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

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

phase separation and coarsening in one dimensional driven diffusive systems local dynamics leading to long range hamiltonians a driven system of three species of particles diffusing on a ring is studied in detail the dynamics is local and conserves the three densities a simple argument suggesting that the model should phase separate and break the translational symmetry is given we show that for the special case where the three densities are equal the model obeys detailed balance and the steady state distribution is governed by a hamiltonian with asymmetric long range interactions this provides an explicit demonstration of a simple mechanism for breaking of ergodicity in one dimension the steady state of finite size systems is studied using a generalized matrix product ansatz the coarsening process leading to phase separation is studied numerically and in a mean field model the system exhibits slow dynamics due to trapping in metastable states whose number is exponentially large in the system size the typical domain size is shown to grow logarithmically in time generalizations to a larger number of species are discussed

stable galerkin reduced order models for linearized compressible flow the galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations the numerical stability of galerkin models is shown to depend on the choice of inner product for the projection for the linearized euler equations a symmetry transformation leads to a stable formulation for the inner product boundary conditions for compressible flow that preserve stability of the reduced order model are constructed preservation of stability for the discrete implementation of the galerkin projection is made possible using a piecewise smooth finite element basis stability of the reduced order model using this approach is demonstrated on several model problems where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation

on the stability of the reduced basis method for stokes equations in parametrized domains we present an application of reduced basis method for stokes equations in domains with affine parametric dependence the essential components of the method are i the rapid convergence of global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n selected points in parameter space ii the off line on line computational procedures decoupling the generation and projection stages of the approximation process the operation count for the on line stage in which given a new parameter value we calculate an output of interest depends only on n typically very small and the parametric complexity of the problem the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation design optimization and real time control particular attention is given i to the pressure treatment of incompressible stokes problem ii to find an equivalent inf sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem iii to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions iv to reduce computational costs in order to allow real time solution of parametrized problem

summation by parts in time we develop a new high order accurate time integration technique for initial value problems we focus on problems that originate from a space approximation using high order finite difference methods on summation by parts form with weak boundary conditions and extend that technique to the time domain the new time integration method is global high order accurate unconditionally stable and together with the approximation in space it generates optimally sharp fully discrete energy estimates in particular it is shown how stable fully discrete high order accurate approximations of the maxwells equations the elastic wave equations and the linearized euler and navier stokes equations can obtained even though we focus on finite difference approximations we stress that the methodology is completely general and suitable for all semi discrete energy stable approximations numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems

n species stochastic models with boundaries and quadratic algebras stationary probability distributions for stochastic processes on linear chains with closed or open ends are obtained using the matrix product ansatz the matrices are representations of some quadratic algebras the algebras and the types of representations considered depend on the boundary conditions in the language of quantum chains we obtain the ground state of n state quantum chains with free boundary conditions or with non diagonal boundary terms at one or both ends in contrast to problems involving the bethe ansatz we do not have a general framework for arbitrary n which when specialized gives the known results for n 2 in fact the n 2 and n 2 cases appear to be very different

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

stochastic models on a ring and quadratic algebras the three species diffusion problem the stationary state of a stochastic process on a ring can be expressed using traces of monomials of an associative algebra defined by quadratic relations we consider only exclusion processes and restrict the type of algebras such that one has recurrence relations for traces of words of different lengths this is possible only if the rates satisfy certain compatibility conditions these conditions are derived and explicit representations of the generators of the quadratic algebras are given

efficient non linear model reduction via a least squares petrov galerkin projection and compressive tensor approximations the first author acknowledges the partial support by a national science foundation graduate fellowship and the partial support by a national defense science and engineering graduate fellowship the second and third authors acknowledge the partial support by the motor sports division of the toyota motor corporation under agreement number 48737 and the partial support by a research grant from the academic excellence alliance program between king abdullah university of science and technology kaust and stanford university all authors also acknowledge the constructive comments received during the review process

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

headway oscillations and phase transitions for diffusing particles with increased velocity a totally asymmetric exclusion process with n particles on a periodic one dimensional lattice of l sites is considered where particles can move one or two sites per infinitesimal timestep an exact analysis for n 2 and a mean field theory in comparison with simulations show even odd oscillations in the headway distribution of particles the expression headway is understood as the number of empty sites in front of a particle oscillations become maximal if particles only move at their maximum possible speed a phase transition separates two density profiles around a generated perturbation that plays the role of a defect the matrix product ansatz is generalized to obtain the exact solution for finite n and l thermodynamically the headway distribution yields the mean field result as n while it is not described generally by a product measure

stabilization of projection based reduced order models summary a rigorous method for stabilizing projection based linear reduced order models without significantly affecting their accuracy is proposed unlike alternative approaches this method is computationally efficient it requires primarily the solution of a small scale convex optimization problem furthermore it is nonintrusive in the sense that it operates directly on readily available reduced order operators these can be precomputed using any data compression technique including balanced truncation balanced proper orthogonal decomposition proper orthogonal decomposition or moment matching the proposed method is illustrated with three applications the stabilization of the reduction of the computational fluid dynamics based model of a linearized unsteady supersonic flow the reduction of a computational structural dynamics system and the stabilization of the reduction of a coupled computational fluid dynamics computational structural dynamics model of a linearized aeroelastic system in the transonic flow regime copyright 2012 john wiley sons ltd

an improved error bound for reduced basis approximation of linear parabolic problems we consider a space time variational formulation for linear para bolic partial dierential equations we introduce an associated petrov galerkin truth nite element discretization with favorable discrete inf sup constant the inverse of which enters into error estimates is unity for the heat equa tion decreases only linearly in time for non coercive but asymptotically stable convection operators the latter in turn permits eective long time a posteriori error bounds for reduced basis approximations in sharp contrast to classical pessimistic exponentially growing energy estimates the paper contains a full analysis and various extensions for the formulation introduced briey in 13 as well as numerical results for a model reaction convection diusion equation

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

exact diffusion constant for the one dimensional partially asymmetric exclusion model we calculate exactly the diffusion constant associated with the fluctuations of the current for the partial asymmetric exclusion model on a ring with an arbitrary number of particles and holes we also give the diffusion constant of a tagged particle on that ring our approach extends using the deformed harmonic oscillator algebra a result already known for the fully asymmetric case in the limit of weak asymmetry we extract from our exact expression the crossover between the edwards wilkinson and the kardar parisi zhang equations in 1 1 dimensions

multilinear maps on products of operator algebras abstract let a1 a2 ar be c algebras with second duals a1 a2 ar and let x be an arbitrary banach space let a 1 a 2 a r x be a bounded r linear map and denote by a 1 a 2 a r x the johnson kadison ringrose extension i e the separately weak to weak continuous r linear extension of the problem of characterising those for which takes its values in x was solved by villanueva when the algebras are all commutative because the dunford pettis property fails for noncommutative c algebras the obvious extension of villanueva s characterisation does not give the correct condition in this paper we solve this problem for general c algebras this result is then applied to obtaining a multilinear generalisation of the normal singular decomposition of a bounded linear operator on a von neumann algebra

the grothendieck inequality for bilinear forms on c algebras abstract the following generalization of grothendieck s inequality is proved for any bounded bilinear form v on a pair of c algebras a b there exist two states 1 2 on a and two states 1 2 on b such that v x y v 1 x x 2 xx 2 1 2 1 y y 2 yy 2 1 2 for all x a and all y b an inequality of this type was proved a few years ago by pisier in the case where one of the c algebras has the bounded approximation property it follows from the above inequality that any bounded linear map t of a c algebra into the dual of a c algebra has a factorization t r s through a hilbert space such that r s 2 t

exact solution of an exclusion process with three classes of particles and vacancies we present an exact solution for an asymmetric exclusion process on a ring with three classes of particles and vacancies using a matrix product ansatz we find exact expressions for the weights of the configurations in the stationary state the solution involves tensor products of quadratic algebras

grothendieck s inequalities for real and complex jbw triples we prove that if m 4 1 2 3 and e 0 if v and w are complex jbw triples with preduals v andw respectively and if u is a separately weak continuous bilinear form on v w then there exist norm one functionals 1 2 v and 1 2 w satisfying u x y m u x 2 2 e 2 x 2 1 1 2 y 2 2 e 2 y 2 1 1 2 for all x y v w here for a norm one functional on a complex jb triple v stands for the prehilertian seminorm on v associated to in bf1 we arrive in this grothendieck s inequality through results of c h chu b iochum and g loupias cil and a corrected version of the little grothendieck s inequality for complex jb triples due to t barton and y friedman bf1 we also obtain extensions of these results to the setting of real jb triples 2000 mathematics subject classification 17c65 46k70 46l05 46l10 and 46l70

high order accurate difference schemes for the hodgkin huxley equations a novel approach for simulating potential propagation in neuronal branches with high accuracy is developed the method relies on high order accurate difference schemes using the summation by parts operators with weak boundary and interface conditions applied to the hodgkin huxley equations this work is the first demonstrating high accuracy for that equation several boundary conditions are considered including the non standard one accounting for the soma presence which is characterized by its own partial differential equation well posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions gains in terms of cpu times are observed when high order operators are used demonstrating the advantage of the high order schemes for simulating potential propagation in large neuronal trees

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

on matrix product states for periodic boundary conditions the possibility of a matrix product representation for eigenstates with energy and momentum zero of a general m state quantum spin hamiltonian with nearest neighbour interaction and periodic boundary condition is considered for this representation we use the quadratic algebra defined in krebs and sandow 1997 j phys a math gen 30 3165 which is generated by 2m operators which fulfil m 2 quadratic relations endowed with a trace like function reflecting periodic boundary conditions it is shown that not every eigenstate with energy and momentum zero can be written in this way an explicit counter example is given this is in contrast to the case of open boundary conditions where every zero energy eigenstate can be written as a matrix product state using a fock like representation of the same quadratic algebra

well posed boundary conditions for the navier stokes equations in this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations with those well posedness can be proven for a general initial boundary value problem the procedure is exemplified on the linearized navier stokes equations in two and three space dimensions on a general domain

weakly compact operators and the strong topology for a banach space the strong topology s x of a banach space x is defined as the locally convex topology generated by the seminorms x 7 ksxk for bounded linear maps s from x into hilbert spaces the w right topology for x x is a stronger locally convex topology which may be analogously characterised by taking reflexive banach spaces in place of hilbert spaces for any banach space y a linear map t x y is known to be weakly compact precisely when t is continuous from the w right topology to the norm topology of y the main results deal with conditions for and consequences of the coincidence of these two topologies on norm bounded sets a large class of banach spaces including all c algebras and more generally all jb triples exhibit this behaviour

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

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

summation by parts projections and stability ii we have derived stability results for high order finite difference approximations of mixed hyperbolic parabolic initial boundary value problems ibvp the results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection by slightly rearranging the analytic equations we can prove strict stability for hyperbolic parabolic ibvp furthermore we generalize our technique so as to yield strict stability on curvilinear non smooth domains in two space dimensions finally we show how to incorporate inhomogeneous boundary data while retaining strict stability using the same procedure one can prove strict stability in higher dimensions as well

topological characterisation of weakly compact operators let x be a banach space then there is a locally convex topology for x the right topology such that a linear map t from x into a banach space y is weakly compact precisely when t is a continuous map from x equipped with the right topology into y equipped with the norm topology when t is only sequentially continuous with respect to the right topology it is said to be pseudo weakly compact this notion is related to pelczynski s property v

matrix product eigenstates for one dimensional stochastic models and quantum spin chains we show that all zero energy eigenstates of an arbitrary m state quantum spin chain hamiltonian with nearest neighbour interaction in the bulk and single site boundary terms which can also describe the dynamics of stochastic models can be written as matrix product states this means that the weights in these states can be expressed as expectation values in a fock representation of an algebra generated by 2m operators fulfilling quadratic relations which are defined by the hamiltonian

on matrix product ground states for reaction diffusion models we discuss a new mechanism leading to a matrix product form for the stationary state of one dimensional stochastic models the corresponding algebra is quadratic and involves four different matrices for the example of a coagulation decoagulation model explicit four dimensional representations are given and exact expressions for various physical quantities are recovered we also find the general structure of n point correlation functions at the phase transition