brief paper identification and data driven model reduction of state space representations of lossless and dissipative systems from noise free data we illustrate procedures to identify a state space representation of a lossless or dissipative system from a given noise free trajectory important special cases are passive systems and bounded real systems computing a rank revealing factorization of a gramian like matrix constructed from the data a state sequence can be obtained the state space equations are then computed by solving a system of linear equations this idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced order model

coercive boundary value problems for regular degenerate differential operator equations abstract this study focuses on nonlocal boundary value problems bvp for degenerate elliptic differential operator equations doe that are defined in banach valued function spaces where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients several conditions obtained that guarantee the maximal l p regularity and fredholmness these results are also applied to nonlocal bvp for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties

passivity preserving model reduction via interpolation of spectral zeros abstract an algorithm is developed for passivity preserving model reduction of linear time invariant systems implementation schemes are described for both medium scale dense and large scale sparse applications the algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros these interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain block matrix which has the spectral zeros as its spectrum explicit interpolation is avoided and passivity of the reduced model is established instead through satisfaction of the necessary conditions of the positive real lemma it is also shown that this procedure indirectly solves the associated controllability and observability riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations from these a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable

structure preserving truncation of nonlinear port hamiltonian systems in this paper we present a novel balancing method for nonlinear port hamiltonian systems based on the hamiltonian and the controllability function this corresponding balanced truncation method results in a reduced order model that is still in port hamiltonian form in contrast to the traditional balanced truncation method based on the controllability and observability functions

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

structure preserving reduced order modeling for gradient systems abstract minimization of energy in gradient systems leads to formation of oscillatory and turing patterns in reaction diffusion systems these patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state in this paper a reduced order model rom is developed which preserves the gradient dissipative structure the coupled system of reaction diffusion equations are discretized in space by the symmetric interior penalty discontinuous galerkin sipg method the resulting system of ordinary differential equations odes are integrated in time by the average vector field avf method which preserves the energy dissipation of the gradient systems the roms are constructed by the proper orthogonal decomposition pod with galerkin projection the nonlinear reaction terms are computed efficiently by discrete empirical interpolation method deim preservation of the discrete energy of the foms and roms with pod deim ensures the long term stability of the steady state solutions numerical simulations are performed for the gradient dissipative systems with two specific equations real ginzburg landau equation and swift hohenberg equation numerical results demonstrate that the pod deim reduced order solutions preserve well the energy dissipation over time and at the steady state

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

stokes equations with small parameters in half plane the stationary and instationary abstract stokes problems are studied in half plane the equations contain an abstract operator and small parameters the uniform well posedness of these problems in vector valued l p spaces are derived

an implicit explicit runge kutta chebyshev finite element method for the nonlinear lithium ion battery equations abstract we introduce a numerical method to integrate the nonlinear system of equations that model the physical chemical laws of the lithium ion batteries the mathematical model formulated in doyle et al j electrochem soc 140 1993 1526 1533 is a system of strongly coupled nonlinear parabolic and elliptic equations our numerical method combines a second order implicit explicit runge kutta chebyshev scheme for time discretization of the parabolic equations governing the dynamics of the variables ce and cs with linear finite element space discretization of the system the implicit explicit numerical formulation of the parabolic equations allows us to decouple the strong nonlinear reaction terms which are treated implicitly from the linear diffusion terms treated explicitly this approach is computationally efficient because a it has an extended stability region and b the coupled system of algebraic equations becomes a single variable nonlinear equation per mesh point which is easily solved by newton method

a posteriori error estimation for deim reduced nonlinear dynamical systems in this work an efficient approach for a posteriori error estimation for pod deim reduced nonlinear dynamical systems is introduced the considered nonlinear systems may also include time and parameter affine linear terms as well as parametrically dependent inputs and outputs the reduction process involves a galerkin projection of the full system and approximation of the system s nonlinearity by the deim method s chaturantabut and d c sorensen siam j sci comput 32 2010 pp 2737 2764 the proposed a posteriori error estimator can be efficiently decomposed in an offline online fashion and is obtained by a one dimensional auxiliary ode during reduced simulations key elements for efficient online computation are partial similarity transformations and matrix deim approximations of the nonlinearity jacobians the theoretical results are illustrated by application to an unsteady burgers equation and a cell apoptosis model

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

information geometry approach to parameter estimation in markov chains we consider the parameter estimation of markov chain when the unknown transition matrix belongs to an exponential family of transition matrices then we show that the sample mean of the generator of the exponential family is an asymptotically efficient estimator further we also define a curved exponential family of transition matrices using a transition matrix version of the pythagorean theorem we give an asymptotically efficient estimator for a curved exponential family primary 62m05

nonlinear abstract boundary value problems in vector valued function spaces and applications abstract the nonlocal boundary value problems for linear and nonlinear differential operator equations of second order with dependent coefficients are studied the principal part of the differential operators generated by linear problems are nonself adjoint several conditions for the maximal regularity r positivity and the fredholmness in banach valued l p spaces are given by using these results the existence and uniqueness of the maximal regular solutions of nonlocal boundary value problems for nonlinear differential operator equations are established in applications nonlocal boundary value problems for nonlinear partial differential equations and their finite or infinite systems on cylindrical domains are studied

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

a new look at proper orthogonal decomposition we investigate some basic properties of the proper orthogonal decomposition pod method as it is applied to data compression and model reduction of finite dimensional nonlinear systems first we provide an analysis of the errors involved in solving a nonlinear ode initial value problem using a pod reduced order model then we study the effects of small perturbations in the ensemble of data from which the pod reduced order model is constructed on the reduced order model we explain why in some applications this sensitivity is a concern while in others it is not we also provide an analysis of computational complexity of solving an ode initial value problem and study the computational savings obtained by using a pod reduced order model we provide several examples to illustrate our theoretical results

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

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

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

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

multiscale empirical interpolation for solving nonlinear pdes in this paper we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations the proposed method combines empirical interpolation techniques and local multiscale methods such as the generalized multiscale finite element method gmsfem to solve nonlinear equations the gmsfem is used to represent the solution on a coarse grid with multiscale basis functions computed offline computing the gmsfem solution involves calculating the system residuals and jacobians on the fine grid we use empirical interpolation concepts to evaluate these residuals and jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse scale problem rather than the fully resolved fine scale one the empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times the coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system the proposed multiscale empirical interpolation techniques 1 divide computing the nonlinear function into coarse regions 2 evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced order representation of the solution and 3 introduce multiscale proper orthogonal decomposition techniques to find appropriate interpolation vectors we demonstrate the effectiveness of the proposed methods on several nonlinear multiscale pdes that are solved with newton s methods and fully implicit time marching schemes our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale pdes on a coarse grid with bounded error and significant computational cost reduction

model reduction for systems with inhomogeneous initial conditions abstract we consider the model reduction problem for linear time invariant dynamical systems having nonzero but otherwise indeterminate initial conditions building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a forced system having null initial conditions we develop a new approach that involves reducing these component responses independently and then combining the reduced responses into an aggregate reduced system response this approach allows greater flexibility and offers better approximation properties than other comparable methods

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

data driven model reduction of monotone systems by nonlinear dc gains in this paper we develop data driven model reduction methods for monotone nonlinear control systems based on a nonlinear version of the dc gain the nonlinear dc gain is a function of the amplitude of the input and can be used to evaluate the importance of each state variable in fact the nonlinear dc gain is directly related to the infinity induced norm of the system as well as a notion of output reachability given the dc gain model reduction is performed by either truncating not so important state variables or aggregating state variables having similar importance under such truncation and clustering monotonicity and boundedness of the nonlinear dc gain are preserved moreover these two operations can be approximately performed based on simulation or experimental data alone this empirical model reduction approach is illustrated by an example of a gene regulatory network

the localized reduced basis multiscale method for two phase flows in porous media summary in this work we propose a novel model order reduction approach for two phase flow in porous media by introducing a global pressure formulation in which the total mobility which realizes the coupling between saturation and pressure is regarded as a parameter to the pressure equation in our approach the mobility itself is an optimal fit of mobility profiles that are precomputed using the time of flight for the initial saturation using this formulation and the localized reduced basis multiscale method we obtain a low dimensional surrogate of the high dimensional pressure equation by applying ideas from model order reduction for parametrized partial differential equations we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two phase flow simulation which is thereby largely accelerated usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online steps to reach an ideal runtime at acceptable error increase for the two phase flow simulation this is achieved by constructing reduced dimensional local spaces that lead to a non conforming global approximation space as one example for a coupling on the global space we introduce a discontinuous galerkin scheme copyright 2014 john wiley sons ltd

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

asymptotic efficiency of the sample mean in markov chain monte carlo schemes in many cases a univariate time series derived from a time reversible stationary markov chain has the property that there is a distribution function on 1 1 0 1 such that for all non negative integers l its lth moment is the ith lag autocorrelation of the time series we discuss some consequences of this property effects of the asymptotic efficiency of the sample mean on the efficiency of subsampling a stationary markov chain are discussed the stationarity assumption is then relaxed we show that under certain regularity conditions the sample mean of a strongly asymptotically stationary process is asymptotically efficient in that it has the smallest mean square error asymptotically in a broad class of linear estimators

the efficiency of one long run versus independent replication in steady state simulation we evaluate the efficiency of one long run versus independent replications in steady state discrete event simulation assuming that an initial portion of each replication will be deleted to allow the process to approach steady state we provide supporting evidence in favor of one long run but we also show that multiple replications can be more efficient the advantage of one long run increases if the amount deleted increases or if the covariance function decreases more quickly assuming it is nonnegative and decreasing thus assuming that the amount deleted depends on the way the process approaches steady state one long run tends to be efficient when the covariance function decays rapidly compared to the rate the process approaches steady state we also discuss ways to determine the initial portion to delete we consider the case of an exponential covariance function in detail and use it as a basis for approximations we also consider the m g queueing model and reflected brownian motion the latter

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

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

linear and nonlinear abstract equations with parameters abstract this paper focuses on nonlocal boundary value problems for linear and nonlinear abstract elliptic equations in banach spaces here equations and boundary conditions contain certain parameters the uniform separability of the linear problem and the existence and uniqueness of maximal regular solution of nonlinear problem are obtained in l p spaces for linear case the discreteness of spectrum of corresponding parameter dependent differential operator is obtained the behavior of solution when the parameter approaches zero and its smoothness with respect to the parameter is established moreover we show the estimate for analytic semigroups in terms of interpolation spaces this fact can be used to obtain maximal regularity properties for abstract boundary value problems

non linearly stable reduced order models for incompressible flow with energy conserving finite volume methods abstract a novel reduced order model rom formulation for incompressible flows is presented with the key property that it exhibits non linearly stability independent of the mesh of the full order model the time step the viscosity and the number of modes the two essential elements to non linear stability are 1 first discretise the full order model and then project the discretised equations and 2 use spatial and temporal discretisation schemes for the full order model that are globally energy conserving in the limit of vanishing viscosity for this purpose as full order model a staggered grid finite volume method in conjunction with an implicit runge kutta method is employed in addition a constrained singular value decomposition is employed which enforces global momentum conservation the resulting velocity only rom is thus globally conserving mass momentum and kinetic energy for non homogeneous boundary conditions a one time poisson equation is solved that accounts for the boundary contribution the stability of the proposed rom is demonstrated in several test cases furthermore it is shown that explicit runge kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

structure preserving model reduction of port hamiltonian systems by moment matching at infinity model reduction of port hamiltonian systems by means of the krylov methods is considered aiming at port hamiltonian structure preservation it is shown how to employ the arnoldi method for model reduction in a particular coordinate system in order to preserve not only a specific number of the markov parameters but also the port hamiltonian structure for the reduced order model furthermore it is shown how the lanczos method can be applied in a structure preserving manner to a subclass of port hamiltonian systems which is characterized by an algebraic condition in fact for the same subclass of port hamiltonian systems the arnoldi method and the lanczos method turn out to be equivalent in the sense of producing reduced order port hamiltonian models with the same transfer function

dissipativity preserving model reduction for large scale distributed control systems we propose a dissipativity preserving structured model reduction method for distributed control systems as a fundamental tool to develop structured model reduction we first establish dissipativity preserving model reduction for general linear systems on the basis of a singular perturbation approximation to this end by deriving a tractable expression of singular perturbation models we characterize dissipativity preservation in terms of a projection like transformation of storage functions and we show that the resultant approximation error is relevant to the sum of neglected eigenvalues of an index matrix next utilizing this dissipativity preserving model reduction we develop a structured controller reduction method for distributed control systems the major significance of this method is to preserve the spatial distribution of dissipative controllers and to provide an a priori bound for the performance degradation of closed loop systems in terms of the cal h 2 norm the efficiency of the proposed method is verified through a numerical example of vibration suppression control for interconnected second order systems

surrogate modeling of multiscale models using kernel methods this work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods the key element is to learn the interface between the different scales using a fast surrogate for the microscale model which is given by multivariate kernel expansions the expansions are computed using statistically representative samples of in and output of the microscale model as learning methods we apply both support vector machines and a vectorial kernel greedy algorithm we demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines first we consider a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and second a model for simulation of saturation overshoots in porous media involving nonclassical shock waves

families of moment matching based structure preserving approximations for linear port hamiltonian systems in this paper we propose a solution to the problem of moment matching with preservation of the port hamiltonian structure in the framework of time domain moment matching we characterize several families of parameterized port hamiltonian models that match the moments of a given port hamiltonian system at a set of finite interpolation points we also discuss the problem of markov parameters matching for linear systems as a moment matching problem for descriptor representations associated with the given system at zero interpolation points solving this problem yields families of parameterized reduced order models that achieve markov parameter matching finally we apply these results to the port hamiltonian case resulting in families of parameterized reduced order port hamiltonian approximations

stability and passivity preserving petrov galerkin approximation of linear infinite dimensional systems this contribution presents two approximation methods for linear infinite dimensional systems that ensure the preservation of stability and passivity the first approach allows one to approximate internal source free infinite dimensional systems such that the resulting approximation is a port controlled hamiltonian system with dissipation the second method deals with the class of systems that are not required to have conjugated outputs but only a dissipative system operator it yields approximations with a dissipative system matrix for which bounds of their stability margin are provided both approaches are based on a state space formulation of the infinite dimensional system this makes it possible to use the petrov galerkin approximation whose free parameters are partly used for achieving the structure preservation since still free parameters remain further application specific objectives such as e g moment matching can be achieved both approaches are applied to the approximation of an euler bernoulli beam

structure preserving moment matching for port hamiltonian systems arnoldi and lanczos structure preserving model reduction of single input single output port hamiltonian systems is considered by employing the rational krylov methods the rational arnoldi method is shown to preserve for the reduced order model not only a specific number of the moments at an arbitrary point in the complex plane but also the port hamiltonian structure furthermore it is shown how the rational lanczos method applied to a subclass of port hamiltonian systems characterized by an algebraic condition preserves the port hamiltonian structure in fact for the same subclass of port hamiltonian systems the rational arnoldi method and the rational lanczos method turn out to be equivalent in the sense of producing reduced order port hamiltonian models with the same transfer function

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

a dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction in this article we introduce a new dimensional reduction approach which is based on the application of reduced basis rb techniques in the hierarchical model reduction hmr frame work considering problems that exhibit a dominant spatial direction the idea of hmr is to perform a galerkin projection onto a reduced space which combines the full solution space in the dominant direction with a reduction space in the transverse direction the latter is spanned by modal or thonormal basis functions while so far the basis functions in the hmr approach have been chosen a priori s perotto a ern and a veneziani multiscale model simul 8 2010 pp 1102 1127 for instance as legendre or trigonometric polynomials in this work a highly nonlinear approximation is employed for the construction of the reduction space to this end we first derive a lower dimen sional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction exploiting the good ap proximation properties of rb methods we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation for an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article we introduce our method for general elliptic problems such as advection diffusion equations in two space dimensions numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach

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

unit canonical correlations between future and past on considere des processus arma stationnaires x t de rang complet et le cas ou il y a des fonctions lineaires du futur x t t 0 et du passe x t t 0 qui ont une correlation unite on montre que le nombre de paires de ce type lineairement independantes est le nombre de zeros du determinant des fonctions de transfert sur le cercle unite

irregular boundary value problems for second order elliptic differential operator equations in umd banach spaces we prove coerciveness with a defect and fredholmness of nonlocal irregular boundary value problems for second order elliptic differential operator equations in umd banach spaces then we prove coerciveness with a defect in both the space variable and the spectral parameter of the problem with a linear parameter in the equation the results do not imply maximal lp regularity in contrast to previously considered regular case in fact a counterexample shows that there is no maximal lp regularity in the irregular case when studying fredholmness the boundary conditions may also contain unbounded operators in perturbation terms finally application to nonlocal irregular boundary value problems for elliptic equations of the second order in cylindrical domains are presented equations and boundary conditions may contain differential integral parts the spaces of solvability are sobolev type spaces w p q 2 2

linear and nonlinear degenerate differential operators and applications abstract the boundary value and mixed value problems for linear and nonlinear singular degenerate abstract elliptic and parabolic equations are studied the linear problems involve some parameters the uniform l p separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained the equations include linear operators defined in banach spaces in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems

nonlinear model reduction via a locally weighted pod method summary in this article we propose a new approach for model reduction of parameterized partial differential equations pdes by a locally weighted proper orthogonal decomposition lwpod method the presented approach is particularly suited for large scale nonlinear systems characterized by parameter variations instead of using a global basis to construct a global reduced model lwpod approximates the original system by multiple local reduced bases each local reduced basis is generated by the singular value decomposition of a weighted snapshot matrix compared with global model reduction methods such as the classical proper orthogonal decomposition lwpod can yield more accurate solutions with a fixed subspace dimension as another contribution we combine lwpod with the chord iteration to solve elliptic pdes in a computationally efficient fashion the potential of the method for achieving large speedups while maintaining good accuracy is demonstrated for both elliptic and parabolic pdes in a few numerical examples copyright 2016 john wiley sons ltd

pod galerkin reduced order methods for combined navier stokes transport equations based on a hybrid fv fe solver abstract the purpose of this work is to introduce a novel pod galerkin strategy for the semi implicit hybrid high order finite volume finite element solver introduced in bermudez et al 2014 and busto et al 2018 the interest is into the incompressible navier stokes equations coupled with an additional transport equation the full order model employed in this article makes use of staggered meshes this feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes the reduced order model presented herein accounts for velocity pressure and a transport related variable the pressure term at both the full order and the reduced order level is reconstructed making use of a projection method more precisely a poisson equation for pressure is considered within the reduced order model results are verified against three dimensional manufactured test cases moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

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

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

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

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

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

effort and flow constraint reduction methods for structure preserving model reduction of port hamiltonian systems abstract the geometric formulation of general port hamiltonian systems is used in order to obtain two structure preserving reduction methods the main idea is to construct a reduced order dirac structure corresponding to zero power flow in some of the energy storage ports this can be performed in two canonical ways called the effort and the flow constraint methods we show how the effort constraint method can be regarded as a projection based model reduction method both the effort and flow constraint reduction methods preserve the stability and passivity properties of the original system as a consequence of preserving the port hamiltonian structure

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

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

h2 model reduction for negative imaginary systems this paper studies the h2 model reduction problem for negative imaginary ni systems for a given high order system that is stable and ni our goal is to approximate it by a low order ni system so

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

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

a new result on passivity preserving model reduction abstract the problem of model reduction with preservation of passivity is investigated the approach is based on positive real interpolation and is inspired by the similarity between lowner and pick matrices the former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions it follows that interpolation of the original set of data together with an appropriately defined mirror image set of data yields automatically positive real interpolants subsequently we show how this result can be implemented using a krylov projection procedure the ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large scale systems

efficient model reduction of parametrized systems by matrix discrete empirical interpolation in this work we apply a matrix version of the so called discrete empirical interpolation mdeim for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced order models roms however in many cases such an affine decomposition is not readily available and must be recovered through often intrusive procedures such as the empirical interpolation method eim and its discrete variant deim in this paper we show that mdeim represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non intrusive efficient and purely algebraic way we propose different strategies to combine mdeim with a state approximation resulting either from a reduced basis greedy approach or proper orthogonal decomposition a posteriori error estimates accounting for the mdeim error are also developed in the case of parametrized elliptic and parabolic equations finally the capability of mdeim to generate accurate and efficient roms is demonstrated on the solution of two computationally intensive classes of problems occurring in engineering contexts namely pde constrained shape optimization and parametrized coupled problems

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea