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

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

krylov subspace methods for model order reduction of bilinear control systems we discuss the use of krylov subspace methods with regard to the problem of model order reduction the focus lies on bilinear control systems a special class of nonlinear systems which are closely related to linear systems while most existent approaches are based on series expansions around zero we will extend the underlying ideas to a more general context and show that there exist several ways to reduce bilinear systems besides we will briefly address the problem of stability preserving model reduction and further explain the benefit of using two sided projection methods by means of some numerical examples we will illustrate the performance of the presented reduction methods

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

reduced order modeling of parameterized pdes using time space parameter principal component analysis this paper presents a methodology for constructing low order surrogate models of finite element finite volume discrete solutions of parameterized steady state partial differential equations the construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high dimensional discrete solutions using only a few coefficients an incremental greedy approach is developed for efficiently tackling problems with high dimensional parameter spaces for numerical experiments and validation several non linear steady state convection diffusion reaction problems are considered first in one spatial dimension with two parameters and then in two spatial dimensions with two and five parameters in the two dimensional spatial case with two parameters it is shown that a 7 7 coefficient matrix is sufficient to accurately reproduce the expected solution while in the five parameters problem a 13 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy the proposed methodology is expected to find applications to parameter variation studies uncertainty analysis inverse problems and optimal design

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

brief accuracy enhancing methods for balancing related frequency weighted model and controller reduction we consider the solution of the balancing related frequency weighted model and controller reduction problems using accuracy enhanced numerical algorithms we propose first new stability enforcing choices of the frequency weighted grammians which can guarantee the stability of reduced models for two sided frequency weights then we show that for the frequency weighted controller reduction problems with standard stability and performance enforcing frequency weights the computation of the frequency weighted grammians can be done by solving reduced order lyapunov equations for both frequency weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their cholesky factors this allows the extension of the square root and balancing free accuracy enhancing techniques to the frequency weighted case

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

the mm me ml el ef and gmm approaches to estimation a synthesis abstract the 20th century began on an auspicious statistical note with the publication of karl pearson s philos mag ser 50 1900 157 goodness of fit test which is regarded as one of the most important scientific breakthroughs the basic motivation behind this test was to see whether an assumed probability model adequately described the data at hand pearson philos trans roy soc london ser a 185 1894 71 also introduced a formal approach to statistical estimation through his method of moments mm estimation ronald a fisher while he was a third year undergraduate at the gonville and caius college cambridge suggested the maximum likelihood estimation mle procedure as an alternative to pearson s mm approach in 1922 fisher published a monumental paper that introduced such basic concepts as consistency efficiency sufficiency and even the term parameter with its present meaning fisher philos trans roy soc london ser a 222 1922 309 provided the analytical foundation of mle and studied its efficiency relative to the mm estimator fisher j roy statist soc 87 1924a 442 established the asymptotic equivalence of minimum 2 and ml estimators and wrote in favor of using minimum 2 method rather than pearson s mm approach recently econometricians have found working under assumed likelihood functions restrictive and have suggested using a generalized version of pearson s mm approach commonly known as the gmm estimation procedure as advocated in hansen econometrica 50 1982 1029 earlier godambe ann math statist 31 1960 1208 and durbin j roy statist soc ser b 22 1960 139 developed the estimating function ef approach to estimation that has been proven very useful for many statistical models a fundamental result is that score is the optimum ef ferguson ann math statist 29 1958 1046 considered an approach very similar to gmm and showed that estimation based on the pearson 2 statistic is equivalent to efficient gmm golan et al maximum entropy econometrics robust estimation with limited data wiley new york 1996 developed entropy based formulation that allowed them to solve a wide range of estimation and inference problems in econometrics more recently imbens et al econometrica 66 1998 333 kitamura and stutzer econometrica 65 1997 861 and mittelhammer et al econometric foundations cambridge university press cambridge 2000 put gmm within the framework of empirical likelihood el and maximum entropy me estimation it can be shown that many of these estimation techniques can be obtained as special cases of minimizing cressie and read j roy statist soc ser b 46 1984 440 power divergence criterion that comes directly from the pearson 1900 2 statistic in this way we are able to assimilate a number of seemingly unrelated estimation techniques into a unified framework

on adaptive choice of shifts in rational krylov subspace reduction of evolutionary problems we compute u t exp ta varphi using rational krylov subspace reduction for 0 leq t infty where u t varphi in mathbf r n and 0 a a in mathbf r n times n a priori optimization of the rational krylov subspaces for this problem was considered in v druskin l knizhnerman and m zaslavsky siam j sci comput 31 2009 pp 3760 3780 there was suggested an algorithm generating sequences of equidistributed shifts which are asymptotically optimal for the cases with uniform spectral distributions here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum the algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost the effectiveness of the developed approach is demonstrated in an example of the three dimensional diffusion problem for maxwell s equation arising in geophysical exploration we compare our approach with the one using the above mentioned equidistributed sequences of shifts numerical examples show that our algorithm is able to adapt to the spectral density of operator a for examples with near uniform spectral distributions both algorithms show the same convergence rates but the new algorithm produces superior convergence for cases with nonuniform spectra

pseudo bosons arising from standard ladder operators pseudo bosons in the form as sa sa bs sa sa with as bs a a i are considered the s and s being real coefficients which depend on real parameters s1 sn the eigenstates of the two number operators and their norm are explicitly obtained pseudo bosons in bagarello s sense are recovered the states form two sets of biorthogonal bases of the full hilbert space but riesz bases are obtained only in the ordinary bosonic case some examples of this setting are analyzed in detail

provably convergent multifidelity optimization algorithm not requiring high fidelity derivatives functions does not alter the con vergence proof of the optimization algorithm and is shown to be robust to poor low delity information the algorithm is compared with an unconstrained single delity quasi newton algorithm and two

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

lyapunov equations energy functionals and model order reduction of bilinear and stochastic systems we discuss the relation of a certain type of generalized lyapunov equations to gramians of stochastic and bilinear systems together with the corresponding energy functionals while gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems the relation of gramians and energy functionals for bilinear systems is less obvious we discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic gramians satisfying generalized lyapunov equations in any of the considered cases the definition of algebraic gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems we illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

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

nonlinear n pseudo fermions nonlinear pseudo fermions of degree n n pseudo fermions are introduced as pseudo particles with creation and annihilation operators b and a b a obeying the simple nonlinear anticommutation relation ab bnan 1 the n 1 order nilpotency of these operators follows from the existence of unique up to a bi normalization factor a vacuum supposing appropriate n 1 order nilpotent para grassmann variables and integration rules the sets of n pseudo fermion number states right and left ladder operator bi overcomplete sets of coherent states are constructed explicit examples of n pseudo fermion ladder operators are provided and the relation of pseudo fermions to finite level pseudo hermitian systems is briefly considered this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

analysis of the rational krylov subspace and adi methods for solving the lyapunov equation for large scale problems an effective approach for solving the algebraic lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation although krylov subspaces have been used for a long time only more recent developments have shown that rational krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit adi recurrence in this paper we develop a convergence analysis of the rational krylov subspace method rksm based on the kronecker product formulation and on potential theory moreover we propose new enlightening relations between this approach and the adi method our results provide solid theoretical ground for recent numerical evidence of the superiority of rksm over adi when the involved parameters cannot be computed optimally as is the case in many practical application problems

balancing and model reduction for second order form linear systems model reduction of second order form linear systems is considered where a second order form reduced model is desired the focus is on reduction methods that employ or mimic moore s balance and truncate 1981 first we examine second order form model reduction by conversion to first order form and obtain a complete solution for this problem then new gramians and input output i o invariants for second order systems are motivated and defined based on these direct second order balancing methods are developed this leads naturally to direct second order form analogs for the well known first order form balance and truncate model reduction method explicit algorithms are given throughout the paper

data driven parametrized model reduction in the loewner framework we introduce a new method for constructing reduced order models that accurately approximate the frequency domain behavior of large scale parametrized linear dynamical systems using multivariate rational interpolation we compute reduced order models that match transfer function measurements of the large scale system the main tools are a new barycentric formula and loewner matrices formed directly from measurements this data driven approach introduces a new degree of freedom for parametrized model reduction the ability to choose separate reduced orders for each parameter more precisely each reduced order is determined by computing the rank of appropriate loewner matrices moreover we show how to control the pointwise approximation error through a new formula involving the barycentric form and the smallest singular value of a loewner matrix finally we also give new state space realizations for multivariate rational functions and conclude with several numerical examples showcasing the effectivene

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

low rank solution of lyapunov equations this paper presents the cholesky factor alternating direction implicit cf adi algorithm which generates a low rank approximation to the solution x of the lyapunov equation ax xat bbt the coefficient matrix a is assumed to be large and the rank of the right hand side bbt is assumed to be much smaller than the size of a the cf adi algorithm requires only matrix vector products and matrix vector solves by shifts of a hence it enables one to take advantage of any sparsity or structure in a this paper also discusses the approximation of the dominant invariant subspace of the solution x we characterize a group of spanning sets for the range of x a connection is made between the approximation of the dominant invariant subspace of x and the generation of various low order krylov and rational krylov subspaces it is shown by numerical examples that the rational krylov subspace generated by the cf adi algorithm where the shifts are obtained as the solution of a rational minimax problem often gives the best approximation to the dominant invariant subspace of x

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

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

orbit optimization by radial basis function interpolation in trust regions we present a new derivative free algorithm orbit for unconstrained local optimization of computationally expensive functions a trust region framework using interpolating radial basis function rbf models is employed the rbf models considered often allow orbit to interpolate nonlinear functions using fewer function evaluations than the polynomial models considered by present techniques approximation guarantees are obtained by ensuring that a subset of the interpolation points is sufficiently poised for linear interpolation the rbf property of conditional positive definiteness yields a natural method for adding additional points we present numerical results on test problems to motivate the use of orbit when only a relatively small number of expensive function evaluations are available results on two very different application problems calibration of a watershed model and optimization of a pde based bioremediation plan are also encouraging and support orbit s effectiveness on blackbox functions for which no special mathematical structure is known or available

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

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

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

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

model problems for the multigrid optimization of systems governed by differential equations we discuss a multigrid approach to the optimization of systems governed by differential equations such optimization problems appear in many applications and are of a different nature than systems of equations our approach uses an optimization based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids our goal is not to argue for a particular optimization based multigrid algorithm but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms furthermore using several model problems we give evidence both theoretical and numerical that the optimization setting is well suited to multigrid algorithms some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation in addition we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization based multigrid algorithm

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

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

model reduction for large scale systems with high dimensional parametric input space a model constrained adaptive sampling methodology is proposed for the reduction of large scale systems with high dimensional parametric input spaces our model reduction method uses a reduced basis approach which requires the computation of high fidelity solutions at a number of sample points throughout the parametric input space a key challenge that must be addressed in the optimization control and probabilistic settings is the need for the reduced models to capture variation over this parametric input space which for many applications will be of high dimension we pose the task of determining appropriate sample points as a pde constrained optimization problem which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters the methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21 which describe thermal analysis and design of a heat conduction fin and compared with statistically based sampling methods for these examples the model constrained adaptive sampling leads to reduced models that for a given basis size have error several orders of magnitude smaller than that obtained using the other methods

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

generalised tangential interpolation for model reduction of discrete time mimo bilinear systems in this article we discuss a model order reduction method for multiple input and multiple output discrete time bilinear control systems similar to the continuous time case we will show that a system can be characterised by a series of generalised transfer functions this will be achieved by a multivariate z transform of kernels corresponding to an explicit solution formula for discrete time systems we will further address the problem of generalised tangential interpolation which naturally comes along with this approach we will introduce a reasonable generalisation of the linear 2 norm based on this concept we discuss the choice of interpolation points furthermore an efficient discretisation of continuous time systems is provided the performance of the proposed method is evaluated in some numerical examples and compared with the method of balanced truncation for bilinear systems

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

parameter multi domain hp empirical interpolation summary in this paper we introduce two parameter multi domain hp techniques for the empirical interpolation method eim in both approaches we construct a partition of the original parameter domain into parameter subdomains h refinement we apply the standard eim independently within each subdomain to yield local in parameter approximation spaces p refinement further for a particularly simple case we introduce a priori convergence theory for the partition procedure we show through two numerical examples that our approaches provide significant reduction in the eim approximation space dimension and thus significantly reduce the computational cost associated with eim approximations copyright 2012 john wiley sons ltd

recycling bicg with an application to model reduction science and engineering problems frequently require solving a sequence of dual linear systems besides having to store only a few lanczos vectors using the biconjugate gradient method bicg to solve dual linear systems has advantages for specific applications for example using bicg to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework using bicg to evaluate bilinear forms for example in quantum monte carlo qmc methods for electronic structure calculations leads to a quadratic error bound since our focus is on sequences of dual linear systems we introduce recycling bicg a bicg method that recycles two krylov subspaces from one pair of dual linear systems to the next pair the derivation of recycling bicg also builds the foundation for developing recycling variants of other bi lanczos based methods such as cgs bicgstab qmr and tfqmr we develop an augmented bi lanczos algorithm and a modified two te

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

model reduction with mapreduce enabled tall and skinny singular value decomposition we present a method for computing reduced order models of parameterized partial differential equation solutions the key analytical tool is the singular value expansion of the parameterized solution which we approximate with a singular value decomposition of a parameter snapshot matrix to evaluate the reduced order model at a new parameter we interpolate a subset of the right singular vectors to generate the reduced order model s coefficients we employ a novel method to select this subset that uses the parameter gradient of the right singular vectors to split the terms in the expansion yielding a mean prediction and a prediction covariance similar to a gaussian process approximation the covariance serves as a confidence measure for the reduced order model we demonstrate the efficacy of the reduced order model using a parameter study of heat transfer in random media the high fidelity simulations produce more than 4tb of data we compute the singular value decomposition and evaluate the reduced or

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 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

hessian based model reduction for large scale systems with initial condition inputs reduced order models that are able to approximate output quantities of interest of high fidelity computational models over a wide range of input parameters play an important role in making tractable large scale optimal design optimal control and inverse problem applications we consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions and pose the task of determining appropriate training sets for reduced basis construction as a sequence of optimization problems we show that under certain assumptions these optimization problems have an explicit solution in the form of an eigenvalue problem yielding an efficient model reduction algorithm that scales well to systems with states of high dimension furthermore tight upper bounds are given for the error in the outputs of the reduced models the reduction methodology is demonstrated for a large scale contaminant transport problem

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

reduced basis method for parametrized elliptic optimal control problems we propose a suitable model reduction paradigm the certied reduced basis method rb for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations pdes in particular we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable firstly we recast the optimal control problem in the framework of saddle point problems in order to take advantage of the already developed rb theory for stokes type problems then the usual ingredients of the rb methodology are called into play a galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure an efficient and rigorous a posteriori error estimate on the state control and adjoint variables as well as on the cost functional finally we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

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

interpolatory projection methods for structure preserving model reduction abstract we present a framework for interpolatory model reduction that treats systems having a generalized coprime factorization c s s 1 b s d this includes rational krylov based interpolation methods as a special case the broader framework allows retention of special structure in reduced models such as symmetry second and higher order structure state constraints internal delays and infinite dimensional subsystems two numerical examples are provided for systems associated with viscoelastic dynamics and with an internal state delay that demonstrate the effectiveness and flexibility of the approach

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

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

partial realization of descriptor systems abstract we address the partial realization problem for linear descriptor systems a general solution to this problem using the markov parameters of the system defined via its laurent series is provided for proper descriptor systems we also discuss numerically feasible algorithms for computing a partial minimal realization based on the unsymmetric lanczos and arnoldi processes applications to model reduction for examples from computational fluid dynamics and mechanical systems with holonomic constraints are also given

regular pseudo bosons versus bosons we discuss in which sense the so called regular pseudo bosons recently introduced by trifonov and analyzed in some detail by the author are related to ordinary bosons we repeat the same analysis for pseudo bosons and analyze the role played by certain intertwining operators which may be bounded or not

h2 norm optimal model reduction for large scale discrete dynamical mimo systems modeling strategies often result in dynamical systems of very high dimension it is then desirable to find systems of the same form but of lower complexity whose input output behavior approximates the behavior of the original system here we consider linear time invariant discrete time dynamical systems the cornerstone of this paper is a relation between optimal model reduction in the h 2 norm and tangential rational hermite interpolation first order necessary conditions for h 2 optimal model reduction are presented for discrete multiple input multiple output mimo systems these conditions suggest a specific choice of interpolation data and a novel algorithm aiming for anh 2 optimal model reduction for mimo systems it is also shown that the conditions are equivalent to two known gramian based first order necessary conditions numerical experiments demonstrate the approximation quality of the method

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

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

construction of response surface approximations for design optimization using response surface approximations for design constraints in design optimization provides the designer with an overall perspective of the system response within the design space response surface approximations also reduce the numerical noise inherent in many numerical models and simplify the process of integrating several design codes as is typically required in multidisciplinary optimization procedures are discussed for constructing accurate response surface approximations to represent design constraints in design optimization response surface approximations are constructed for the stresses and buckling loads of an isotropic plate with an abrupt change of thickness these response surface approximations are constructed from numerical experiments conducted with a finite element analysis procedure and are used for minimum weight optimum design of the plate nondimensional variahles and stepwise regression are used to reduce the complexity and increase the accuracy of the response surface approximations additionally higher order polynomials cubic and quartic instead of the more traditional quadratic are used as response surface approximations and a detailed error analysis using an independent data set is performed finally it is shown that by making use of response surface approximations the optimum weight of the plate may be presented in the form of a design chart for a wide range of geometric loading and material constants

localized discrete empirical interpolation method this paper presents a new approach to construct more efficient reduced order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method deim whereas deim projects the nonlinear term onto one global subspace our localized discrete empirical interpolation method ldeim computes several local subspaces each tailored to a particular region of characteristic system behavior then depending on the current state of the system ldeim selects an appropriate local subspace for the approximation of the nonlinear term in this way the dimensions of the local deim subspaces and thus the computational costs remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes ldeim uses machine learning methods in the offline computational phase to discover these regions via clustering local deim approximations are then computed for each cluster in the online computational phase machine

projective and coarse projective integration for problems with continuous symmetries temporal integration of equations possessing continuous symmetries e g systems with translational invariance associated with traveling solutions and scale invariance associated with self similar solutions in a co evolving frame i e a frame which is co traveling co collapsing or co exploding with the evolving solution leads to improved accuracy because of the smaller time derivative in the new spatial frame the slower time behavior permits the use of projective and coarse projective integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems respectively these methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries the ideas of projective integration in a co evolving frame are illustrated on the one dimensional translationally invariant nagumo partial differential equation pde a corresponding kinetic monte carlo model motivated from the nagumo kinetics is used to illustrate the coarse grained method a simple one dimensional diffusion problem is used to illustrate the scale invariant case the efficiency of projective integration in the co evolving frame for both the macroscopic diffusion pde and for a random walker particle based model is again demonstrated

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

positive real and bounded real balancing for model reduction of descriptor systems we present an extension of the positive real and bounded real balanced truncation model reduction methods to large scale descriptor systems these methods are based on balancing the solutions of the projected lur e matrix equations important properties of these methods are that respectively passivity and contractivity are preserved in the reduced order models and that there exist approximation error bounds we also discuss the numerical solution of the projected lur e equations numerical examples are given

stabilized reduced basis method for parametrized advection diffusion pdes the aim of this master thesis is to study and develop a stabilized reduced basis method suitable for the approximation of the solution of parametrized advection diffusion pdes with high peclet number that is roughly the ratio between the advection coefficient and the diffusion one advection diffusion equations are very important in many engineering applications because they are used to model for example heat transfer phenomena or the diffusion of pollutants in the atmosphere in such applications we often need very fast evaluations of the approximated solution depending on some input parameters this happens for example in the case of real time simulation moreover we need rapid evaluations also if we have to perform repeated approximation of the solution for different input parameters an important case of this many query situation is represented by some optimization problems in which the objective function to optimize depends on the parameters through the solution of a pde the reduced basis rb method meets our need for rapidity and it is also able to guarantee the reliability of the solution thanks to sharp a posteriori error bounds we can find in literature many works about the application of the rb method to advection diffusion problems but they mainly deal with equations in which the peclet number is low the need for stabilization arise from the fact that the finite element fe approximated solution that the rb method aims to recover shows strong instability problems that have to be fixed in this work we want to go further in the study of the stabilization of the rb method for advection dominated problem in both steady and unsteady case as regards the steady case we first compare two possible stabilization strategies by testing them on some test problems in order to design an efficient stabilized reduced basis method we will then test this method using the piecewise quadratic fe approximation as reference solution instead of the usual piecewise linear one we extend the method designed for the steady case to the time dependent case and we will carry out some numerical tests

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

model reduction of descriptor systems by interpolatory projection methods in this paper we investigate an interpolatory projection framework for model reduction of descriptor systems with a simple numerical example we first illustrate that directly applying the subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded mathcal h 2 or mathcal h infty errors due to the mismatch of the polynomial parts of the full and reduced order transfer functions we then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error for the special cases of index 1 and index 2 descriptor systems we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation the question of how to choose interpolation points optimally naturally arises as in the standard state space setting we answer this question in the framework of the mathcal h 2 norm by extending the iterative rational krylov algorithm to descriptor systems

goal oriented inference approach linear theory and application to advection diffusion inference of model parameters is one step in an engineering process often ending in predictions that support decision in the form of design or control incorporation of end goals into the inference process leads to more efficient goal oriented algorithms that automatically target the most relevant parameters for prediction in the linear setting the control theoretic concepts underlying balanced truncation model reduction can be exploited in inference through a dimensionally optimal subspace regularizer the inference for prediction method exactly replicates the prediction results of either truncated singular value decomposition tikhonov regularized or gaussian statistical inverse problem formulations independent of data it sacrifices accuracy in parameter estimate for online efficiency the new method leads to low dimensional parameterization of the inverse problem enabling solution on smartphones or laptops in the field

on h2 model reduction of bilinear systems the h 2 model reduction problem for continuous time bilinear systems is studied in this paper by defining the h 2 norm of bilinear systems in terms of the state space matrices the h 2 model reduction error is computed via the reachability or observability gramian necessary conditions for the reduced order bilinear models to be h 2 optimal are given the gradient flow approach is used to obtain the solution of the h 2 model reduction problem the formulation allows certain properties of the original models to be preserved in the reduced order models the model reduction procedure developed can also be applied to finite dimensional linear time invariant systems a numerical example is employed to illustrate the effectiveness of the proposed method

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

a two directional arnoldi process and its application to parametric model order reduction we consider a two directional krylov subspace k k a j b j where besides the dimensionality k of the subspace increases the matrix a j and vector b j which induce the subspace may also augment specifically we consider the case where the matrix a j and the vector b j are augmented by block triangular bordering we present a two directional arnoldi process to efficiently generate a sequence of orthonormal bases q k j of the krylov subspaces the concept of a two directional krylov subspace and an arnoldi process is triggered by the need of a multiparameter moment matching based model order reduction technique for parameterized linear dynamical systems numerical examples illustrate computational efficiency and flexibility of the proposed two directional arnoldi process

balanced truncation model reduction for a class of descriptor systems with application to the oseen equations we discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete oseen equations with time independent advection and the linearized navier stokes equations linearized around a steady state the purpose of this paper is twofold first we show how to apply standard balanced truncation model reduction techniques which apply to dynamical systems given by ordinary differential equations to this class of descriptor systems this is accomplished by eliminating the algebraic equation using a projection the second objective of this paper is to demonstrate how the important class of adi smith type methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection instead we utilize the solution of saddle point problems we demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete oseen equations