linear time variable systems balancing and model reduction a uniformly balanced realization for linear time variable systems is defined this representation is characterized by the fact that its controllability and observability gramians are equal and diagonal existence and uniqueness of the uniformly balanced realization is studied such a framework has many remarkable properties and leads to a novel method for approximating time variable systems where the subsystems of the balanced realization can be taken as a reduced model the reduced model is examined from the point of view of stability controllability and observability

dynamic analysis of finite linear elastic solids containing small elastic imperfections theory with application to asymmetric circular plates abstract eigensolutions green s functions and steady state responses of three dimensional homogeneous isotropic finite linear elastic solids containing small linear elastic imperfections are presented in this paper the inhomogeneities are treated as perturbations to otherwise homogeneous solids the spectral eigensolutions of which are known the exact eigensolutions of the imperfect solid formulated through the betti reciprocal theorem satisfy a set of integral equations from which both analytic perturbation formulas and a numerical iteration scheme are developed special emphasis is given to analysis of perfect homogeneous elastic solids that are spectrally degenerate finally the vibration of circular plates with evenly spaced radial inclusions which cause splitting of some of the repeated vibration modes is analyzed both analytically and numerically for the eigensolutions

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

a novel technique for inverse identification of distributed stiffness factor in structures abstract a computational inverse technique for identifying stiffness distribution in structures is proposed in this paper using structural dynamics response in the frequency domain in the present technique element stiffness factors of the finite element model of a structure are taken to be the parameters and explicitly expressed in a linear form in the system equation for forward analysis of the harmonic response of the structure this offers great convenience in applying newton s method to search for the parameters of stiffness factor inversely as the jacobian matrix can be obtained simply by solving sets of linear algebraic equation derived from the system equation examples of identifying stiffness factor distribution which is often related to damage in the elements of the structure are presented to demonstrate the present technique the advantages of the present technique for inverse parameter identification problem are 1 the number of the parameters can be very large 2 the identification process is very fast and 3 the accuracy is very high the efficiency of the proposed technique is compared with genetic algorithms

nondestructive identification of multiple flaws using xfem and a topologically adapting artificial bee colony algorithm summary we present a novel algorithm based on the extended finite element method xfem and an enhanced artificial bee colony eabc algorithm to detect and quantify multiple flaws in structures the concept is based on recent work that have shown the excellent synergy between xfem used to model the forward problem and a genetic type algorithm to solve an inverse identification problem and converge to the best flaw parameters in this paper an adaptive algorithm that can detect multiple flaws without any knowledge on the number of flaws beforehand is proposed the algorithm is based on the introduction of topological variables into the search space used to adaptively activate deactivate flaws during run time until convergence is reached the identification is based on a limited number of strain sensors assumed to be attached to the structure surface boundaries each flaw is approximated by a circular void with the following three variables center coordinates xc yc and radius rc within the xfem framework in addition the proposed eabc scheme is improved by a guided to best solution updating strategy and a local search ls operator of the nelder mead simplex type that show fast convergence and superior global ls abilities compared with the standard abc or classic genetic algorithms several numerical examples with increasing level of difficulty are studied in order to evaluate the proposed algorithm in particular we consider identification of multiple flaws with unknown a priori information on the number of flaws which makes the inverse problem harder the proximity of flaws flaws having irregular shapes similar to artificial noise and the effect of structured unstructured meshes the results show that the proposed xfem eabc algorithm is able to converge on all test problems and accurately identify flaws hence this methodology is found to be robust and efficient for nondestructive detection and quantification of multiple flaws in structures copyright 2013 john wiley sons ltd

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

complementary eigenvalues of graphs abstract in this paper we study the eigenvalue complementarity problem eicp when its matrix a belongs to the class s g a a i j a i j a j i 0 iff i j e where g v e is a connected graph it is shown that if all nondiagonal elements of a s g are nonpositive then a has a unique complementary eigenvalue which is the smallest eigenvalue of a in particular zero is the unique complementary eigenvalue of the laplacian and the normalized laplacian matrices of a connected graph the number c g of complementary eigenvalues of the adjacency matrix of a connected graph g is shown to be bounded above by the number b g of induced nonisomorphic connected subgraphs of g furthermore c g b g if the perron roots of the adjacency matrices of these subgraphs are all distinct finally the maximum number of complementary eigenvalues for the adjacency matrices of graphs is shown to grow faster than any polynomial on the number of vertices

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

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

natural norm a posteriori error estimators for reduced basis approximations we present a technique for the rapid and reliable prediction of linear functional outputs of coercive and noncoercive linear elliptic partial differential equations with affine parameter dependence the essential components are i rapidly convergent global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n judiciously selected points in parameter space ii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest and iii offiine 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 bound depends only on n typically very small and the parametric complexity of the problem in this paper we propose a new natural norm formulation for our reduced basis error estimation framework that a greatly simplifies and improves our inf sup lower bound construction offline and evaluation online a critical ingredient of our a posteriori error estimators and b much better controls significantly sharpens our output error bounds in particular through deflation for parameter values corresponding to nearly singular solution behavior we apply the method to two illustrative problems a coercive laplacian heat conduction problem which becomes singular as the heat transfer coefficient tends to zero and a non coercive helmholtz acoustics problem which becomes singular as we approach resonance in both cases we observe very economical and sharp construction of the requisite natural norm inf sup lower bound rapid convergence of the reduced basis approximation reasonable effectivities even for near singular behavior for our deflated output error estimators and significant several order of magnitude online computational savings relative to standard finite element procedures

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

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

a semi continuous formulation for goal oriented reduced order models 1d problems a semi continuous formulation is introduced for finding bases that minimise the error in a specific output functional of a reduced order model the formulation is advantageous in that it can be used with arbitrary reference data and can be easily applied to nonlinear models and functionals a general description of the approach is given then explicit formulations are derived for the advection diffusion and burgers equations numerical results are given for both linear and nonlinear functionals these show substantial reductions in error when compared with pod modes depending on the functional considered the optimisation of bases for a reduced order model using an approximated governing equation is also described for which large increases in accuracy are obtained relative to pod modes copyright c 2015 john wiley sons ltd

reduced basis isogeometric methods rb iga for the real time simulation of potential flows about parametrized naca airfoils abstract we present a reduced basis rb method based on isogeometric analysis iga for the rapid and reliable evaluation of pde systems characterized by complex geometrical features at the current state of the art this is the first case of coupling between rb and iga methods the construction of the rb method relies on an isogeometric boundary element method iga bem as the high fidelity technique allowing a direct interface with computer aided design cad tools a suitable empirical interpolation method eim ensures an efficient offline online decomposition between the construction and the evaluation of the rb method we consider the real time simulation of potential flows past airfoils parametrized with respect to the angle of attack and the naca number identifying their shape and we provide a validation of our methodology with respect to experimental data and reference numerical codes showing in both cases a very good agreement

algorithms for model reduction of large dynamical systems three algorithms for the model reduction of large scale continuous time time invariant linear dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described they rely on low rank approximations to the controllability and observability gramians which can efficiently be computed by adi based iterative low rank methods the first two model reduction methods are closely related to the well known square root method and schur method which are balanced truncation techniques the third method is a heuristic balancing free technique the performance of the model reduction algorithms is studied in numerical experiments

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

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

a quadratic layer element for analyzing stress waves in fgms and its application in material characterization abstract a novel method is presented for investigating elastic waves in functionally graded material fgm plates excited by plane pressure waves the fgm plate is first divided into quadratic layer elements qles a general solution for the equation of motion governing the qle has been derived the general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the frequency domain for an arbitrary fgm plate the response of the plate to an incident pressure wave is obtained using the fourier transform techniques results obtained by the present method are compared with an existing method using homogeneous layer elements numerical examples are presented to investigate stress waves in fgm plates the relationship between the surface displacement response and the material property of quadratic fgm plates has been analytically obtained for the material characterization a computational inverse technique is also presented for characterizing material property of an arbitrary fgm plate from the surface displacement response data using present qle method as forward solver and genetic algorithm as the inverse operator this technique is utilized to reconstruct the material property of an actual sic c fgm

modeling and identification of an arbitrarily shaped scatterer using dynamic xfem with cubic splines abstract localization and shape identification of an arbitrarily shaped scatterer embedded in elastic heterogeneous media is investigated using the dynamic extended finite element method xfem the scatterer s geometry is represented using cubic splines whose parameters are adaptively updated in a gradient based minimization framework that is used to solve the inverse identification problem the dynamic xfem is employed to solve the forward wave propagation problem because it enables direct parametric modeling of the moving boundary of the scatterer over a stationary background mesh the use of open close cubic splines combined with xfem enables the effective construction of the complicated scatterer geometry by minimizing the number of unknown shape parameters a two phase divide and conquer approach is adopted to alleviate complications due to the potential manifestation of multiple solutions to the inverse problem and for increased computational efficiency in first phase the scatterer is localized using a simple geometric representation and the search is carried out using multiple independent inversions starting from different initial estimates incidentally this strategy is perfectly scalable in parallel computation in the second phase the geometric representation is adaptively refined using cubic splines to obtain an accurate estimate of the scatterer s shape several numerical experiments are provided which clearly show that the proposed approach is robust and can yield highly accurate results

a domain decomposition approach for uncertainty analysis this paper proposes a decomposition approach for uncertainty analysis of systems governed by partial differential equations pdes the system is split into local components using domain decomposition our domain decomposed uncertainty quantification dduq approach performs uncertainty analysis independently on each local component in an offline phase and then assembles global uncertainty analysis results using precomputed local information in an online phase at the heart of the dduq approach is importance sampling which weights the precomputed local pde solutions appropriately so as to satisfy the domain decomposition coupling conditions to avoid global pde solves in the online phase a proper orthogonal decomposition reduced model provides an efficient approximate representation of the coupling functions

a combined genetic algorithm and nonlinear least squares method for material characterization using elastic waves abstract the inverse problem of material characterization is formulated as a parameter identification problem in which a set of parameters corresponding to the material property can be found by minimizing error functions formulated using the measured displacement response and the one computed by a forward solver based on projected candidates of parameters a hybrid numerical method is employed as the forward solver to calculate the dynamic displacement response on the surface of the composite plate for given material property a combined method is used as the inverse operator to determine the material property of composite plate in this method genetic algorithm is first used to select a set of better solutions close to the optima then the nonlinear least squares method is applied using these better solutions as the initial guesses finally the identification results can be determined from the solutions of nonlinear least squares method by comparing their corresponding error function values actual material characterizations of composites demonstrate the higher efficiency of the present method

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

eigenvalue eigenvector sensitivity analysis of linear time invariant singular systems this note refers to the eigenvalue eigenvector sensitivity analysis of linear time invariant multivariable singular systems procedures that can be easily programmed on a digital computer are presented for the determination of the deviation of the eigenvalues and the corresponding right and left eigenvectors which are caused by the system s parameter changes about their nominal values the method is illustrated by two examples

accelerating pde constrained optimization by the reduced basis method application to batch chromatography summary in this work we show that the reduced basis method accelerates a partial differential equation constrained optimization problem where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization such an optimization problem arises for example from batch chromatography to reduce the computational burden of repeatedly solving the large scale system under parameter variations a parametric reduced order model with a small number of equations is derived by using the reduced basis method as a result the small reduced order model rather than the full system is solved at each step of the optimization process an adaptive technique for selecting the snapshots is proposed so that the complexity and runtime for generating the reduced basis are largely reduced an output oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically an early stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient numerical examples show that the adaptive technique is very efficient in reducing the offline time the optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution copyright 2015 john wiley sons ltd

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

rapid inverse parameter estimation using reduced basis approximation with asymptotic error estimation this paper presents a rapid and reliable approach to solve inverse problems of parameter estimation for structural systems using reduced basis method rbm a reduced basis model is first developed with asymptotic error estimation and is used for fast computation of solving forward mechanics problems of solids and structures a genetic algorithm ga is then used in the inverse search procedure for parameter estimation the approach is applied to a typical inverse problem of estimating the crack location length and orientation inside a cantilever beam the displacements measured at five points on the lower surface of the beam which can also be evaluated by our fast rbm solver are used as inputs for constructing objective functions of error the genetic algorithm is used to search these parameters of the crack inside cantilever beam that minimize the objective function an example has been presented it is found that the estimated results are very accurate and reliable due to the use of rbm forward model with an effective and robust error estimation and detailed sensitivity analysis the present procedure is 460 times faster than the full fem model used inverse procedure

reduced basis approximation and a posteriori error estimation for stress intensity factors we develop reduced basis approximations rigorous a posteriori error bounds and offline online computational procedures for the accurate fast and reliable prediction of stress intensity factors or strain energy release rates for mode i linear elastic crack problems relevant to fracture mechanics we demonstrate the efficiency and rigor of our numerical method in several examples

identification of a scatterer embedded in elastic heterogeneous media using dynamic xfem abstract a numerical approach for identifying a scatterer embedded in elastic heterogeneous media is described this scatterer may be a crack a void or an inclusion with properties that have detectable contrasts from those of the host medium the forward wave propagation problem is solved using the dynamic extended finite element method xfem which allows the boundary of the scatterer to be easily relocated over a stationary background mesh the inverse problem is cast as a minimization problem whereby the unknown shape parameters of the scatterer e g a line crack s center coordinates size and orientation are the updating parameters a gradient based method is utilized to solve the minimization problem in order to alleviate the potential manifestation of multiple solutions to the inverse problem various deployment patterns for multiple sensors are investigated and also a divide and conquer approach is adopted wherein a set of independent inversions using multiple initial guesses are carried out

enhanced linearized reduced order models for subsurface flow simulation trajectory piecewise linearization tpwl represents a promising approach for constructing reduced order models using tpwl new solutions are represented in terms of expansions around previously simulated and saved solutions high degrees of efficiency are achieved when the representation is projected into a low dimensional space using a basis constructed by proper orthogonal decomposition of snapshots generated in a training run in recent work a tpwl procedure applicable for two phase subsurface flow problems was presented the method was shown to perform well for many cases such as those with no density differences between phases though accuracy and robustness were found to degrade in other cases in this work these limitations are shown to be related to model accuracy at key locations and model stability enhancements addressing both of these issues are introduced a new tpwl procedure referred to as local resolution tpwl enables key grid blocks such as those containing injection or production wells to be represented at full resolution i e these blocks are not projected into the low dimensional space this leads to high accuracy at selected locations and will be shown to improve the accuracy of important simulation quantities such as injection and production rates next two techniques for enhancing the stability of the tpwl model are presented the first approach involves a basis optimization procedure in which the number of columns in the basis matrix is determined to minimize the spectral radius of an appropriately defined amplification matrix the second procedure incorporates a basis matrix constructed using snapshots from a simulation with equal phase densities both approaches are compatible with the local resolution procedure results for a series of test cases demonstrate the accuracy and stability provided by the new treatments finally the tpwl model is used as a surrogate in a direct search optimization algorithm and comparison with results using the full order model demonstrates the efficacy of the enhanced tpwl procedures for this application

new results on pareto spectra abstract a pareto eigenvalue of a matrix a of order n is a scalar r for which the complementarity problem 0 x a x x 0 admits a nonzero solution x r n pareto eigenvalues are also known as complementarity eigenvalues they have found applications in graph theory cone constrained dynamical systems and mathematical modeling in general in this paper we continue our study on theoretical properties of pareto spectra special attention is paid to classification of pareto eigenvalues and cardinality issues

an optimal projection method for the reduced order modeling of incompressible flows a new method to build reduced order models roms for incompressible flows is addressed in this paper proper orthogonal decompositions pods are used to efficiently represent both the fluid velocity and pressure fields with a small number of spatial modes a minimum residual projection is then developed for building the reduced models in which the velocity and pressure temporal coefficients are fully coupled comparisons of numerical calculations based on the proposed approach with results arising from the classical galerkin projection are exposed on three fluid flows a transient ventilated cavity a periodic lid driven cavity and a transient mixed convection flow it is shown that both stability and accuracy of the roms are strongly enhance when the minimum residual projection is used

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

complementarity eigenvalue analysis of connected graphs abstract this work concerns the spectral analysis of connected graphs from a non traditional point of view instead of the usual eigenvalues of the adjacency matrix a g of a graph g under consideration we compute and analyze the complementarity eigenvalues of a g the complementarity eigenvalues of a general square matrix are defined in terms of a certain complementarity system relative to the componentwise ordering the complementarity eigenvalues of a g form the so called complementarity spectrum of g in general the structure of a connected graph is better discriminated in terms of its complementarity spectrum than in terms of its usual spectrum this observation is one of the leading motivation behind our work

an online coupled state input parameter estimation approach for structural dynamics abstract in many practical structural applications unknown states inputs and parameters are present however most methods require one or more of these variables to be known in order to estimate the other s in this research an estimation technique which employs physical models is proposed to perform coupled state input parameter estimation in order to obtain a modeling technique which allows the estimation of a wide range of parameters in a generic fashion at a minimal computational cost even real time the use of a parametric model reduction technique is proposed the reduced model is coupled to an extended kalman filter ekf with augmented states for the unknown inputs and parameters this leads to a very efficient framework for estimation in structural dynamics problems special attention is also given to the measurement requirements in order to obtain an adequate observability of all unknown quantities and the necessity for at least one displacement level measurement is shown the proposed methodology is validated numerically and experimentally the approach is shown to be easy to tune and provides good results with different measurement methods

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

a novel reduced basis method with upper and lower bounds for real time computation of linear elasticity problems abstract this paper presents a novel reduced basis method for analyzing problems of linear elasticity in a systematical rapid and reliable fashion for solutions with both upper and lower bounds to the exact solution in the form of energy norm or compliance output the lower bound of the solution output is obtained form the well known reduced basis method based on the galerkin projection used in the finite element method which is termed as gp rbm for the upper bound a new reduced basis approach is developed by the combination of the reduced basis method and a smoothed galerkin projection used in the linearly conforming point interpolation method and it is thus termed as sgp rbm to examine the present sgp rbm we first conduct a theoretical study on the very important upper bound property reduced basis models for both gp rbm and sgp rbm are constructed with the aid of an asymptotic error estimation and greedy adaptive procedure the gp rbm and the newly proposed sgp rbm are applied to analyze a cantilever beam with an oblique crack to verify the proposed rbm technique in terms of accuracy convergence bound properties and computational savings both theoretical analysis and numerical results have demonstrated that the present method is a very efficient method for real time solutions providing exact output bounds

the sylvester equation and approximate balanced reduction abstract the purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems the resulting approach is completely automatic once an error tolerance is specified and also yields an error bound this is to be contrasted with existing projection methods namely pade via lanczos pvl and rational krylov which do not satisfy these properties our approach is based on the computation and approximation of the cross gramian of the system the cross gramian is the solution of a sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system it is well suited to large scale problems because there are no matrix factorizations of the large sparse system matrix only matrix vector products are required

taylor approximation and variance reduction for pde constrained optimal control under uncertainty abstract in this work we develop a scalable computational framework for the solution of pde constrained optimal control problems under high dimensional uncertainty specifically we consider a mean variance formulation of the control objective and employ a taylor expansion with respect to the uncertain parameter field either to directly approximate the control objective or as a control variate for variance reduction the expressions for the mean and variance of the taylor approximation are known analytically although their evaluation requires efficient computation of the trace of the preconditioned hessian of the control objective we propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the hessian on a small number of random directions then the computational work does not depend on the nominal dimension of the uncertain parameter but depends only on the effective dimension i e the rank of the preconditioned hessian thus ensuring scalability to high dimensional problems moreover to increase the estimation accuracy of the mean and variance of the control objective by the taylor approximation we use it as a control variate for variance reduction which results in considerable computational savings several orders of magnitude compared to a plain monte carlo method in summary our approach amounts to solving an optimal control problem constrained by the original pde the generalized eigenvalue equations at a small number of eigenfunctions and a set of linearized pdes that arise from the computation of the gradient and hessian of the control objective with respect to the uncertain parameter we use the lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient based optimization method to solve the problem we demonstrate the accuracy efficiency and scalability of the proposed computational method for two examples with high dimensional uncertain parameters subsurface flow in a porous medium modeled as an elliptic pde and turbulent jet flow modeled by the reynolds averaged navier stokes equations coupled with a nonlinear advection diffusion equation characterizing model uncertainty in particular for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field

eigenvalue tracking due to parameter variation this note describes an efficient eigenvalue tracking algorithm which is applicable in many engineering systems where the effect of parameter variation on system stability is to be determined starting with the original system eigenvalues the algorithm uses determinants to compute first order eigenvalue sensitivities which are used in an iterative method that converges rapidly to the new eigenvalues corresponding to the parameter change the algorithm tracks all the system eigenvalues starting from a given base case it is shown to be less costly than repeated eigenvalue evaluation and is illustrated by means of a simple power system example

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

bounds for the spectral radius of a graph when nodes are removed abstract we present a new type of lower bound for the spectral radius of a graph in which m nodes are removed as a corollary cioab s theorem 4 which states that the maximum normalized principal eigenvector component in any graph never exceeds 1 2 with equality for the star appears as a special case of our more general result

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

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

inverse identification of thermal parameters using reduced basis method a novel computational method is presented to inversely identify heat convection constants for complex engineering systems a reduced basis approach is developed as a forward solver for heat transfer analysis in order to significantly reduce the computational complexity in each forward analysis which is crucial for the inverse analysis of complex systems an intergeneration projection genetic algorithm ip ga is introduced as the inverse procedure to speed up the process of finding the desired global minimum of the fitness function of error that leads to the parameters to be identified as an example identification of the heat convection constants of a typical microelectronic package are performed using the present method and the performance is examined comparing with other inverse identification methods it is found that the reduced basis method combined with the ip ga significantly outperforms the conventional methods the proposed inverse identification method is efficient enough even for online analysis of inverse problems due to both the use of the reduced basis approach and the present inverse procedure

an inverse procedure for determination of material constants of composite laminates using elastic waves a procedure is proposed to determine the material constants of composite laminates from dynamic displacement responses wave response obtained at only one receiving point of laminate surfaces a hybrid numerical method is used for forward computation that relates the material constants to the displacement responses a uniform micro genetic algorithm is adopted to develop an inverse procedure and its effectiveness to the present type of applications is shown the robustness of the procedure to the effect of measurement noise is investigated by adding gauss noise to the input displacement response numerical simulations are performed on laminates with different composite materials layer orientations layer number and thickness the satisfactory identification results for all cases demonstrate the effectiveness of the present procedure as well as its robustness to the noises effects

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

reduced order modelling of parameter dependent linear and nonlinear dynamic partial differential equation models in this paper we develop reduced order models for dynamic parameter dependent linear and nonlinear partial differential equations using proper orthogonal decomposition pod the main challenges are to accurately and efficiently approximate the pod bases for new parameter values and in the case of nonlinear problems to efficiently handle the nonlinear terms we use a bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values computational efficiency is ensured by using manifold learning to perform the emulation in a low dimensional space the accuracy of the method is demonstrated on a linear and a nonlinear example with comparisons with a global basis approach

xfem based crack detection scheme using a genetic algorithm a new computational tool is developed for the accurate detection and identification of cracks in structures to be used in conjunction with non destructive testing of specimens it is based on the solution of an inverse problem based on some measurements typically along part of the boundary of the structure that describe the response of the structure to vibration in a chosen frequency or a combination of frequencies the goal is to estimate whether the structure contains a crack and if so to find the parameters location size orientation and shape of the crack that produces a response closest to the given measurement data in some chosen norm the inverse problem is solved using a genetic algorithm ga the ga optimization process requires the solution of a very large amount of forward problems the latter are solved via the extended finite element method xfem this enables one to employ the same regular mesh for all the forward problems performance of the method is demonstrated via a number of numerical examples involving a cracked flat membrane various computational aspects of the method are discussed including the a priori estimation of the ill posedness of the crack identification problem copyright 2007 john wiley sons ltd

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

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

a point interpolation method with least square strain field pim lss for solution bounds and ultra accurate solutions using triangular mesh abstract a point interpolation method with least square strain field pim lss is developed for solid mechanics problems using triangular background mesh in the pim lss pim shape functions are used for displacement field construction that may or may not be compatible and a least square fitting technique is adopted to construct the strain field a strain constructed galerkin sc galerkin weak formulation is then proposed for establishing discretized pim lss models that have a number of special properties we proved theoretically 1 the pim lss provides a softening effect to the fem model and a stiffing effect to the node based smoothed point interpolation method ns pim model 2 the exact solution is bounded from both by pim lss solutions with strain field of zero order fitting and that with higher order fitting 3 there exists a preferred order of fitting for the strain field such that ultra accurate one order higher accuracy solution can be obtained using the pim lss these theorems and properties have been confirmed in numerical examples

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

reduced basis approximation and a posteriori error estimation for a boltzmann model abstract we present a reduced basis approximation and associated rigorous a posteriori output flow rate error bound for a simple linearized bgk boltzmann model for flow in a channel we consider two parameters the knudsen number and the accommodation coefficient the reduced basis approximation is built upon and the reduced basis error is measured relative to a truth mapped weighted spectral discretization of a stabilized petrov galerkin weak formulation we exploit the affine nature of the parametric dependence to provide an offline online computational procedure that decouples the spectral and reduced basis quantities the computational complexity of the online stage in which given a new parameter value we evaluate the reduced basis flow rate prediction and associated error bound depends only on the typically very small dimension of the reduced basis space n and not on the typically very large dimension of the truth spectral approximation n we demonstrate that the reduced basis approximation converges very rapidly with n and that the error estimator effectivities remain bounded albeit somewhat large in the online stage the reduced basis approximation achieves a certifiable relative accuracy of 10 2 with respect to the truth spectral result at only 1 100th the computational effort the method is appropriate in the many query and real time e g parameter estimation contexts in which there is a premium on reduced marginal cost and a tolerance for significant offline pre processing effort

on spectral maps induced by convex cones abstract let l e be the vector space of linear endomorphisms on a euclidean space e and k be a closed convex cone in e the spectral map k l e 2 r induced by k is defined as follows to each linear endomorphism a on e it associates the set k a of all r for which the complementarity problem k x a x x k has a nonzero solution x e here k is the dual cone of k and stands for orthogonality the set k a is known as the k spectrum of a the purpose of this work is twofold firstly we wish to compare algebraic properties of k and geometric properties of k secondly we wish to identify classes of linear transformations f l e l e that leave the k spectrum invariant i e such that k f a k a for all a l e the set of linear transformations preserving the k spectrum depends of course on the cone k under consideration

flaw detection in sandwich plates based on time harmonic response using genetic algorithm an inverse identification technique is proposed to assess the presence location size and degree of flaw in the core layer of sandwich plates the time harmonic response of the sandwich plates is used as the input for inverse analysis the responses of the plates in forward analysis are predicted using finite element method fem based on mindlin plate theory the flaw is characterized by a set of parameters the genetic algorithm ga is applied to search the parameters to improve the efficiency of ga in searching for the true solution a two stage searching method is proposed numerical examples are given to illustrate the proposed technique

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems