an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

data driven operator inference for nonintrusive projection based model reduction abstract this work presents a nonintrusive projection based model reduction approach for full models based on time dependent partial differential equations projection based model reduction constructs the operators of a reduced model by projecting the equations of the full model onto a reduced space traditionally this projection is intrusive which means that the full model operators are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a given vector however in many situations the full model is given as a black box that computes trajectories of the full model states and outputs for given initial conditions and inputs but does not provide the full model operators our nonintrusive operator inference approach infers approximations of the reduced operators from the initial conditions inputs trajectories of the states and outputs of the full model without requiring the full model operators our operator inference is applicable to full models that are linear in the state or have a low order polynomial nonlinear term the inferred operators are the solution of a least squares problem and converge with sufficient state trajectory data in the frobenius norm to the reduced operators that would be obtained via an intrusive projection of the full model operators our numerical results demonstrate operator inference on a linear climate model and on a tubular reactor model with a polynomial nonlinear term of third order

energy stable model reduction of neurons by nonnegative discrete empirical interpolation the accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences this work develops a novel structure preserving model reduction technique to address this problem based on galerkin projection and nonnegative operator approximation it is first shown that the corresponding reduced order model is guaranteed to be energy stable thanks to both the structure preserving approach that constructs a distinct reduced order basis for each cable in the network and the preservation of nonnegativity furthermore a posteriori error estimates are provided showing that the model reduction error can be bounded and controlled finally the application to the model reduction of a large scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups

asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis abstract in this paper we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed by a change of variable from the well known keller segel model describing cell bacteria movement toward the concentration gradient of the chemical that is consumed by the cells we prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

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

global existence and decay rate of the boussinesq burgers system with large initial data abstract in this paper we study the global existence and the long time asymptotic behavior of classical solutions to the cauchy problem for the boussinesq burgers system with large initial data more precisely we first show that the classical solutions exist globally in time with large initial data by using the l p p 2 estimates other than the conventional l 2 estimate then we prove that the global classical solutions converge to constant equilibrium states with an algebraic decay rate as time approaches infinity

matrix rational h 2 approximation a gradient algorithm based on schur analysis this paper deals with the rational approximation of specified order n to transfer functions which are assumed to be matrix valued functions in the hardy space for the complement of the closed unit disk endowed with the l2 norm an approach is developed leading to a new algorithm the first one to our knowledge which concerns matrix transfer functions in l2 norm this approach generalizes the ideas presented in l baratchart m cardelli and m olivi automatica 27 1991 pp 413 418 in the scalar case but involves substantial new difficulties using the douglas shapiro shields factorization of transfer functions the criterion for the rational approximation problem above is expressed in terms of inner matrix functions of mcmillan degree n these functions which possess a manifold structure are represented by means of local coordinate maps obtained in d alpay l baratchart and a gombani oper theory adv appl 73 1994 pp 30 66 from a tangential schur algorithm and for which the coordinates range over n copies of the unit ball a gradient algorithm is then employed to solve the approximation problem using the coordinate maps to describe the manifold locally and changing from one coordinate map to another when required however while processing the gradient algorithm a boundary point can be reached it is proved that such a point can be considered as an initial point for searching for a local minimum of lower degree while a local minimum of mcmillan degree k n provides a starting point for searching for a local minimum at degree k 1 the minimization process then pursues through different degrees the convergence of this algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example

an adaptive proper orthogonal decomposition method for model order reduction of multi disc rotor system abstract the proper orthogonal decomposition pod method is a main and efficient tool for order reduction of high dimensional complex systems in many research fields however the robustness problem of this method is always unsolved although there are some modified pod methods which were proposed to solve this problem in this paper a new adaptive pod method called the interpolation grassmann manifold igm method is proposed to address the weakness of local property of the interpolation tangent space of grassmann manifold itgm method in a wider parametric region this method is demonstrated here by a nonlinear rotor system of 33 degrees of freedom dofs with a pair of liquid film bearings and a pedestal looseness fault the motion region of the rotor system is divided into two parts simple motion region and complex motion region the adaptive pod method is compared with the itgm method for the large and small spans of parameter in the two parametric regions to present the advantage of this method and disadvantage of the itgm method the comparisons of the responses are applied to verify the accuracy and robustness of the adaptive pod method as well as the computational efficiency is also analyzed as a result the new adaptive pod method has a strong robustness and high computational efficiency and accuracy in a wide scope of parameter

a locally parametrized reduced order model for the linear frequency domain approach to time accurate computational fluid dynamics for transonic flows governed by the time accurate navier stokes equations small approximately periodic perturbations can be calculated accurately by transition to the frequency domain and truncating the fourier expansion after the first harmonic this is referred to as the linear frequency domain lfd method in this paper a parametric trajectory of reduced order models roms for the lfd solver is presented to this end several local projection based roms which are essentially specified by suitable low order subspaces are computed by the method of proper orthogonal decomposition pod in an offline stage the claimed trajectory is obtained locally by interpolating the given local subspaces considered as sample points in the grassmann manifold it is shown that the manifold interpolation technique is subject to certain restrictions moreover it turns out that the application of computing accurate roms for the lfd solver requires a special choice of underlying inner product necessitating a non eu

mathematical and numerical results on the sensitivity of the pod approximation relative to the burgers equation from mathematical and numerical points of view we study the sensitivity with respect to parametric evolutions of the error obtained by approximating a given parametric partial differential equation using a proper orthogonal decomposition pod basis determined once and for all with association to a fixed parameter more precisely we will be considering the one dimensional burgers equation with a source term

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

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

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 riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

fast low rank modifications of the thin singular value decomposition this paper develops an identity for additive modivations of a singular value decomposition svd to reflect updates downdates shifts and edits of the data matrix this sets the stage for fast and ememory efficient sequential algorithms for tracking singular values and subspaces in conjunction with a fast solution for the pseudo inverse of a submatrix of an orthogonal matrix we develop a scheme for computing a thin svd of streaming data in a single pass with linear time complexity a rank r think svd of a p x q matrix can be computed in o pqr time for r less than or equal sqroot min p q linear algebra and its applications

nonlinear model order reduction based on local reduced order bases summary a new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced order bases is presented it is particularly suited for problems characterized by different physical regimes parameter variations or moving features such as discontinuities and fronts instead of approximating the solution of interest in a fixed lower dimensional subspace of global basis vectors the proposed model order reduction method approximates this solution in a lower dimensional subspace generated by most appropriate local basis vectors to this effect the solution space is partitioned into subregions and a local reduced order basis is constructed and assigned to each subregion offline during the incremental solution online of the reduced problem a local basis is chosen according to the subregion of the solution space where the current high dimensional solution lies this is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower dimensional solution space because it is also applicable to the process of hyper reduction the proposed method for nonlinear model order reduction is computationally efficient its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid structure electric interaction problems copyright 2012 john wiley sons ltd

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

order reduction of large scale second order systems using krylov subspace methods in order reduction of large scale linear time invariant systems krylov subspace methods based on moment matching are among the best choices today however in many technical fields models typically consist of sets of second order differential equations and krylov subspace methods cannot directly be applied two methods for solving this problem are presented in this paper 1 an approach by su and craig is generalized and the number of matching moments is increased 2 a new approach via first order models is presented resulting in an even higher number of matching moments both solutions preserve the specific structure of the second order type model

a matrix deim technique for model reduction of nonlinear parametrized problems in cardiac mechanics abstract when relying on newton iterations to solve nonlinear problems in the context of reduced basis rb methods the assembling of the rb arrays during the online stage depends on the dimension of the underlying high fidelity approximation this is more of an issue when dealing with fully nonlinear problems for which the global jacobian matrix has to be entirely reassembled at each newton step in this paper the discrete empirical interpolation method deim and its matrix version mdeim are combined to evaluate both the residual vector and the jacobian matrix very efficiently in the case of complex parametrized nonlinear mechanical problems we compare this strategy with the classical deim approach and we derive a posteriori error estimates on the solution accounting for the contribution of deim mdeim errors the effectiveness of the proposed framework is assessed on quasi static nonlinear problems in particular we consider a nonlinear elasticity problem defined on a cube and a mechanical model describing heart contraction for an idealized left ventricle geometry the latter is a coupled problem in which the activation of the heart contraction is given by the solution of an electrophysiology model our numerical results show that mdeim is preferable to the classical deim both in terms of efficiency and accuracy

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

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

a new proof of the garoufalidis le zeilberger quantum macmahon master theorem abstract we propose a new proof of the quantum version of macmahon s master theorem established by garoufalidis le and zeilberger

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

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis abstract the asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper by a cole hopf type transformation the singular chemotaxis is converted into a non singular hyperbolic system then we study the transformed system and establish the global existence asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates the main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established

an on line time dependent parametric model order reduction scheme with focus on dynamic stress recovery in view of the tendency towards ever lighter and more powerful machines and even shorter design cycles it becomes essential to have virtual prototyping tools that allow for fast and reliable numerical simulations current state of the art structural dynamics and flexible multibody simulation techniques usually involve the solution of matrix systems with thousands to millions of variables model order reduction schemes are used to keep computational effort affordable at the expense of a minimal loss of accuracy these techniques typically face difficulties with systems in which flexible bodies can be loaded in many degrees of freedom and rarely allow for accurate local stress and strain evaluation the present work proposes to address both issues for the particular but frequent case of moving loads or boundary conditions this behavior is found in most of the contact problems and systems that include sliding components for which many loading or boundary locations are possible but only a few of them are active at a certain moment in time the proposed scheme exploits a reduction vector space that continuously varies in time by means of a parametric definition of the external load position contrary to the majority of the parametric model order reduction schemes that allow mainly for quasi static parametric variations the proposed approach can be used efficiently for time simulation of dynamically varying parametric models this is achieved by considering the implicit time dependency of the reduction vector space using galerkin projections or alternatively by direct substitution of the reduced kinetic energy potential energy and generalized forces in the lagrange equations it is shown by developing a consistent mathematical framework that the price to pay for the very compact reduction space obtained is the evaluation of some extra terms in the equations of motion numerical examples are used to assess the accuracy of the proposed method results show the potential of this strategy with particular focus on displacement and stress fields and furthermore highlight its real time potential moreover the developed framework together with the numerical results allow for a deeper physical understanding of the complex phenomena related to this category of time varying multiple input multiple output systems

a method for interpolating on manifolds structural dynamics reduced order models a rigorous method for interpolating a set of parameterized linear structural dynamics reduced order models roms is presented by design this method does not operate on the underlying set of parameterized full order models hence it is amenable to an online real time implementation it is based on mapping appropriately the rom data onto a tangent space to the manifold of symmetric positive definite matrices interpolating the mapped data in this space and mapping back the result to the aforementioned manifold algorithms for computing the forward and backward mappings are offered for the case where the roms are derived from a general galerkin projection method and the case where they are constructed from modal reduction the proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input in all cases good accuracy is demonstrated at real time processing speeds copyright 2009 john wiley sons ltd

proper orthogonal decomposition based model order reduction via radial basis functions for molecular dynamics systems summary model order reduction for molecular dynamics md systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi atomic interactions in high dimensions in the present work we introduce a proper orthogonal decomposition based method in conjunction with the radial basis function rbf approximation of the nonlinear and nonlocal potential energies and inter atomic forces for md systems this approach avoids coordinate transformation between the physical and reduced order coordinates and allows the potentials and inter atomic forces to be calculated directly in the reduced order space the rbf approximated potential energies and inter atomic forces in the reduced order space are discretized on the basis of the smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method the good approximation properties of rbfs in interpolating scattered data make them ideal candidates for the reduced order approximation of md inter atomic force calculations the proposed approach is validated by performing the reduced order simulations of dna molecules under various external loadings copyright 2013 john wiley sons ltd

an approximate approach to h sup 2 optimal model reduction deals with the problem of computing an h sub 2 optimal reduced order model for a given stable multivariable linear system by way of orthogonal projection the problem is formulated as that of minimizing the h sub 2 model reduction cost over the stiefel manifold so that the stability constraint on reduced order models is automatically satisfied and thus totally avoided in the new problem formulation the closed form expression for the gradient of the cost over the manifold is derived from which a gradient flow results as an ordinary differential equation ode a number of nice properties about such a flow are established furthermore two explicit iterative convergent algorithms are developed from the flow one has a constant step size and the other has a varying step size and is much more efficient both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates a procedure for closing the gap between the original and modified problem is proposed in the symmetric case the two problems are shown to be equivalent numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence

scattered manifold valued data approximation we consider the problem of approximating a function f from an euclidean domain to a manifold m by scattered samples f xi i i in mathcal i f i i i where the data sites xi i i in mathcal i i i i are assumed to be locally close but can otherwise be far apart points scattered throughout the domain we introduce a natural approximant based on combining the moving least square method and the karcher mean we prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart the analysis also tells us that the use of karcher s mean dependent on a riemannian metric and the associated exponential map is inessential and one can replace it by a more general notion of center of mass based on a general retraction on the manifold consequently we can substitute the karcher mean by a more computationally efficient mean we illustrate our work with numerical results which confirm our theoretical findings

krylov subspace methods for reduced order modeling in circuit simulation the simulation of electronic circuits involves the numerical solution of very large scale sparse in general nonlinear systems of differential algebraic equations often the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approximate models of much smaller state space dimension in this paper we describe the use of krylov subspace methods for generating such reduced order models of linear subcircuits particular emphasis is on reduced order modeling techniques that preserve the passivity of linear rlc subcircuits

a boussinesq system for two way propagation of nonlinear dispersive waves abstract in this report we study the system n t w x nw x 1 6 n xxt 0 w t n x ww x 1 6 w xxt 0 which describes approximately the two dimensional propagation of surface waves in a uniform horizontal channel of length l0 filled with an irrotational incompressible inviscid fluid which in its undisturbed state has depth h the non dimensional variables n x t and w x t represent the deviation of the water surface from its undisturbed position and the horizontal velocity at water level 2 3h respectively the natural initial boundary value problem corresponding to the situation wherein the channel is fitted with a wavemaker at both ends is formulated and analyzed theoretically we then present a numerical algorithm for the approximation of solutions of the system and prove the algorithm is fourth order accurate both in time and in space is unconditionally stable and has optimal computational complexity which is to say the operation cost per time step is of order m where m is the number of grid points in the spatial discretization further we implement the algorithm as a computer code and use it to study head on collisions of solitary waves our numerical simulations are compared with existing theoretical numerical and experimental results we have tentatively concluded that the system is a good candidate for modeling two dimensional surface water waves

global existence and asymptotic behavior of the boussinesq burgers system abstract this paper is concerned with the boussinesq burgers system which models the propagation of bores by combing the dissipation dispersion and nonlinearity we establish the global existence and asymptotical behavior of classical solutions of the initial value boundary problem of the boussinesq burgers system with the help of a lyapunov functional and the technique of moser iteration particularly we show that the solution converges to the unique constant stationary solution exponentially as time tends to infinity

boundary layer problem on a hyperbolic system arising from chemotaxis abstract this paper is concerned with the boundary layer problem for a hyperbolic system transformed via a cole hopf type transformation from a repulsive chemotaxis model with logarithmic sensitivity proposed in 23 34 modeling the biological movement of reinforced random walkers which deposit a non diffusible or slowly moving signal that modifies the local environment for succeeding passages by prescribing the dirichlet boundary conditions to the transformed hyperbolic system in an interval 0 1 we show that the system has the boundary layer solutions as the chemical diffusion coefficient e 0 and further use the formal asymptotic analysis to show that the boundary layer thickness is e 1 2 our work justifies the boundary layer phenomenon that was numerically found in the recent work 25 however we find that the original chemotaxis system does not possess boundary layer solutions when the results are reverted to the pre transformed system

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

two sided grassmann manifold algorithm for optimal ℋ2 model reduction summary we consider an optimal 2 model reduction problem for large scale dynamical systems the problem is formulated as a minimization problem over grassmann manifold with two variables this formulation allows us to develop a two sided grassmann manifold algorithm which is numerically efficient and suitable for the reduction of large scale systems the resulting reduced system preserves the stability of the original system numerical examples are presented to show that the proposed algorithm is computationally efficient and robust with respect to the selection of initial projection matrices copyright 2015 john wiley sons ltd

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

reduced order techniques for sensitivity analysis and design optimization of aerospace systems this work proposes a new method for using reduced order models in lieu of high fidelity analysis during the sensitivity analysis step reduced order models are developed using a combination of proper orthogonal decomposition and radial basis functions interpolation using reduced order models based on proper orthogonal decomposition is compared against normal optimization on a general airfoil shape optimization problem the interpolation procedure does not require additional high fidelity evaluations to construct new reduced order models the errors associated with the reduced order models themselves as well as the gradients calculated from them are compared the effects of each approach on the overall optimization paths times and function counts are also examined

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

initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis abstract this paper is devoted to the analytical study of initial boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis model with logarithmic sensitivity in the first part of the paper we show that subject to the dirichlet boundary conditions classical solutions exist globally in time for large initial data asymptotically in time the solutions are shown to converge to their boundary data at an exponential rate as time goes to infinity numerical simulations are supplied to corroborate the analytical results the analytic approach developed herein can be utilized to handle a family of initial and initial boundary value problems of the model and related models with similar mathematical structure as a demonstration of the effectiveness of our approach in the second part of the paper we show that subject to the neumann dirichlet boundary conditions classical solutions exist and converge to constant equilibrium states for large initial data and for arbitrary values of the chemical diffusion coefficient this improves a previous result obtained in 30 where the smallness of the chemical diffusion coefficient was required

model order reduction for dynamic simulation of a flexible multibody system via absolute nodal coordinate formulation abstract the absolute nodal coordinate formulation ancf can be used to model a flexible multibody system subject to both overall motions and large deformations with an increasing number of finite elements of ancf to mesh flexible bodies however the computation cost will become extremely high to improve the computational efficiency of ancf for a large scaled flexible multibody system a systematic method is proposed in this study for the model order reduction based on the proper orthogonal decomposition and the galerkin projection at first an approach for the selection of reduced constraint equations is proposed to deal with the singularity of the coefficient matrix of the reduced order model rom then the computation of the reduced stiffness matrix and generalized force vector of the rom are parallelized via the openmp directives afterwards two parametric approaches are presented to make the rom be adaptive to the change of system parameters one is the interpolation approach of reduced order basis vectors on a manifold to obtain the parametric rom and the other is the fast to slow rom approach via the reduced order basis vectors extracted from the fast dynamic responses to simulate slow dynamic responses finally four numerical examples are given to validate the efficacy of the proposed method for the dynamic simulations of both rigid and flexible multibody systems

specializations and extensions of the quantum macmahon master theorem abstract we study some specializations and extensions of the quantum version of the macmahon master theorem derived by garoufalidis le and zeilberger in particular we obtain a t q analogue for the cartier foata noncommutative version and a semi strong t q analogue for the contextual algebra

a real time procedure for affinely dependent parametric model order reduction using interpolation on grassmann manifolds summary model order reduction helps to reduce the computational time in dealing with large dynamical systems for example during simulation control optimization in many cases the considered model depends on parameters model order reduction techniques are therefore preferred to symbolically preserve this dependence or to be adaptive to the change of the model caused by the variation in the values of the parameters in this paper we first present the application of the interpolation technique on grassmann manifolds to this problem we then improve the method for the models whose system matrices depend affinely on parameters by considerably reducing the computational complexity on the basis of analyzing the structure of sums of singular value decompositions and decomposing the whole procedure into offline and online stages a numerical example is shown to illustrate the method as well as to prove its effectiveness copyright 2012 john wiley sons ltd

a parameterised model order reduction method for parametric systems based on laguerre polynomials abstractthis paper presents a laguerre polynomials based parametrised model order reduction method for the parametric system in time domain the method allows that the parametric dependence in system matrices is nonaffine the method is presented via reducing an approximate polynomial parametric system based on taylor expansion and laguerre polynomials resulting in a parametric reduced system that can accurately approximate the time response of the original parametric system over a wide range of parameter the reduced parametric system obtained by proposed method can be implemented by two algorithms algorithm 1 is a direct way that is suitable for single input multi output parametric systems algorithm 2 is presented based on a connection to the krylov subspace which is efficient and suitable for multi input multi output parametric systems the effectiveness of the proposed method is illustrated with two benchmarks in practical applications

real time modelling of diastolic filling of the heart using the proper orthogonal decomposition with interpolation abstract in this research a reduced order method rom called the proper orthogonal decomposition with interpolation podi is used to drastically reduce computation time of highly complex and non linear problems as encountered in simulating the heart the idea behind this method is to first construct a database of pre computed full scale solutions using the element free galerkin method efg and then project a selected subset of these solutions to a low dimensional space using the moving least square method an interpolation is carried out for the problem at hand before the resulting coefficients are projected back to the original high dimensional solution space computations are carried out on a bi ventricle model to investigate the performance and accuracy varying the material parameters and to determine the sensitivity with respect to the parametric values the podi calculations are completed within 1 25 s on a normal desktop machine with the relative l2 error norm not exceeding 2 5 10 3 hence it is demonstrated that real time modelling of the heart can be successfully carried out at acceptable error levels

non commutative extensions of the macmahon master theorem we present several non commutative extensions of the macmahon master theorem further extending the results of cartier foata and garoufalidis le zeilberger the proofs are combinatorial and new even in the classical cases we also give applications to the s extension and krattenthaler schlosser s q analogue

reduced order output feedback control design for pde systems using proper orthogonal decomposition and nonlinear semidefinite programming the design of an optimal output feedback reduced order control roc law for a dynamic control system is an important example of a difficult and in general non convex nonlinear optimal control problem in this paper we present a novel numerical strategy to the solution of the roc design problem if the control system is described by partial differential equations pde the discretization of the roc problem with pde constraints leads to a large scale non convex nonlinear semidefinite program nsdp for reducing the size of the high dimensional control system first we apply a proper orthogonal decomposition pod method to the discretized pde the pod approach leads to a low dimensional model of the control system thereafter we solve the corresponding small sized nsdp by a fully iterative interior point constraint trust region ipctr algorithm ipctr is designed to take advantage of the special structure of the nsdp finally the solution is a roc for the low dimensional approximation of the control system in our numerical examples we demonstrate that the reduced order controller computed from the small scaled problem can be used to control the large scale approximation of the pde system

adaptation of aeroelastic reduced order models and application to an f 16 configuration the proper orthogonal decomposition method has been shown to produce accurate reduced order models for the aeroelastic analysis of complete aircraft configurations at fixed flight conditions however changes in the mach number or angle of attack often necessitate the reconstruction of the reduced order model to maintain accuracy which destroys the sought after computational efficiency straightforward approaches for reduced order model adaptation such as the global proper orthogonal decomposition method and the direct interpolation of the proper orthogonal decomposition basis vectors that have been attempted in the past have been shown to lead to inaccurate proper orthogonal decomposition bases in the transonic flight regime alternatively a new reduced order model adaptation scheme is described in this paper and evaluated for changes in the freestream mach number and angle of attack this scheme interpolates the subspace angles between two proper orthogonal decomposition subspaces then generates a new proper orthogonal decomposition basis through an orthogonal transformation based on the interpolated subspace angles the resulting computational methodology is applied to a complete f 16 configuration in various airstreams the predicted aeroelastic frequencies and damping coefficients are compared with counterparts obtained from full order nonlinear aeroelastic simulations and flight test data good correlations are observed including in the transonic regime the obtained computational results reveal a significant potential of the adapted reduced order model computational technology for accurate near real time aeroelastic predictions

non blowup of a generalized boussinesq burgers system with nonlinear dispersion relation and large data abstract we study the qualitative behavior of classical solutions to the cauchy problem of a generalized boussinesq burgers system in one space dimension assuming initial data belong to h 2 r and utilizing energy methods we show that there exist unique global in time classical solutions to the cauchy problem of the model and the solutions converge to constant equilibrium states as time goes to infinity regardless of the magnitude of the initial data moreover it is shown that the viscous and inviscid models are consistent in the process of vanishing viscosity limit

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

differentiable families of subspaces abstract we show how the differentiable families of subspaces can be studied from a geometrical point of view by means of their representation as a differentiable mapping from the manifold of parameters m into the grassmann manifold gr k n of all k dimensional subspaces of k n and the consideration of principal bundles over gr k n and m in order to achieve this first we present a survey of the topology the differentiable structure and the bundle structure of gr k n in particular we show that the topology induced by the gap metric is equivalent to the usual quotient topology second we use these structures to prove two local characterizations of the differentiability of a family of subspaces and we state that these characterizations can be globalized if m is contractible finally we apply these properties to study the existence of global jordan basis of class c r

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

reduced order models for computational fluid dynamics based nonlinear aeroelastic problems in this work a nonlinear state space based identification method is proposed to describe compactly unsteady aerodynamic responses such a reduced order model is trained on a series of signals that implicitly represent the relationship between the structural motion and the aerodynamic loads the determination of the model parameters is obtained through a two level training procedure where in the first stage the matrices associated to the linear part of the model are computed by a robust subspace projection technique whereas the remaining nonlinear terms are determined by an output error minimization procedure in the second stage the present approach is tested on two different problems proving the convergence toward the reference results obtained by a computational fluid dynamics solver in linear and nonlinear aerodynamic and aeroelastic applications whereas the aerodynamic reduced order models are coupled with the related structural mechanical systems demonstrating the ability of capturing the ma

dynamics of large scale coupled structural mechanical systems a singular perturbation proper orthogonal decomposition approach we have combined the theories of geometric singular perturbation and proper orthogonal decomposition to study systematically the dynamics of coupled systems in mechanics involving coupling between continuous structures and nonlinear oscillators here we analyze a prototypical structural mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material we cast the equations of motion in a singularly perturbed set of oscillators and compute analytic approximations to an attractive global invariant manifold in phase space of the coupled system the invariant manifold two dimensional for the unforced system and three dimensional for the forced system carries a continuum of slow motions for a sufficiently stiff rod a proper orthogonal decomposition of any long time motion extracts a single structure for the spatial coherence of the dynamics which is a realization of the slow invariant manifold as the flexibility of the rod increases the energy of

a new algorithm for l 2 optimal model reduction in this paper the quadratically optimal model reduction problem for single input single output systems is considered the reduced order model is determined by minimizing the integral of the magnitude squared of the transfer function error it is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and on this basis a two step iterative algorithm is developed combining a least squares solver with a gradient minimizer convergence of the proposed algorithm to stationary values of the quadratic cost function is proved the formulation is extended to handle the frequency weighted optimal model reduction problem three examples demonstrate the optimization algorithm

model order reduction for temperature dependent nonlinear mechanical systems a multiple scales approach abstract the thermal dynamics in thermo mechanical systems exhibits a much slower time scale compared to the structural dynamics in this work we use the method of multiple scales to reduce the thermo mechanical structural models with a slowly varying temperature distribution in a systematic manner in the process we construct a reduction basis that adapts according to the instantaneous temperature distribution of the structure facilitating an efficient reduction in the number of unknowns as a proof of concept we demonstrate the method on a range of linear and nonlinear beam examples and obtain a consistently better accuracy and reduction in the number of unknowns than the standard galerkin projection using a constant basis

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