fast frequency sweep of fem models via the balanced truncation proper orthogonal decomposition a fast frequency sweep method for wideband antennas and infinite arrays based on a singular value decomposition svd krylov model reduction method for frequency domain tangential vector finite elements tvfems is presented reduced models are constructed using balanced congruence transformations constructed from the dominant invariant subspace of the system s hankel matrix traditionally forming such matrix requires the intensive computation of gramians the proposed method only forms their low rank cholesky factors via a novel adaptive proper orthogonal decomposition pod sampling strategy leading to significant savings unlike some other model reduction methods balanced truncation pod bt pod is directly applicable to lossy and dispersive electromagnetic models numerical studies on large scale wideband antennas and infinite arrays show that the method is stable error controllable and without memory overheads capable of up to two orders of magnitude speed ups

using automatic differentiation to create a nonlinear reduced order model aerodynamic solver a novel nonlinear reduced order modeling technique for computational aerodynamics and aeroelasticity is presented the method is based on a taylor series expansion of a frequency domain harmonic balance computational fluid dynamicsolver residual the first andsecond order gradientmatrices andtensorsof the taylorseries expansion are computed using automatic differentiation via fortran 90 95 operator overloading a ritz type expansion using proper orthogonal decomposition shapes is then used in the taylor series expansion to create the nonlinearreduced order model the nonlinear reduced order modeling technique is applied to a viscous flow about anaeroelastic nlr 7301 airfoil modelto determine limit cycle oscillations computational times are decreased from hours to seconds using the nonlinear reduced order model

geometric programming for aircraft design optimization formulating conceptual stage aircraft design problems as geometric programs which are a specific type of convex optimization problem is proposed recent advances in convex optimization offer sign

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

stabilization of explicit flow solvers using a proper orthogonal decomposition technique a new numerical technique for the stabilization of explicit computational fluid dynamic solvers is presented when using computational fluid dynamic codes to solve practical problems one sometimes finds the solution fails to converge due to the presence of a small number of eigenvalues of the flow solver outside the unit circle in this paper we use the proper orthogonal decomposition technique to estimate the unstable eigenvalues and eigenmodes of the flow solver as the solution diverges for linear problems or exhibits limit cycle oscillations for nonlinear problems we use the resulting eigeninformation to construct a preconditioner that repositions the unstable eigenvalues from outside the unit circle to a point inside the unit circle close to the origin resulting in a stable flow solver in this work the proposed method is applied to a nonlinear steady cascade solver however the technique can be easily applied to external flow solvers and to unsteady frequency domain time linearized and harmonic ba

proper orthogonal decomposition with high number of linear constraints for aerodynamical shape optimization shape optimization involving finite element analysis in engineering design is frequently hindered by the prohibitive cost of function evaluations reduced order models based on proper orthogonal decomposition pod constitute an economical alternative however the truncation of the pod basis implies an error in the calculation of the global values used as objectives and constraints which in turn affects the optimization results in our former contribution xiao and breitkopf 2013 we have introduced a constrained pod projector allowing for exact linear constraint verification for a reduced order model nevertheless this approach was limited a to relatively low numbers constraints therefore in the present paper we propose an approach for a high number of constraints the main idea is to extend the snapshot pod by introducing a new constrained projector in order to reduce both the physical field and the constraint space this allows us to search for the pareto set of best compromises between the projection and the constraint verification errors thereby enabling fine tuning of the reduced model for a particular purpose we illustrate the proposed approach with the reduced order model of the flow around an airfoil parameterized with shape variables

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

approximate solution of large sparse lyapunov equations describes a simple method for efficiently estimating the dominant eigenvalues and eigenvectors of the solution to a lyapunov equation without first solving the equation explicitly the method is based on the power method and matrix vector multiplications and is particularly suitable for problems where those multiplications can be done efficiently such as where the coefficient matrices are large and sparse or low rank the same idea is directly applicable to balanced truncation order reduction of linear systems

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

pod deim model order reduction for strain softening viscoplasticity we demonstrate a model order reduction technique for a system of nonlinear equations arising from the finite element method fem discretization of the three dimensional quasistatic equilibrium equation equipped with a perzyna viscoplasticity constitutive model the procedure employs the proper orthogonal decomposition galerkin pod g in conjunction with the discrete empirical interpolation method deim for this purpose we collect samples from a standard full order fem analysis in the offline phase and cluster them using a novel kk means clustering algorithm the pod and the deim algorithms are then employed to construct a corresponding reduced order model in the online phase a sample from the current state of the system is passed at each time step to a nearest neighbor classifier in which the cluster that best describes it is identified the force vector and its derivative with respect to the displacement vector are approximated using deim and the system of nonlinear equations is projected onto a lower dimensional subspace using the pod g the constructed reduced order model is applied to two typical solid mechanics problems showing strain localization a tensile bar and a wall under compression and a three dimensional square footing problem

a randomized balanced proper orthogonal decomposition technique abstract in this paper we consider the model reduction problem of large scale linear systems such as systems obtained through the discretization of partial differential equations we propose a randomized proper orthogonal decomposition rpod technique to obtain the reduced order model by perturbing the primal and adjoint system using gaussian white noise we show that computations required by rpod algorithm is orders of magnitude lower while its performance is much better than other state of the art algorithms we also relate the rpod algorithm to krylov subspace methods and show that it constitutes a randomized approach to computational linear algebra problems that utilize krylov subspace methods

efficient surrogate modelling of nonlinear aerodynamics in aerostructural coupling schemes this paper considers the substitution of the computational fluid dynamics solver within an aerostructural coupling scheme by an efficient reduced order surrogate model to realize high fidelity nonlinear aeroelastic analyses within a fraction of the time compared to the corresponding coupled computational fluid dynamics analysis the presented surrogate model approach is a combination of parameter reduction via proper orthogonal decomposition and system identification methods designed to cover nonlinear aerodynamic effects such as viscosity as well as transient effects the investigated test case is the thick nlr7301 airfoil that shows significant nonlinear aerodynamic behavior along with nonnegligible viscous effects the model is identified from a set of transient forced motion computational fluid dynamics analyses after identification the model is used to predict the discrete surface force distribution of the nlr7301 airfoil in static as well as transient coupled analyses it is shown that the static

towards simultaneous reduction of both input and output spaces for interactive simulation based structural design abstract engineering design problems generally involve a high dimensional input space of design variables yielding an output space by means of costly high fidelity evaluations in order to decrease the overall cost reduced order models for the output space such as proper orthogonal decomposition pod and proper generalized decomposition pgd are an active area of research however little research has been conducted into alleviating the problems associated with a high dimensional input space in addition to higher dimensionality being an impediment to efficient design by itself complex shapes involve a high number of explicit implicit constraints restricting the design space geometric parameterization methods in traditional cad present difficulties in expressing these constraints leading to a high failure rate and the generation of inadmissible shapes in this paper we propose a simultaneous meta modeling protocol for both input and output spaces we perform a reparametrization of the input space using constrained shape interpolation by introducing the concept of an manifold of admissible meshed shapes the output space is reduced using constrained proper orthogonal decomposition by simultaneously using meta modeling for both spaces we facilitate interactive design space exploration for the purpose of design the proposed approach is applied to the industrial problem of designing a car engine intake port

a zonal galerkin free pod model for incompressible flows abstract a domain decomposition method which couples a high and a low fidelity model is proposed to reduce the computational cost of a flow simulation this approach requires to solve the high fidelity model in a small portion of the computational domain while the external field is described by a galerkin free proper orthogonal decomposition pod model we propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models this zonal approach can be used to study multi body configurations or to perform detailed local analyses in the framework of shape optimisation problems the efficiency of the method to perform predictive low cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem

experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model 1 an optimal experimental design algorithm is developed to select locations for a network of observation wells that provide maximum information about unknown groundwater pumping in a confined anisotropic aquifer the design uses a maximal information criterion that chooses among competing designs the design that maximizes the sum of squared sensitivities while conforming to specified design constraints the formulated optimization problem is non convex and contains integer variables necessitating a combinatorial search given a realistic large scale model the size of the combinatorial search required can make the problem difficult if not impossible to solve using traditional mathematical programming techniques genetic algorithms gas can be used to perform the global search however because a ga requires a large number of calls to a groundwater model the formulated optimization problem still may be infeasible to solve as a result proper orthogonal decomposition pod is applied to the groundwater model to reduce its dimensionality then the information matrix in the full model space can be searched without solving the full model results from a small scale test case show identical optimal solutions among the ga integer programming and exhaustive search methods this demonstrates the ga s ability to determine the optimal solution in addition the results show that a ga with pod model reduction is several orders of magnitude faster in finding the optimal solution than a ga using the full model the proposed experimental design algorithm is applied to a realistic two dimensional large scale groundwater problem the ga converged to a solution for this large scale problem

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

nonlinear proper orthogonal decomposition based model of forced convection heat transfer in pulsating flow a nonlinear low order physics based model for the dynamics of forced convection wall heat transfer in pulsating flow is formulated based on the proper orthogonal decomposition technique in a multivariate approach proper orthogonal decomposition modes are constructed from computational fluid dynamics data for laminar flow and heat transfer over a flat plate in pulsating flow spanning a range of pulsation frequencies and amplitudes then the conservation equations for mass momentum and energy are projected onto the proper orthogonal decomposition modes such that a system of ordinary differential equations for the modal amplitudes is obtained the forcing at the inlet is written explicitly in the ordinary differential equations of the low order model the contribution of the nonvanishing pressure term resulting from the incompressible navier stokes equation is included with a calibration method the accuracy and stability of the low order model are evaluated by comparison with computational fluid dy

constraint reduction procedures for reduced order subsurface flow models based on pod tpwl summary the properties and numerical performance of reduced order models based on trajectory piecewise linearization tpwl and proper orthogonal decomposition pod are assessed the target application is subsurface flow modeling although our findings should be applicable to a range of problems the errors arising at each step in the pod tpwl procedure are described the impact of constraint reduction on accuracy and stability is considered in detail constraint reduction entails projection of the overdetermined system into a low dimensional subspace in which the system is solvable optimality conditions for constraint reduction in terms of error minimization are derived galerkin and petrov galerkin projections are shown to correspond to optimality in norms that involve weighting with the jacobian matrix two new treatments inverse projection and weighted inverse projection are suggested these methods minimize error in appropriate norms although they require substantial preprocessing computations numerical results are presented for oil reservoir simulation problems galerkin projection provides reasonable accuracy for simpler oil water systems although it becomes unstable in more challenging cases petrov galerkin projection is observed to behave stably in all cases considered weighted inverse projection also behaves stably and it provides the highest accuracy runtime speedups of 150 400 are achieved using these pod tpwl models copyright 2015 john wiley sons ltd

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

reduced order model for viscous aerodynamic flow past an airfoil a method is presented to obtain reduced order models of multiparametric steady transonic aerodynamic flows around two dimensional airfoils the method is based on proper orthogonal decomposition and appropriate projection on the proper orthogonal decomposition manifold of the governing equations and boundary conditions and it includes various ingredients that are new in this field proper orthogonal decomposition modes are obtained from a set of snapshots calculated by computational fluid dynamics based on the compressible navier stokes equations and a turbulence model but projection on the proper orthogonal decomposition manifold is made using the inviscid euler equations which makes the method independent of the computational fluid dynamics scheme projection is made using only a limited number of mesh points obtaining quite good results shock waves are treated specifically in the proper orthogonal decomposition description to avoid the need of using a too large number of snapshots the method is checked and discussed on a specific aerodynamic problem

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

an hp proper orthogonal decomposition moving least squares approach for molecular dynamics simulation abstract the application of model order reduction for molecular dynamic systems exhibits intrinsic complexities due to the highly nonlinear and nonlocal properties of such systems the most costly computational part of such work is the calculation of potential energy and inter atomic force fields we introduce the so called hp proper orthogonal decomposition moving least squares hp pod mls method not only to tackle this force computation but also to improve the accuracy of reduced displacement approximations in an adaptive and controllable manner the approach combines the proper orthogonal decomposition pod method with the moving least squares mls one to interpolate the inter atomic force field at an arbitrary time instance this approximated force field is then fed into the reduced molecular dynamic md governing equation to compute the reduced order displacement field in fact this pod mls approach is already able to handle efficiently the force computation in the online computational stage i e the online force computational cost is independent of the dimension of the full md space however its corresponding approximated displacement error will be very large if the considered total time interval is long the proposed hp pod mls algorithm will divide adaptively and automatically the global temporal interval into many smaller local subintervals h refinement and then build the sets of local reduced basis functions over each of these subintervals p refinement the goal is to reduce the corresponding displacement errors to a desired tolerance over these subintervals the effectiveness of the algorithm is demonstrated by performing reduced order simulations of several molecular dynamic systems

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

projection based reduction of fluid structure interaction systems using monolithic space time modes abstract the focus of this work is the development of reduced models for engineering applications in complex bidirectional fluid structure interaction in the simultaneous solution procedure velocity variables are used for both fluid and solid and the whole set of model equations is discretized by a stabilized time discontinuous space time finite element method flexible structures are modeled using a three dimensional continuum approach in a total lagrangian setting considering large displacements and rotations in the flow domain the incompressible navier stokes equations describe the newtonian fluid a continuous finite element mesh is applied to the entire spatial domain and the discretized model equations are assembled in a single set of algebraic equations considering the two field problem as a whole the continuous fluid structure mesh with identical orders of approximation for both solid and fluid in space and time automatically yields conservation of mass momentum and energy at the fluid structure interface a mesh moving scheme is used to adapt the nodal coordinates of the fluid space time finite element mesh to the structural deformation the computational approach for strongly coupled fluid structure interaction is used to create suitable reduced models of generic nonlinear problems reduction is performed with monolithic projection based space time modes ensuring strong coupling of fluid and structure in the reduced model the contribution discusses results using proper orthogonal decomposition pod for determination of monolithic space time modes in the reduction of fluid structure systems

a posteriori error estimators for linear reduced order models using krylov based integrators summary reduced order models for linear time invariant dynamical systems are considered and the error between the full order model and the reduced order model solutions is characterized based on the analytical representation of the error an a posteriori error indicator is proposed that combines a krylov based exponential integrator and an a posteriori residual based estimate numerical experiments illustrate the quality of the error estimator copyright 2014 john wiley sons ltd

nonlinear operations on a class of modulation spaces abstract we discuss when the nonlinear operation f f f maps the modulation space m s p q r n 1 p q to the same space again it is known that m s p q r n is a multiplication algebra when s n n q hence it is true for this space if f is entire we claim that it is still true for non analytic f when q 4 3

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

pod deim reduced order strategies for efficient four dimensional variational data assimilation this work studies reduced order modeling rom approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models it is shown that a key requirement for a successful reduced order solution is that reduced order karush kuhn tucker conditions accurately represent their full order counterparts in particular accurate reduced order approximations are needed for the forward and adjoint dynamical models as well as for the reduced gradient new strategies to construct reduced order based are developed for proper orthogonal decomposition pod rom data assimilation using both galerkin and petrov galerkin projections for the first time pod tensorial pod and discrete empirical interpolation method deim are employed to develop reduced data assimilation systems for a geophysical flow model namely the two dimensional shallow water equations numerical experiments confirm the theoretical framework for galerkin projection in the case of petrov galerkin projection stabilization strategies must be considered for the reduced order models the new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time

finite time balanced truncation for linear systems via shifted legendre polynomials abstract in this paper we present a finite time model order reduction method for linear systems via shifted legendre polynomials the main idea of the approach is to use finite time empirical gramians which are constructed from impulse responses by solving block tridiagonal linear systems to generate approximate balanced system for the large scale system the balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted legendre polynomials without individual reduction of the gramians and a separate eigenvector solve then the reduced order model is constructed by truncating the states corresponding to the small approximate hankel singular values hsvs the stability preservation of the reduced model is briefly discussed and in combination with the dominant subspace projection method we modify the reduction procedure to alleviate the shortcomings of the above method which may unexpectedly lead to unstable systems even though the original one is stable furthermore the properties of the resulting reduced models are considered finally numerical experiments are given to demonstrate the effectiveness of the proposed methods

an efficient output error estimation for model order reduction of parametrized evolution equations in this work we present an a posteriori output error bound for model order reduction of parametrized evolution equations with the help of the dual system and a simple representation of the relationship between the field variable error and the residual of the primal system a sharp output error bound is derived such an error bound successfully avoids the accumulation of the residual over time which is a common drawback in the existing error estimations for time stepping schemes an estimation needs to be performed for practical computation of the error bound and as a result the output error bound reduces to an output error estimation the proposed error estimation is applied to four kinds of problems the first one is a linear convection diffusion equation which is used to compare the performance of the new error estimation and an existing primal dual error bound the second one is the unsteady viscous burgers equation an academic benchmark of nonlinear evolution equations in fluid dynamics often u

constrained proper orthogonal decomposition based on qr factorization for aerodynamical shape optimization while performing numerical simulations for design optimization one of the major issues is reaching a good compromise between accuracy and computational effort simulation methods such as finite elements finite volumes etc provide high fidelity numerical solutions at a cost which may be prohibitive in optimization problems requiring frequent calls to the discretized equations solver the proper orthogonal decomposition pod constitutes an economical and efficient option to decrease the cost of the solution however the truncation of the pod basis implies an error in the calculation of the global quantities used as objectives and optimization constraints which in turn might bias the optimization results our idea is thus to improve the snapshot pod by means of a reduced basis constructed to provide an exact interpolation of the quantities of interest obtained by integration of the physical fields to this end we reformulate the pod as a minimization problem where the desired properties are expressed as a set of constraints impacting the calculation of both the modes and the coefficients the main contribution of this paper is to provide a detailed mathematical justification for our constrained variant of the pod including a graphical interpretation of the proposed approach the constrained pod method is then applied to the problem of representing the pressure field around a 2d wing and is compared with the traditional pod

eigenvalue calculation procedure for an euler navier stokes solver with application to flows over airfoils abstract a lanczos procedure is applied to a navier stokes solver for computing eigenvalues and eigenvectors these eigenvalues and eigenvectors are associated with small perturbation analysis of a finite difference representation of the navier stokes equations for flows over airfoils a combination of block tridiagonal matrices is converted into a two dimensional matrix for this eigensystem calculation this matrix is very large sparse real and nonsymmetric a separate procedure based on lopsided iteration is also used to determine the eigensystem the results from these two procedures are compared the lanczos procedure provides complete spectral information about the eigenvalues whereas the lopsided iteration provides only a few of the eigenvalues which are largest in magnitude and the corresponding eigenvectors such eigensystem information is central to transient stability analysis of navier stokes solvers for determining the modal behavior of fluid in a fluid structure interaction problem and for development of reduced order models based on variational principles for navier stokes solvers

a nonlinear modeling approach using weighted piecewise series and its applications to predict unsteady flows to preserve nonlinearity of a full order system over a range of parameters of interest we propose an accurate and robust nonlinear modeling approach by assembling a set of piecewise linear local solutions expanded about some sampling states the work by rewienski and white 1 on micromachined devices inspired our use of piecewise linear local solutions to study nonlinear unsteady aerodynamics these local approximations are assembled via nonlinear weights of radial basis functions the efficacy of the proposed procedure is validated for a two dimensional airfoil moving with different pitching motions specifically agard s ct2 and ct5 problems 27 in which the flows exhibit different nonlinear behaviors furthermore application of the developed aerodynamic model to a two dimensional aero elastic system proves the approach is capable of predicting limit cycle oscillations lcos by using agard s ct6 28 as a benchmark test all results based on inviscid solutions confirm that our nonlinear model is stable and accurate against the full model solutions and measurements and for predicting not only aerodynamic forces but also detailed flowfields moreover the model is robust for inputs that considerably depart from the base trajectory in form and magnitude this modeling provides a very efficient way for predicting unsteady flowfields with varying parameters because it needs only a tiny fraction of the cost of a full order modeling for each new condition the more cases studied the more savings rendered hence the present approach is especially useful for parametric studies such as in the case of design optimization and exploration of flow phenomena

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

on l 1 approximation preface 1 preliminaries 2 approximation from finite dimensional subspaces of l1 3 approximation from finite dimensional subspaces in c1 k 4 unicity subspaces and property a 5 one sided l1 approximation 6 discrete lm1 approximation 7 algorithms appendices references author index subject index

reduced order model for the bgk equation based on pod and optimal transport we present a reduced order approximation of the bgk equation leading to fast and accurate computations the bgk model describes the dynamics of a gas flow in both hydrodynamic and rarefied regimes the particles of the gas are represented by a density distribution function depending on physical space velocity space and time in this work the density distribution function is approximated in the velocity space by a small number of basis functions computed offline in the offline phase the bgk equation is sampled in order to collect information on the density distribution function to complete this sampling optimal transport is used to add new information by interpolating the samples of the density distribution function finally the basis functions are built by proper orthogonal decomposition during the online phase the offline knowledge is used to compute approximations of the density distribution function at low cost to do so the bgk equation is projected onto the basis functions leading to a system of partial differential equations the system obtained is hyperbolic by construction and is solved by an imex runge kutta scheme in time and a finite volume scheme in space to improve the accuracy the reduced oder model is modified to conserve mass momentum and energy of the gas numerical illustrations of 1d and 2d flows are given in particular we investigate the reconstruction and the prediction of shock waves boundary layers and vortices the results demonstrate the accuracy of the reduced order model and the significant reduction of the computational cost

frequency domain karhunen loeve method and its application to linear dynamic systems forthee rsttime thekarhunen loeve kl procedureisderivedinthefrequencydomainasatoolforcalculating eigenmodes of linear systems the new derivation is based on the discrete fourier transform representation of a time average of proe le energy including a proper system response and a proe le function taking the variational problem posed as such with respect to the proe le function leads to an eigenformulation in the frequency domain choice of a system response for efe cient kl mode calculation and construction of reduced order systems using the kl eigenmodes are also discussed to demonstrate themethod both mechanical and e uid dynamic models are considered the method is equally useful in extracting eigenmodes of an experimentally generated database nomenclature c wing chord length f snapshot matrix as dee ned in eq 13 f fourier operator gik impulse response for the kth input gsk step response for the kth input i

proper orthogonal decomposition based reduced order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible boussinesq equations abstract in this study we employ a proper orthogonal decomposition pod method to establish a pod based reduced order stabilized mixed finite volume element smfve extrapolating model with very few degrees of freedom for the nonstationary incompressible boussinesq equations we provide error estimates of the pod based reduced order smfve solutions and the algorithm implementation for the pod based reduced order smfve extrapolating model in addition we present some numerical experiments that verify the validity and reliability of the pod based reduced order smfve extrapolating model

comparative numerical analysis using reduced order modeling strategies for nonlinear large scale systems we perform a comparative analysis using three reduced order strategies missing point estimation mpe method gappy pod method and discrete empirical interpolation method deim applied to a biological model describing the spatio temporal dynamics of a predator prey community the comparative study is focused on the efficiency of the reduced order approximations and the complexity reduction of the nonlinear terms different variants are discussed related to the projection based model reduction framework combined with selective spatial sampling to efficiently perform the online computations numerical results are presented

a manifold learning based reduced order model for springback shape characterization and optimization in sheet metal forming abstract the parameters of a stamping process include the geometry of the tools the shape of the initial sheet blank the material constitutive law and the process parameters when designing the overall process one has to also take into account the springback effect that appears when the tools are removed and additional surfaces are cut off the goal then is to obtain a final shape as close as possible to the desired shape while satisfying the admissibility constraints on the variable parameters as well as the feasibility constraints frequently expressed in the form of forming limit diagrams in the present paper we represent the post springback shape by a level set function then rather than rely on arbitrarily selected case dependent measurement locations as in the numisheet benchmark problems we build a reduced order shape space where this level set evolves by extending our recent shape manifold approach to the problem of springback assessment for 3d shapes next we propose an optimization algorithm designed to minimize the gap between the post springback and the desired final shapes the required level set functions are generated from a corresponding set of springback shapes predicted by finite element simulations using our approach we determine the minimal number of parameters needed in order to uniquely characterize the final formed shape regardless of complexity finally we demonstrate the approach using an industrial test case springback assessment of the deep drawing operation of an automotive strut tower

real time parameter identification and solution reconstruction from experimental data using the proper generalized decomposition abstract some industrial processes are modelled by parametric partial differential equations integrating computational modelling and data assimilation into the control process requires obtaining a solution of the numerical model at the characteristic frequency of the process real time this paper introduces a computational strategy allowing to efficiently exploit measurements of those industrial processes providing the solution of the model at the required frequency this is particularly interesting in the framework of control algorithms that rely on a model involving a set of parameters for instance the curing process of a composite material is modelled as a thermo mechanical problem whose corresponding parameters describe the thermal and mechanical behaviours in this context the information available measurements is used to update the parameters of the model and to produce new values of the control variables data assimilation the methodology presented here is devised to ensure the possibility of having a response in real time of the problem and therefore the capability of integrating it in the control scheme the proper generalized decomposition is used to describe the solution in the multi parametric space the real time data assimilation requires a further simplification of the solution representation that better fits the data reconstructed solution and it provides an implicit parameter identification moreover the analysis of the assimilated data sensibility with respect to the points where the measurements are taken suggests a criterion to locate the sensors

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

a posteriori error estimation and basis adaptivity for reduced basis approximation of nonaffine parametrized linear elliptic partial differential equations in this paper we extend the earlier work m barrault y maday n c nguyen a t patera an empirical interpolation method application to efficient reduced basis discretization of partial differential equations c r acad sci paris serie i 339 2004 667 672 m a grepl y maday n c nguyen a t patera efficient reduced basis treatment of nonaffine and nonlinear partial differential equations m2an math model numer anal 41 3 2007 575 605 to provide a posteriori error estimation and basis adaptivity for reduced basis approximation of linear elliptic partial differential equations with nonaffine parameter dependence the essential components are i rapidly convergent reduced basis approximations galerkin projection onto a space w n u spanned by n global hierarchical basis functions which are constructed from solutions of the governing partial differential equation at judiciously selected points in parameter space ii stable and inexpensive interpolation procedures methods which allow us to replace nonaffine parameter functions with a coefficient function expansion as a sum of m products of parameter dependent coefficients and parameter independent functions iii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp error bounds for the error in the outputs of interest iv optimal basis construction processes which make use of the error bounds as an inexpensive surrogate for the expensive true error to explore the parameter space in the quest for an optimal sampling set and v offline online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bounds depends only on n m and the affine parametric complexity of the problem the method is thus ideally suited for repeated and reliable evaluation of input output relationships in the many query or real time contexts

stabilized reduced order models for the advection diffusion reaction equation using operator splitting reduced order modeling rom coupled with finite element methods has been used effectively in many disciplines to efficiently solve complex problems however for advection dominated flows numerical simulations often contain spurious nonphysical oscillations which will also be apparent in the rom simulations in this work we consider stabilization methods for rom for the advection diffusion reaction adr equation when it is solved both with and without operator splitting specifically we consider the streamline upwind petrov galerkin supg stabilization method and the spurious oscillations at layers diminishing sold stabilization method we build on these methods by constructing a coherent framework which successfully integrates these model reduction stabilization and operator splitting approaches and we provide numerical examples detailing the application of this framework in the adr setting the stabilized rom results are compared numerically with their corresponding full finite element simulations

model reduction and perturbation analysis of wave finite element formulations for computing the forced response of coupled elastic systems involving junctions with uncertain eigenfrequencies abstract the wave finite element method is investigated for computing the low and mid frequency forced response of coupled elastic systems involving straight structures with junctions the relevance of the method is discussed when a component mode synthesis procedure is used for modeling the junctions a norm wise selection criterion is proposed so as to reduce efficiently the number of junction modes retained in the wave based formulations component wise perturbation bounds of the wave based displacement force solutions are also derived to address slight uncertainties for the junction eigenfrequencies numerical comparisons with standard finite element solutions as well as monte carlo simulations highlight the relevance of the formulation

a multiscale method for model order reduction in pde parameter estimation abstract estimating parameters of partial differential equations pdes is of interest in a number of applications such as geophysical and medical imaging parameter estimation is commonly phrased as a pde constrained optimization problem that can be solved iteratively using gradient based optimization a computational bottleneck in such approaches is that the underlying pdes need to be solved numerous times before the model is reconstructed with sufficient accuracy one way to reduce this computational burden is by using model order reduction mor techniques such as the multiscale finite volume method msfv in this paper we apply msfv for solving high dimensional parameter estimation problems given a finite volume discretization of the pde on a fine mesh the msfv method reduces the problem size by computing a parameter dependent projection onto a nested coarse mesh a novelty in our work is the integration of msfv into a pde constrained optimization framework which updates the reduced space in each iteration we present a computationally tractable way of explicitly differentiating the mor solution that acknowledges the change of basis as we demonstrate in our numerical experiments our method leads to computational savings for large scale parameter estimation problems where iterative pde solvers are necessary and offers potential for additional speed ups through parallel implementation

pod assisted strategies for structural topology optimization abstract we propose a new numerical tool for structural optimization design to cut down the computational burden typical of the solid isotropic material with penalization simp method we apply proper orthogonal decomposition on simp snapshots computed on a fixed grid to construct a rough structure predictor which becomes the input of a simp procedure performed on an anisotropic adapted mesh corrector the benefit of the proposed design tool is to deliver smooth and sharp layouts which require a contained computational effort before moving to the 3d printing production phase

non linear petrov galerkin methods for reduced order modelling of the navier stokes equations using a mixed finite element pair abstract a new nonlinear petrov galerkin approach has been developed for proper orthogonal decomposition pod reduced order modelling rom of the navier stokes equations the new method is based on the use of the cosine rule between the advection direction in cartesian space time and the direction of the gradient of the solution a finite element pair p 1 dg p 2 which has good balance preserving properties is used here consisting of a mix of discontinuous for velocity components and continuous for pressure basis functions the contribution of the present paper lies in applying this new non linear petrov galerkin method to the reduced order navier stokes equations and thus improving the stability of rom results without tuning parameters the results of numerical tests are presented for a wind driven 2d gyre and the flow past a cylinder which are simulated using the unstructured mesh finite element cfd model in order to illustrate the numerical performance of the method the numerical results obtained show that the newly proposed pod petrov galerkin method can provide more accurate and stable results than the pod bubnov galerkin method

certified reduced basis methods for parametrized saddle point problems we present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems first we develop new a posteriori error estimates that unlike earlier approaches provide upper bounds for the errors in the approximations of the primal variable and the lagrange multiplier separately the proposed method is an application of brezzi s theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods second based on an analysis of brezzi s theory we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost finally we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and compared to earlier approaches computationally much more efficient the method is applied to a stokes flow problem in a two dimensional channel with a parametrized rectangular obstacle

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

a variational approach for parameter estimation based on balanced proper orthogonal decomposition abstract the adjoint method has been used significantly for parameter estimation this requires a significant programming effort to build adjoint model code and is computationally expensive as cost of one adjoint simulation often exceeds several original model runs the work proposed here is variational data assimilation based on balanced proper orthogonal decomposition bpod to identify uncertain parameters in numerical models and avoids the implementation of the adjoint with respect to model input parameters an ensemble of model simulations forward and backward is used to determine the model subspace while considering both inputs and outputs of the system by projecting the original model onto this subspace an approximate linear reduced model is obtained the adjoint of the tangent linear model is replaced by the adjoint of linear reduced model and the minimization problem is then solved in the reduced space at very low computational cost the performance of the method are illustrated with a number of data assimilation experiments in a 2d advection diffusion model the results demonstrate that the bpod based estimation approach successfully estimates the diffusion coefficient for both advection and diffusion dominated problems the paper also proposes an efficient method for computing the observable subspace when the number of observations is large is also proposed

finite rank adi iteration for operator lyapunov equations we give an algorithmic approach to the approximative solution of operator lyapunov equations for controllability motivated by the successfully applied alternating direction implicit adi iteration for matrix lyapunov equations we consider this method for the determination of gramian operators of infinite dimensional control systems in the case where the input space is finite dimensional this method provides approximative solutions of finite rank under the assumption of infinite time admissibility and boundedness of the semigroup we analyze convergence in several operator norms we show that under a mild assumption on the shift parameters convergence to the gramian is obtained particular emphasis is placed on systems governed by a heat equation with boundary control we present that adi iteration for the heat equation consists of solving a sequence of helmholtz equations two numerical examples are presented the first showing the benefit of adaptive finite elements and the second illustrating con

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

a priori space time separated representation for the reduced order modeling of low reynolds number flows abstract reduced order models roms in fluid dynamics are nowadays mostly developed by performing a projection of the navier stokes equations onto a low dimensional space basis this basis is usually obtained through proper orthogonal decomposition pod which remains one the most efficient techniques to compress precomputed data the main drawback of a posteriori pod based roms is however their lacks of reliability as their parameters are varied preventing their direct uses within optimization algorithms the goal of the present article is to obtain an a priori low dimensional space time separated representation of the fluid fields without precomputed data the approach is based on the use of space time proper generalized decomposition pgd definitions which are successfully applied in several fields but whose uses in fluid dynamics remain scarce their applications to the navier stokes equations are indeed not straightforward due to the pressure velocity coupling the divergence free constraint and the non linear convective term the roms are built here from a space time weak formulation of the chorin temam prediction correction scheme more particularly a priori space time separated representations are obtained by applying the galerkin based progressive pgd definition the minimax pgd definition is moreover experimented on a linear stokes simplified case the related algorithms are explicitly given and illustrated on a transient lid driven cavity flow it is shown that the a priori space time separated representations converge to the full model solution as the decomposition order increases the ability of the resulting rom to learn iteratively from its own error is highlighted the progressive pgd algorithm can be used to effectively enrich incomplete pod and pgd precomputed space bases such as those obtained with different parameter values this may allow to ensure the accuracy of a rom as the parameter is varied which is of crucial interest for optimization problems

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

projection based model reduction for contact problems large scale finite element analysis requires model order reduction for computationally expensive applications such as optimization parametric studies and control design although model reduction for nonlinear problems is an active area of research a major hurdle is modeling and approximating contact problems this manuscript introduces a projection based model reduction approach for static and dynamic contact problems in this approach non negative matrix factorization is utilized to optimally compress and strongly enforce positivity of contact forces in training simulation snapshots moreover a greedy algorithm coupled with an error indicator is developed to efficiently construct parametrically robust low order models the proposed approach is successfully demonstrated for the model reduction of several two dimensional elliptic and hyperbolic obstacle and self contact problems corresponding author email address maciej balajewicz stanford edu maciej balajewicz 1postdoctoral fellow 2engineering research associate 3vivian church hoff professor of aircraft structures

a stable galerkin reduced order model for coupled fluid structure interaction problems summary a stable reduced order model rom of a linear fluid structure interaction fsi problem involving linearized compressible inviscid flow over a flat linear von karman plate is developed separate stable roms for each of the fluid and the structure equations are derived both roms are built using the continuous galerkin projection approach in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product the mode shapes for the structure rom are the eigenmodes of the governing linear plate equation the fluid rom basis is constructed via the proper orthogonal decomposition for the linearized compressible euler fluid equations a symmetry transformation is required to obtain a stable formulation of the galerkin projection step in the model reduction procedure stability of the galerkin projection of the structure model in the standard l2 inner product is shown the fluid and structure roms are coupled through solid wall boundary conditions at the interface plate boundary an a priori energy linear stability analysis of the coupled fluid structure system is performed it is shown that under some physical assumptions about the flow field the fsi rom is linearly stable a priori if a stabilization term is added to the fluid pressure loading on the plate the stability of the coupled rom is studied in the context of a test problem of inviscid supersonic flow past a thin square elastic rectangular panel that will undergo flutter once the non dimensional pressure parameter exceeds a certain threshold this a posteriori stability analysis reveals that the fsi rom can be numerically stable even without the addition of the aforementioned stabilization term moreover the rom constructed for this problem properly predicts the maintenance of stability below the flutter boundary and gives a reasonable prediction for the instability growth rate above the flutter boundary copyright 2013 john wiley sons ltd

eigenanalysis of unsteady flow about airfoils cascades and wings a general technique for constructing reduced order models of unsteady aerodynamic flows about two dimensional isolated airfoils cascades of airfoils and three dimensional wings is developed the starting point is a the domain computational model of the unsteady small disturbance flow for illustration purposes we apply the technique to an unsteady incompressible vortex lattice model the eigenmodes of the system which may be thought of as aerodynamic states are computed and subsequently used to construct computationally efficient reduced order models of the unsteady flowfield only a handful of the most dominant eigenmodes are retained in the reduced order model the effect of the remaining eigenmodes is included approximately using a static correction technique an important advantage of the present method is that once the eigenmode information has been computed reduced order models can be constructed for any number of arbitrary modes of airfold motion very inexpensively numerical examples are presented that demonstrate the accuracy and computational efficiency of the present method finally we show how the reduced order model may be incorporated into an aeroelastic flutter model

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

brief paper when is the discretization of a spatially distributed system good enough for control this paper describes a new and straightforward method for controlling spatially distributed plants based on low order models obtained from spatial discretization techniques a suitable level of discretization is determined by computing the sequence of n gaps between weighted models of successively finer spatial resolution and bounding this by another sequence with an analytic series it is proved that such a series forms an upper bound on the n gap between a weighted model in the initial sequence and the spatially distributed weighted plant this enables the synthesis on low order models of robust controllers that are guaranteed to stabilize the actual plant a feature not shared by most model reduction methods where the gap between the high order model and plant is often not known and where the gap between high order and reduced models may be too expensive to compute since the calculation of the current bound is based on weighted models of small state dimension the new method avoids the numerical problems inherent in large scale model reduction based approaches the ideas presented in this paper are demonstrated on a disturbance rejection problem for a 1d heat equation

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

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

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

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

a priori estimation of memory effects in reduced order models of nonlinear systems using the mori zwanzig formalism reduced models of nonlinear dynamical systems require closure or the modelling of the unresolved modes the mori zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics in these equations the unclosed terms are recast as a memory integral involving the time history of the resolved variables while this procedure does not reduce the complexity of the original system these equations can serve as a mathematically consistent basis to develop closures based on memory approximations in this scenario knowledge of the memory kernel is paramount in assessing the validity of a memory approximation unravelling the memory kernel requires solving the orthogonal dynamics which is a high dimensional partial differential equation that is intractable in general a method to estimate the memory kernel a priori using full order solution snapshots is proposed the key idea is to solve a pseudo orthogonal dynamics equation which has a convenient liouville form instead this ersatz arises from the assumption that the semi group of the orthogonal dynamics is a composition operator for one observable the method is exact for linear systems numerical results on the burgers and kuramoto sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel

fast time domain simulation for large order linear time invariant state space systems time domain simulation is essential for both analysis and design of complex systems unfortunately high model fidelity leads to large system size and bandwidths often causing excessive computatio

an efficient reduced order modeling approach for non linear parametrized partial differential equations summary for general non linear parametrized partial differential equations pdes the standard galerkin projection is no longer efficient to generate reduced order models this is because the evaluation of the integrals involving the non linear terms has a high computational complexity and cannot be pre computed this situation also occurs for linear equations when the parametric dependence is nonaffine in this paper we propose an efficient approach to generate reduced order models for large scale systems derived from pdes which may involve non linear terms and nonaffine parametric dependence the main idea is to replace the non linear and nonaffine terms with a coefficient function approximation consisting of a linear combination of pre computed basis functions with parameter dependent coefficients the coefficients are determined efficiently by an inexpensive and stable interpolation at some pre computed points the efficiency and accuracy of this method are demonstrated on several test cases which show significant computational savings relative to the standard galerkin projection reduced order approach copyright q 2008 john wiley sons ltd

proper orthogonal decomposition with supg stabilized isogeometric analysis for reduced order modelling of unsteady convection dominated convection diffusion reaction problems abstract we consider reduced order modelling of unsteady convection dominated convection diffusion reaction problems with proper orthogonal decomposition pod in combination with isogeometric analysis isogeometric analysis has potential advantages in exact geometry representations efficient mesh generation different h p and k refinements and smooth bspline nurbs basis functions in order to compensate the oscillations caused by the convection dominated effect the streamline upwind petrov galerkin supg stabilization method is used both in generation of snapshots and pod galerkin method based on the recent novel and promising discretization method isogeometric analysis we propose a new fully discrete supg stabilized scheme the associated numerical error features three components due to spatial discretization by isogeometric analysis with supg stabilization time discretization with the crank nicolson scheme and modes truncation by pod we show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically a variety of two and three dimensional benchmark tests and numerical examples are provided to show the effectiveness accuracy and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis

optimal subcarrier power allocation for ofdm with low precision adc at receiver motivated by the need to reduce power consumption in the receiver analog to digital converter adc in multi gbps communication systems in this paper we study subcarrier power allocation based on channel side information csi at the transmitter we derive a fixed point equation for subcarrier power allocation that minimizes uncoded symbol error rate ser when a finite precision adc is used at the receiver we study the sensitivity of the optimal power allocation with respect to a parameter that depends on the adc precision based on this we propose a simple analytical approximation for the optimal power allocation which performs within 0 5 db of the exact case over a wide range of signal to noise ratio snr the proposed power allocation leads to only a small increase in the peak to average power ratio papr 0 3 db for channel models suggested in ieee 802 15 3c further for a 16 qam input 7 8 rate low density parity check ldpc code and 3 bit precision adc it attains a coded bit error rate ber of 10 5 at a snr of 23 5 db which compares favorably with the 25 db required by a traditional system with equal power allocation across the subcarriers and infinite sampling precision we also show the robustness of the performance gain to channel estimation errors

parametric domain decomposition for accurate reduced order models applications of mp lrom methodology abstract the multivariate predictions of local reduced order models mp lrom methodology recently proposed by the authors moosavi et al 0000 uses machine learning based regression methods to predict the errors of reduced order models this study considers two applications of mp lrom first the error model is used in conjunction with a greedy sampling algorithm to generate decompositions of one dimensional parametric domains with overlapping regions such that the associated local reduced order models meet the prescribed accuracy requirements once a parametric domain decomposition is constructed any parametric configuration belongs to at least one of the partitions the local reduced order model associated with that partition approximates the full order model at the given parameters within an accuracy level that is estimated a priori the parameter domain decomposition creates a database of the available local bases local reduced order and high fidelity models and identifies the most accurate solutions for an arbitrary parametric configuration next this database is used to enhance the accuracy of the reduced order models using 1 lagrange interpolation of reduced bases in the matrix space 2 lagrange interpolation of reduced bases in the tangent space of the grassmann manifold 3 concatenation of reduced bases followed by a gram schmidt orthogonalization process and 4 lagrange interpolation of high fidelity model solutions numerical results with a viscous burgers model illustrate the potential of the mp lrom methodology to improve the design of parametric reduced order models

error bounds for pod expansions of parameterized transient temperatures abstract we focus on the convergence analysis of the pod expansion for the parameterized solution of transient heat equations the parameter of interest is the conductivity coefficient we prove that this expansion converges with exponential accuracy uniformly if the conductivity coefficient remains within a compact set of positive numbers this convergence result is independent of the regularity of the temperature with respect to the space and time variables we present some numerical experiments to show that a reduced number of modes allows to represent with high accuracy the family of solutions corresponding to parameters that lie in the compact set under study

multiparameter analysis of aero icing problems using proper orthogonal decomposition and multidimensional interpolation steady and unsteady three dimensional viscous turbulent aero icing simulations are computationally expensive especially for certification campaigns when broad parametric studies are needed to overcome the computational effort of such investigations a reduced order modeling approach based on proper orthogonal decomposition and kriging interpolation is proposed using a database of high fidelity numerical simulations experimental data or combinations of both the proposed technique allows approximating solutions by linear combination of a limited number of eigenfunctions bayesian kriging a recent variant is used to obtain the scalar coefficients for the expansion the accuracy of the proposed method is assessed against reference solutions from two and three dimensional aero icing simulations as well as against experimental data