a global rational arnoldi method for model reduction abstract in this paper we propose two new approaches for model order reduction of large scale multi input multi output mimo linear time invariant dynamical systems lti these methods are based on a generalization of the global arnoldi algorithm which is used to generate projection subspaces an adaptive procedure for the selection of shift parameters is proposed and some simple arnoldi relations are established in order to compute error bounds on the transfer function numerical experiments showing the effectiveness of these approaches are provided

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

comparison between h sub 2 and h sub infinity controllers the h sub infinity performances of a controller designed by h sub 2 methodology and a controller designed by h sub infinity methodology are quantitatively compared it is shown that the ratio of the h sub infinity performance of the two is no greater than 2k for the one block problem where k is the total number of right half plane transmission zeros for two subsystems namely the subsystem between the disturbance and the measurement and the subsystem between the control signal and the controlled output it is also shown that there are systems which can give a ratio arbitrarily close to the upper bound however those systems are of a pathological type some possible generalizations to two and four block problems are discussed

ergodicity of probabilistic cellular automata a constructive criterion we give a sequence of criteria of increasing complexity for the exponential ergodicity of discrete time interacting particle systems each criterion involves estimating the dependence on initial conditions of the process on finite space time volumes it generalizes and improves the existing single site condition and is the analog of the dobrushin shlosmanc v condition in equilibrium statistical mechanics our dynamic criterion may also be used to prove the uniqueness of gibbs state in situations where thec v condition fails as a converse we prove that if there is a certain form of convergence to the stationary measure faster thann d wheren is the time andd is the dimension of the lattice then our condition holds for some space time volumes and hence the convergence must be exponentially fast

when is an interacting particle system ergodic we consider stochastic flip dynamics for an infinite number of ising spins on the lattice d we find a sequence of constructive criteria for the system to be exponentially ergodic the main idea is to approximate the continuous time process with discrete time processes itseuler polygon and to use an improved version of previous results ms about constructive ergodicity of discrete time processes

nonlinear parametric inversion using interpolatory model reduction nonlinear parametric inverse problems appear in several prominent applications one such application is diffuse optical tomography dot in medical image reconstruction such inverse problems present huge computational challenges mostly due to the need for solving a sequence of large scale discretized parametrized partial differential equations in the forward model in this paper we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in dot by drastically reducing the computational cost of solving the forward problems the key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis a similar observation holds for jacobian evaluations as well this motivates the use of system theoretic model order reduction methods we discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model within the dot s

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

accelerating optimization of parametric linear systems by model order reduction design optimization problems are often formulated as an optimization problem whose objective is a function of the output of a large scale parametric linear system obtained from the discretization of a pde to reduce the high computational cost of the objective and its gradient model order reduction techniques can be used this paper uses interpolatory reduced models as surrogate models in an optimization procedure we replace the standard first order condition by the relaxed first order condition which is more suitable when algebraic reduced models are used as surrogate models the relaxed first order condition imposes that the approximation quality of the surrogate model at the interpolation point can be measured and refined and that the surrogate model is equipped with an error bound on the entire parameter space we propose two optimization algorithms one uses the error bound to define a trust region and the other penalizes the objective with the error bound we prove convergence of both methods un

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

on level set regularization for highly ill posed distributed parameter estimation problems the recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward pdes is fraught with theoretical and practical difficulties better results are obtained for problems where the solution may take on at each point only one of two values thus yielding a shape recovery problem this article considers level set regularization for such problems however rather than explicitly integrating a time embedded pde to steady state which typically requires thousands of iterations methods based on gauss newton are applied more directly one of these can be viewed as damped gauss newton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a tikhonov type regularization with a sharpening sub step at each iteration in practice this method is eclipsed however by a special finite time or levenberg marquardt type method which we call dynamic regularization applied to the output least squares formulation our stopping criterion for the iteration does not involve knowledge of the true solution the regularization functional is applied to the smooth level set function rather than to the discontinuous function to be recovered and the second focus of this article is on selecting this functional typical choices may lead to flat level sets that in turn cause ill conditioning we propose a new quartic non local regularization term that penalizes flatness and produces a smooth level set evolution and compare its performance to more usual choices two numerical test cases are considered a potential problem and the classical eit dc resistivity problem

a new selection operator for the discrete empirical interpolation method improved a priori error bound and extensions this paper introduces a new framework for constructing the discrete empirical interpolation method sf deim projection operator the interpolation node selection procedure is formulated using the qr factorization with column pivoting and it enjoys a sharper error bound for the sf deim projection error furthermore for a subspace mathcal u given as the range of an orthonormal mathsf u the sf deim projection does not change if mathsf u is replaced by mathsf u omega with arbitrary unitary matrix omega in a large scale setting the new approach allows modifications that use only randomly sampled rows of mathsf u but with the potential of producing good approximations with corresponding probabilistic error bounds another salient feature of the new framework is that robust and efficient software implementation is easily developed based on readily available high performance linear algebra packages

inexact solves in interpolatory model reduction we investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem we give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods we show that when a petrov galerkin framework is employed for the inexact solves the associated reduced order model is an exact interpolatory model for a nearby full order system thus demonstrating backward stability we also give evidence that for h2 optimal interpolation points interpolatory model reduction is robust with respect to perturbations due to inexact solves finally we demonstrate the effectiveness of direct use of inexact solves in optimal h2 approximation the result is an effective model reduction strategy that is applicable in realistically large scale settings

large deviations for probabilistic cellular automata we consider a generalized model of a probabilistic cellular automata described by a markov chain on an infinite dimensional space and derive certain large deviations bounds for corresponding occupational measures

a projection method for model reduction of bilinear dynamical systems a krylov subspace based projection method is presented for model reduction of large scale bilinear systems a reduced bilinear system is constructed in such a way that it matches a desired number of moments of multivariable transfer functions corresponding to the kernels of volterra series representation of the original system applications to the simulation of dynamical responses of a nonlinear circuit and a micromachined device are presented to illustrate the efficiency of the new method and compare with an approach recently proposed by phillips

a deim induced cur factorization we derive a cur approximate matrix factorization based on the discrete empirical interpolation method deim for a given matrix bf a such a factorization provides a low rank approximate decomposition of the form bf a approx bf c bf u bf r where bf c and bf r are subsets of the columns and rows of bf a and bf u is constructed to make bf c bf u bf r a good approximation given a low rank singular value decomposition bf a approx bf v bf s bf w t the deim procedure uses bf v and bf w to select the columns and rows of bf a that form bf c and bf r through an error analysis applicable to a general class of cur factorizations we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of bf v and bf w for very large problems bf v and bf w can be approximated well using an incremental qr algorithm that makes only one pass through bf a numer

solution of the time domain inverse resistivity problem in the model reduction framework part i one dimensional problem with siso data many time domain problems in engineering applications can be described by means of parameter dependent time invariant dynamic systems we are interested in parameter estimation by fitting available transient measurements using the nonlinear least square method as the main application we consider the control source electromagnetic csem method of geophysical exploration governed by the diffusion maxwell system where the unknown parameters describe the spatial distribution of electrical resistivity we propose a novel model reduction approach for constructing an efficient approximation of the jacobian the reduction is based on subspace projection of the state variable and it allows us to split the time and space dependence of the derivative we examine several popular rational krylov subspaces and single out the h 2 optimal subspace that not only minimizes the approximation error but completely annuls its influence on the inversion result preliminary numerical experiments with a simplified one dim

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

constructive fractional moment criteria for localization in random operators abstract we present a family of finite volume criteria which cover the regime of exponential decay for the fractional moments of green functions of operators with random potentials such decay is a technically convenient characterization of localization for it is known to imply spectral localization absence of level repulsion dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases the constructive criteria also preclude fast power law decay of the green functions at mobility edges

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

phase diagrams of majority voter probabilistic cellular automata we present a detailed numerical analysis of the phase diagrams for some majority voter probabilistic cellular automata and connect the results with theory enabling us to prove many of the observed features

a non iterative method for computing the infimum in h optimization this paper presents a simple and non iterative procedure for the computation of the exact value of the infimum in the singular h optimization problem and is an extension of our earlier work the problem formulation is general and does not place any restriction on the direct feedthrough terms between the control input and the controlled output variables and between the disturbance input and the measurement output variables our method is applicable to a class of singular h optimization problems for which the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output have no invariant zeros on the jw axis and also satisfy certain geometric conditions the computation of the infimum in our method involves solving two well defined riccati and two lyapunov equations

spectral analysis in linear process identification a new look at an old problem the problem of estimating the transfer function frequency response of a linear process by means of a modified spectral analysis method is treated a method of determining the maximum time lags of the correlation functions which makes use of the available a priori physical knowledge about the process and the input signal is proposed this leads in a natural way to the use of an asymmetrical window technique some simulation results are used to illustrate the method based on the asymptotic properties of the estimate the input design is studied finally it will be shown that this work can be used for the estimation of an upper bound of the model uncertainty in the frequency domain

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

a best points interpolation method for efficient approximation of parametrized functions we present an interpolation method for efficient approximation of parametrized functions the method recognizes and exploits the low dimensional manifold structure of the parametrized functions to provide good approximation basic ingredients include a specific problem dependent basis set defining a low dimensional representation of the parametrized functions and a set of best interpolation points capturing the spatial parameter variation of the parametrized functions the best interpolation points are defined as solution of a least squares minimization problem which can be solved efficiently using standard optimization algorithms the approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure in addition an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost numerical results are presented to demonstrate the accuracy and efficiency of the method copyright 2007 john wiley sons ltd

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

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

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

h2 tf optimality conditions for a finite time horizon abstract in this paper we establish the interpolatory model reduction framework for optimal approximation of mimo dynamical systems with respect to the h 2 norm over a finite time horizon denoted as the h 2 t f norm using the underlying inner product space we derive the interpolatory first order necessary optimality conditions for approximation in the h 2 t f norm then we develop an algorithm which yields a locally optimal reduced model that satisfies the established interpolation based optimality conditions we test the algorithm on various numerical examples to illustrate its performance

approximation of fir by iir digital filters an algorithm based on balanced model reduction an algorithm for the approximation of finite impulse response fir filters by infinite impulse response iir filters is presented the algorithm is based on a concept of balanced model reduction the matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required this difficulty is circumvented here by directly formulating a reduced state space system description which is input output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an interim balanced realization examples of fir by iir filter approximations are included

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

interpolatory projection methods for parameterized model reduction we provide a unifying projection based framework for structure preserving interpolatory model reduction of parameterized linear dynamical systems i e systems having a structured dependence on parameters that we wish to retain in the reduced order model the parameter dependence may be linear or nonlinear and is retained in the reduced order model moreover we are able to give conditions under which the gradient and hessian of the system response with respect to the system parameters is matched in the reduced order model we provide a systematic approach built on established interpolatory mathcal h 2 optimal model reduction methods that will produce parameterized reduced order models having high fidelity throughout a parameter range of interest for single input single output systems with parameters in the input output maps we provide reduced order models that are optimal with respect to an mathcal h 2 otimes mathcal l 2 joint error measure the capabilities of these approaches are illustrated by several numerical examples from technical applications

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

two sided projection methods for nonlinear model order reduction in this paper we investigate a recently introduced approach for nonlinear model order reduction based on generalized moment matching using basic tensor calculus we propose a computationally efficient way of computing reduced order models we further extend the idea of two sided interpolation methods to this more general setting by employing the tensor structure of the hessian we investigate the use of oblique projections in order to preserve important system properties such as stability we test one sided and two sided projection methods for different semi discretized nonlinear partial differential equations and show their competitiveness when compared to proper orthogonal decomposition pod

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

large deviations for probabilistic cellular automata ii we obtain an upper large deviations bound which shows that for some models of probabilistic cellular automata which are far away from the product case the lower large deviation bound derived in eizenberg and kifer j stat phys 108 1255 1280 2002 is sharp and so the corresponding large deviations phenomena cannot be described via the traditional donsker varadhan form of the action functional for models which are close to the product case we derive approximate large deviations bounds using the donsker varadhan functional for the product case

monte carlo investigation of the critical behavior of stavskaya s probabilistic cellular automaton stavskaya s model is a one dimensional probabilistic cellular automaton pca introduced in the end of the 1960s as an example of a model displaying a nonequilibrium phase transition although its absorbing state phase transition is well understood nowadays the model never received a full numerical treatment to investigate its critical behavior in this brief report we characterize the critical behavior of stavskaya s pca by means of monte carlo simulations and finite size scaling analysis the critical exponents of the model are calculated and indicate that its phase transition belongs to the directed percolation universality class of critical behavior as would be expected on the basis of the directed percolation conjecture we also explicitly establish the relationship of the model with the domany kinzel pca on its directed site percolation line a connection that seems to have gone unnoticed in the literature so far

arma spectral estimation of narrow band processes via model reduction the problem of estimating autoregressive moving average arma models for narrowband processes is considered the following approach is proposed estimate a high order autoregressive ar approximation of the process by model reduction based on a truncated internally balanced realization or optimal hankel norm model reduction reduce the order of this high order ar estimate to find a lower order arma model this algorithm gives arma spectral estimates with excellent resolution properties without using iterative numerical minimization methods as for the maximum likelihood method how to take the narrowband assumption into account in the model reduction step is discussed in detail

a multiscale method for model order reduction in pde parameter estimation abstract estimating parameters of partial differential equations pdes is of interest in a number of applications such as geophysical and medical imaging parameter estimation is commonly phrased as a pde constrained optimization problem that can be solved iteratively using gradient based optimization a computational bottleneck in such approaches is that the underlying pdes need to be solved numerous times before the model is reconstructed with sufficient accuracy one way to reduce this computational burden is by using model order reduction mor techniques such as the multiscale finite volume method msfv in this paper we apply msfv for solving high dimensional parameter estimation problems given a finite volume discretization of the pde on a fine mesh the msfv method reduces the problem size by computing a parameter dependent projection onto a nested coarse mesh a novelty in our work is the integration of msfv into a pde constrained optimization framework which updates the reduced space in each iteration we present a computationally tractable way of explicitly differentiating the mor solution that acknowledges the change of basis as we demonstrate in our numerical experiments our method leads to computational savings for large scale parameter estimation problems where iterative pde solvers are necessary and offers potential for additional speed ups through parallel implementation

a regularized gauss newton trust region approach to imaging in diffuse optical tomography we present a new algorithm for the solution of nonlinear least squares problems arising from parameterized imaging problems with diffuse optical tomographic data d boas et al ieee signal process mag 18 2001 pp 57 75 the parameterization arises from the use of parametric level sets for regularization m e kilmer et al proc spie 5559 2004 pp 381 391 a aghasi m e kilmer and e l miller siam j imaging sci 4 2011 pp 618 650 such problems lead to jacobians that have relatively few columns a relatively modest number of rows and are ill conditioned moreover such problems have function and jacobian evaluations that are computationally expensive our optimization algorithm is appropriate for any inverse or imaging problem with those characteristics in fact we expect our algorithm to be effective for more general problems with ill conditioned jacobians the algorithm aims to minimize the total number of function and jacobian evaluations by analyzing which spectral components of the gauss newton direction should be discarded or damped the analysis considers for each component the reduction of the objective function and the contribution to the search direction restricting the computed search direction to be within a trust region the result is a truncated svd like approach to choosing the search direction however we do not necessarily discard components in order of decreasing singular value and some components may be scaled to maintain fidelity to the trust region model our algorithm uses the basic trust region algorithm from a r conn n i m gould and ph l toint trust region methods siam philadelphia 2000 we prove that our algorithm is globally convergent to a critical point our numerical results show that the new algorithm generally outperforms competing methods applied to the dot imaging problem with parametric level sets and regularly does so by a significant factor

system identification via cur factored hankel approximation subspace based system identification for dynamical systems is a sound system theoretic way to obtain linear time invariant system models from data the interplay of data and systems theory is ref

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

h sub infinity fixed lag smoothing filter for scalar systems the solution of the h sub infinity optimal linear fixed lag smoothing problem is considered using a polynomial matrix description for the discrete system the smoother is obtained from the solution of a linear equation and a spectral factorization calculation the pole zero properties of the optimal smoother are obvious in the polynomial representation and insights into the measurement noise rejection properties of the smoother are obtained allowance is made for both dynamic cost weighting and colored measurement noise the polynomial form of the smoothing filter may be incorporated into a self tuning algorithm and a simple adaptive smoother is discussed

finite volume fractional moment criteria for anderson localization a technically convenient signature of localization exhibited by discrete operators with random potentials is exponential decay of the fractional moments of the green function within the appropriate energy ranges known implications include spectral localization absence of level repulsion strong form of dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases we present a family of finite volume criteria which under some mild restrictions on the distribution of the potential cover the regime where the fractional moment decay condition holds the constructive criteria permit to establish this condition at spectral band edges provided there are sufficient lifshitz tail estimates on the density of states they are also used here to conclude that the fractional moment condition and thus the other manifestations of localization are valid throughout the regime covered by the multiscale analysis in the converse direction the analysis rules out fast power law decay of the green functions at mobility edges

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

a lyapunov theory for linear time delay systems some relationships between a lyapunov based stability test and linear control system models with time delays are explored the main result is a generalization of the result that if a b is a reachable pair of matrices then square a is a stability matrix if and only if there is a positive definite matrix k such that ak ka bb

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

ergodicity of quantum cellular automata we define a class of dynamical maps on the quasi local algebra of a quantum spin system which are quantum analoges of probabilistic cellular automata we develop criteria for such a system to be ergodic i e to posses a unique invariant state intuitively ergodicity obtains if the local transition operators exhibit sufficiently large disorder the ergodicity criteria also imply bounds for the exponential decay of correlations in the unique invariant state the main technical tool is a quantum version of oscillation norms defined in the classical case as the sum over all sites of the variations of an observable with respect to local spin flips

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

a new parametrization of h suboptimal controllers elementary state space concepts are used to derive a transparent solution to the h control problem on an infinite horizon the main contribution of this solution is a novel representation of suboptimal controllers in terms of a pair of parametric riccati equations with a coupling constraint unlike the classical parametrization in terms of linear fractional transformation this state space representation has a homogeneous riccati formulation which should help to make the most of the suboptimal controller diversity potential applications include the design of reduced order controllers and more generally the selection of suboptimal controllers to meet or optimize additional constraints

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

uniform and l 2 convergence in one dimensional stochastic ising models we study the rate of convergence to equilibrium of one dimensional stochastic ising models with finite range interactions we donot assume that the interactions are ferromagnetic or that the flip rates are attractive the infinitesimal generators of these processes all have gaps between zero and the rest of their spectra we prove that if one of these processes is observed by means of local observables then the convergence is seen to be exponentially fast with an exponent that is any number less than the spectral gap moreover this exponential convergence is uniform in the initial configuration

a stable state space realization in the formulation of h norm computation in the two block hinftyoptimization problem usually we are given the state space realizations of the proper rational matrices r 1 s and r 2 s whose poles are all the open right half plane two problems are studied in the note the first is the evaluation of phi s r 1 s at s s k k 1 2 n where phi s is an inner function whose zeros s k k 1 2 n are the poles of r 1 s this evaluation is essential if chang and pearson s method is used for computing the optimal hinftynorm the problem is solved in state space via the solutions of lyapunov equations neither polynomial matrix manipulations nor numerical pole zero cancellations are involved in the evaluation the second problem is to find a stable state space realization of s s u s r 2 s where u s is an inner matrix this problem arises in the spectral factorization of gamma 2 r 2 ast r 2 doyle and chu had a method for constructing stable s s based on a minimal realization of r 2 s an alternate method is proposed the alternate method does not require a minimal realization of r 2 s and only a lyapunov equation is involved

optimal hankel norm identification of dynamical systems abstract the problem of optimal approximate system identification is addressed with a newly defined measure of misfit between observed time series and linear time invariant models the behavioral framework is used as a suitable axiomatic setting for a non parametric introduction of system complexity and allows for a notion of misfit of dynamical systems that is independent of system representations the misfit function introduced here is characterized in terms of the induced norm of a hankel operator associated with the data and a co inner kernel representation of a model sets of pareto optimal models are defined as feasible trade offs between complexity and misfit of models and it is shown how all pareto optimal models are characterized as exact models of compressed data sets obtained from hankel norm approximations of data matrices this leads to new conceptual algorithms for optimal approximate identification of time series 1997 elsevier science ltd

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

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

brief paper balanced truncation model reduction for systems with inhomogeneous initial conditions we present a rigorous approach to extend balanced truncation model reduction btmr to systems with inhomogeneous initial conditions we provide an estimate for the error between the input output maps of the original and of the reduced initial value system and we illustrate numerically the superiority of our approach over the naive application of btmr when btmr is applied to linear time invariant systems with inhomogeneous initial conditions it is crucial that the initial data are well represented by the subspaces generated by btmr this requirement is often ignored or it is avoided by making the restrictive assumption that the initial data are zero to ensure that the initial data are well represented by the btmr subspaces we add auxiliary inputs determined by the initial data

the gacs kurdyumov levin automaton revisited we study a one dimensional cellular automaton that was originally proposed as a candidate for exhibiting nonergodic behavior under noise we prove that the deterministic model has the eroder property for two and only two invariant states moreover we give the best possible estimates for the corresponding erosion times we then review the results we have obtained from extensive computer simulations for the stochastic model and for a mixed model and argue that they suggest numerical and heuristic evidence in favour of ergodic behavior for all nonzero values of the noise parameter

partially ordered models we provide a formal definition and study the basic properties of partially ordered random fields porf these systems were proposed to model textures in image processing and to represent independence relations between random variables in statistics in the latter case they are known as bayesian networks our random fields are a generalization of probabilistic cellular automata pca and their theory has features intermediate between that of discrete time processes and the theory of statistical mechanical lattice fields its proper definition is based on the notion of partially ordered specification pos in close analogy to the theory of gibbs measures this paper contains two types of results first we present the basic elements of the general theory of porfs basic geometrical issues definition in terms of conditional probability kernels extremal decomposition extremality and triviality reconstruction starting from single site kernels relations between pom and gibbs fields second we prove three uniqueness criteria that correspond to the criteria known as uniform boundedness dobrushin uniqueness and disagreement percolation in the theory of gibbs measures

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

optimal sensitivity theory for multivariate distributed plants in this note we generalized the results of foias and tannenbaum 1986 a and foias et al 1987 on optimal sensitivity minimization to a much broader class of distributed plants we relate our results to certain kinds of hankel norm approximation problems that frequently appear in control engineering applications

from pca s to equilibrium systems and back stationary measures for probabilistic cellular automata pca s ind dimensions give rise to space time histories whose statistics may naturally be described by gibbs states ind 1 dimensions for an interaction energy obtained from the pca in this note we study the converse question do all gibbs states for this correspond to statistical space time histories for the pca our main result states that the answer is yes at least for translation invariant or periodic gibbs states thus ergodicity questions for pca s can at least partially be formulated as questions of uniqueness of gibbs states

symbols and orbits for 3 step nilpotent lie groups abstract sufficient conditions are derived for l 2 boundedness of a convolution operator on a 3 step nilpotent lie group this is achieved by producing estimates on the kirillov symbol of the operator and is closely linked to the co adjoint orbit structure of the group a structure theorem for 3 step nilpotent lie groups with 1 dimensional center is proved

a quantization of the cartan domain bd i q 2 and operators on the light cone abstract the weyl definition of pseudodifferential operators lends itself in a natural way to generalizations to hermitian symmetric spaces the boundedness of operators obtained in the case of the domain bd i q 2 is proven as a major step towards the development of a symbolic calculus of operators on the light cone

macrodynamics large scale structures in turbulent media we develop a method to derive the macroscopic equations governing the evolution of the mean field in continuous turbulent media the approach is based on the concept of local equilibrium which enables one to evaluate averages of nonlinear terms and to close the averaged equation examples include the kuramoto sivashinsky equation and its modifications

invariant measures and decay of correlations for a class of ergodic probabilistic cellular automata we give new sufficient ergodicity conditions for two state probabilistic cellular automata pca of any dimension and any radius the proof of this result is based on an extended version of the duality concept under these assumptions in the one dimensional case we study some properties of the unique invariant measure and show that it is shift mixing also the decay of correlation is studied in detail in this sense the extended concept of duality gives exponential decay of correlation and allows to compute explicitly all the constants involved

weyl quantization of symmetric spaces i hyperbolic matrix domains abstract it is shown that the generalized weyl calculus on the hyperbolic matrix domains defined by integration over a field of symmetries leads to bounded operators when the symbol is laplace bounded i e remains bounded after certain invariant differential operators higher laplacians are applied