchronization are discussed