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

decay property for second order hyperbolic systems of viscoelastic materials we study a class of second order hyperbolic systems with dissipation which describes viscoelastic materials the considered dissipation is given by the sum of the memory term and the damping term when the dissipation is effective over the whole system we show that the solution decays in l 2 at the rate t n 4 as t provided that the corresponding initial data are in l 2 l 1 w heren is the space dimension the proof is based on the energy method in the fourier space also we discuss similar systems with weaker dissipation by introducing the operator 1 2 with 0 in front of the dissipation terms and

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

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

multidimensional a priori hyper reduction of mechanical models involving internal variables this paper concerns the adaptation of reduced order models during simulations of series of elastoviscoplastic problems in continuation with previous works this paper aimed at extending the a priori hyper reduction method aphr method for nonlinear thermal problems to nonlinear mechanical problems involving internal variables this method is an a priori approach because full incremental responses of detailed models are not forecasted in order to build reduced order models the recent extension of the hyper reduction method to reduction of mechanical models involving internal variables makes possible the reduction of degrees of freedom and the reduction of integration points a multi level formulation is introduced to focus on the capability of the method to perform efficient parallel computations to adapt reduced order models

a multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

a reduced basis model with parametric coupling for fluid structure interaction problems we present a new model reduction technique for steady fluid structure interaction problems when the fluid domain deformation is suitably parametrized the coupling conditions between the fluid and the structure can be formulated in the low dimensional space of geometric parameters moreover we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid structure iterations in this way a reduced order model with reliable a posteriori error bounds is obtained the proposed method is validated with an example of steady stokes flow in an axisymmetric channel where the structure is described by a simple one dimensional generalized string model we demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased

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

estimation of the error in the reduced basis method solution of nonlinear equations the reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension in this paper the reduced basis error is shown to be dominated by an approximation error this in turn leads to error estimates for projection onto specific subspaces for example subspaces related to taylor lagrange and discrete least squares approximation

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

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

free divisors in prehomogeneous vector spaces we study linear free divisors that is free divisors arising as discriminants in prehomogeneous vector spaces and in particular in quiver representation spaces we give a characterization of the prehomogeneous vector spaces containing such linear free divisors for reductive linear free divisors we prove that the numbers of geometric and representation theoretic irreducible components coincide as a consequence we find that a quiver can only give rise to a linear free divisor if it has no oriented or unoriented cycles we also deduce that the linear free divisors which appear in sato and kimura s list of irreducible prehomogeneous vector spaces are the only irreducible reductive linear free divisors furthermore we show that all quiver linear free divisors are strongly euler homogeneous that they are locally weakly quasihomogeneous at points whose corresponding representation is not regular and that all tame quiver linear free divisors are locally weakly quasihomogeneous in particular the latter satisfy the logarithmic comparison theorem

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

certified reduced basis methods for parametrized parabolic partial differential equations with non affine source terms abstract we present rigorous a posteriori output error bounds for reduced basis approximations of parametrized parabolic partial differential equations with non affine source terms the method employs the empirical interpolation method in order to construct affine coefficient function approximations of the non affine parametrized functions our a posteriori error bounds take both error contributions explicitly into account the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation to this end we employ recently developed rigorous error bounds for the empirical interpolation method and develop error estimation and primal dual formulations to provide rigorous bounds for the error in specific outputs of interest we present an efficient offline online computational procedure for the calculation of the reduced basis approximation and associated error bound the method is thus ideally suited for many query or real time contexts as a specific motivational example we consider a three dimensional mathematical model of a welding process our numerical results show that we obtain efficient and reliable mathematical models which may be gainfully employed in manufacturing and product development

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

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

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

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

cohomology of the complement of a free divisor we prove that if d is a strongly quasihomogeneous free divisor in the stein manifold x and u is its complement then the de rham cohomology of u can be computed as the cohomology of the complex of meromorphic differential forms on x with logarithmic poles along d with exterior derivative the class of strongly quasihomogeneous free divisors introduced here includes free hyperplane arrangements and the discriminants of stable mappings in mather s nice dimensions and in particular the discriminants of coxeter groups

decay estimate and asymptotic profile for a plate equation with memory abstract in this paper the initial value problem for a plate equation with a memory is considered under weaker assumptions on the memory function we obtain the decay estimates of solutions which is of the regularity loss type by applying the energy method in fourier space moreover if the memory kernel is an exponential function we derive the large time behavior and asymptotic profile of solutions in the one dimensional space by virtue of the technique of spectral analysis

evaluation of proper orthogonal decomposition based decomposition techniques applied to parameter dependent nonturbulent flows the proper orthogonal decomposition pod method as a systematic technique to analyze parameter dependent problems may be inappropriate as a practical tool for generating low dimensional models we propose a weighted pod w pod as an alternative to give higher priority to low energetic or important modes by simply weighting a predefined pod p pod is suggested where modes are selected not only on the basis of energy but also on some a priori knowledge of the system the techniques are tested on a flow problem undergoing steady or unsteady transition but may in general be applied to nonlinear systems of discretized pdes closed by additional linear algebraic equations various residual functions to measure the quality of the reduced models are introduced they provide lower limits of residual energy to ensure an adequate physical representation for a given modeling it is shown that modes extracted locally may generally not contain information regarding the global dynamics and the dynamics may not be correctly predicted over a longer range of model parameters it is demonstrated that by weighting and predefining base vectors it is possible to improve the pod technique s capability to generate low dimensional models

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

optimal decay estimates of a regularity loss type system with constraint condition abstract in the paper 14 the authors formulated a new structural condition which includes the kawashima shizuta condition and analyzed the weak dissipative structure called the regularity loss type for general systems which contain the timoshenko system and the euler maxwell system however this new structural condition can not cover all of dissipative systems indeed we introduce a dissipative system which does not satisfy the new condition and analyze the weaker dissipative structure in this paper precisely we first derive the l 2 decay estimate of solutions and discuss the type of the corresponding regularity loss structure moreover in order to show the optimality of the decay estimate we analyze the expansion for the corresponding eigenvalue of our problem and derive that the solution approaches the diffusion wave as time tends to infinity

global existence and asymptotic decay of solutions to the nonlinear timoshenko system with memory abstract in this paper we consider the initial value problem for the nonlinear timoshenko system with a memory term due to the regularity loss property and weak dissipation we have to assume stronger nonlinearity than usual by virtue of the semi group arguments we obtain the global existence and optimal decay of solutions to the nonlinear problem under smallness and enough regularity assumptions on the initial data where we employ a time weighted l 2 energy method combined with the optimal l 2 decay of lower order derivatives of solutions

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

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

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

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

a reduced order method for simulation and control of fluid flows this article presents a reduced order modeling approach for simulation and control of viscous incompressible flows the reduced order models suitable for control and which capture the essential physics are developed using the reduced basis method the so called lagrange approach is used to define reduced bases and the basis functions in this approach are obtained from the numerical solutions the feasibility of this method for flow control is demonstrated on boundary control problems in closed cavity and in wall bounded channel flows control action is effected through boundary surface movement on a part of the solid wall our formulation of the reduced order method applied to flow control problems leads to a constrained minimization problem and is solved by applying newton like methods to the necessary conditions of optimality through our computational experiments we demonstrate the feasibility and applicability of the reduced order method for simulation and control of fluid flows

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

reduced order modeling of time dependent pdes with multiple parameters in the boundary data abstract the computational approximation of solutions to nonlinear partial differential equations pdes such as the navier stokes equations is often a formidable task for this reason there is significant interest in the development of very low dimensional models that can be used to determine reasonably accurate approximations in simulation and control problems for pdes many concrete tests have been reported on in the literature that show that several classes of reduced order models roms are effective i e an accurate approximate solution can be obtained using very low dimensional roms most of these tests involved problems for which the solution depends on only a single parameter appearing in the boundary data the extension of the techniques used in those tests to the case of multiple parameters is not a straightforward matter here we present test and compare two methods for treating within the context of a class of roms inhomogeneous dirichlet type boundary conditions that contain multiple parameters in this study we focus on the proper orthogonal decomposition pod approach to rom however the issues and results discussed here in the pod context apply equally well to other rom approaches

energy decay rates for the timoshenko system of thermoelastic plates abstract we consider a model describing nonlinear dynamical motions of an unbounded thermoelastic plate we prove the well posedness of the above system and analyse the behaviour of the total energy e t as t our main result shows that the total energy of the system satisfies the following estimate there exist a constant e 0 0 such that e t 4 e 0 exp t for all t 0 the result is proved by constructing a lyapunov function which is a suitable perturbation of the energy and satisfies a differential inequality leading to the desired decay estimate

parametric free form shape design with pde models and reduced basis method we present a coupling of the reduced basis methods and free form deformations for shape optimization and design of systems modelled by elliptic pdes the free form deformations give a parameterization of the shape that is independent of the mesh the initial geometry and the underlying pde model the resulting parametric pdes are solved by reduced basis methods an important role in our implementation is played by the recently proposed empirical interpolation method which allows approximating the non affinely parameterized deformations with affinely parameterized ones these ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization the proposed approach is demonstrated on an airfoil inverse design problem

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

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

decay of solutions to an inertial model for a semilinear plate equation with memory abstract in this paper we focus on the initial value problem of an inertial model for a semilinear plate equation with memory in multi dimensions n 1 which is of regularity loss property by using fourier transform and laplace transform we obtain fundamental solutions to the corresponding linear problem and their pointwise estimates in the fourier space appealing to this pointwise estimates we obtain the global existence and the optimal decay estimates of solutions to the semilinear problem by employing the contraction mapping theorem

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

a posteriori error analysis of the reduced basis method for nonaffine parametrized nonlinear pdes in this paper we present the a posteriori error analysis for the reduced basis method rbm applied to nonlinear variational problems that depend on a parameter in a nonaffine manner to this end we generalize the analysis by veroy and patera int j numer methods fluids 47 2005 pp 773 788 to nonaffine parametrized partial differential equations we use the empirical interpolation method eim in order to approximate the nonaffine parameter dependencies by a linear combination of affine functions we also investigate a standard dual problem formulation in particular for the computation of a general output functional also in combination with the eim first we study the well posedness in terms of the brezzi rappaz raviart theory then we develop a posteriori error estimates and investigate offline online decompositions the a posteriori error analysis allows us to introduce an adaptive sampling procedure for the choice of the modes numerical experiments for a convection diffusion problem around a rotating propeller show the effectivity of the scheme

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

asymptotic behavior of the energy for a class of weakly dissipative second order systems with memory a class of second order abstract systems with memory and dirichlet boundary conditions is investigated by suitable liapunov functionals existence of solutions as well as asymptotic behavior are determined in particular when the memory kernel decays exponentially the polynomially decay of the solutions is proved

characterizing normal crossing hypersurfaces the objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds it is shown that a divisor hypersurface has normal crossings if and only if it is a free divisor has a radical jacobian ideal and a smooth normalization using k saito s theory of free divisors also a characterization in terms of logarithmic differential forms and vector fields is found finally we give another description of a normal crossing divisor in terms of the logarithmic residue using recent results of m granger and m schulze

decay structure for symmetric hyperbolic systems with non symmetric relaxation and its application this paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation when the relaxation matrix is symmetric the dissipative structure of the systems is completely characterized by the kawashima shizuta stability condition formulated in umeda et al jpn j appl math 1 435 457 1984 and shizuta and kawashima hokkaido math j 14 249 275 1985 and we obtain the asymptotic stability result together with the explicit time decay rate under that stability condition however some physical models which satisfy the stability condition have non symmetric relaxation term for example the timoshenko system and the euler maxwell system moreover it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity loss type see duan in j hyperbolic differ equ 8 375 413 2011 ide et al in math models meth appl sci 18 647 667 2008 ide and kawashima in math models meth appl sci 18 1001 1025 2008 ueda et al in siam j math anal 2012 ueda and kawashima in methods appl anal 2012 therefore our purpose in this paper is to formulate a new structural condition which includes the kawashima shizuta condition and to analyze the weak dissipative structure for general systems with non symmetric relaxation

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

manifold learning for the emulation of spatial fields from computational models repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible for partial differential equation pde models the outputs of interest are often spatial fields leading to high dimensional output spaces although emulators can be used to find faithful and computationally inexpensive approximations of computer models there are few methods for handling high dimensional output spaces for gaussian process gp emulation approximations of the correlation structure and or dimensionality reduction are necessary linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies manifold learning can overcome the limitations of linear methods if an accurate inverse map is available in this paper we use kernel pca and diffusion maps to construct gp emulators for very high dimensional output spaces arising from pde model simulations for diffusion maps we develop a new inverse map approximation several examples are presented to demonstrate the accuracy of our approach