preconditioning techniques for reduced basis methods for parameterized elliptic partial differential equations the reduced basis methodology is an efficient approach to solve parameterized discrete partial differential equations when the solution is needed at many parameter values an offline step approximates the solution space and an online step utilizes this approximation the reduced basis to solve a smaller reduced problem which provides an accurate estimate of the solution traditionally the reduced problem is solved using direct methods however the size of the reduced system needed to produce solutions of a given accuracy depends on the characteristics of the problem and it may happen that the size is significantly smaller than that of the original discrete problem but large enough to make direct solution costly in this scenario it may be more effective to use iterative methods to solve the reduced problem we construct preconditioners for reduced iterative methods which are derived from preconditioners for the full problem this approach permits reduced basis methods to be practical for larger bas

majorization as a tool for optimizing a class of matrix functions the problem of minimizing a general matrix trace function possibly subject to certain constraints is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found it is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively three algorithms based on majorization for solving certain least squares problems are shown to be special cases in addition by means of several examples it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available

geometric programming for aircraft design optimization formulating conceptual stage aircraft design problems as geometric programs which are a specific type of convex optimization problem is proposed recent advances in convex optimization offer sign

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

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

on the stability and convergence of a galerkin reduced order model rom of compressible flow with solid wall and far field boundary treatment a reduced order model rom based on the proper orthogonal decomposition pod galerkin projection method is proposed as an alternative discretization of the linearized compressible euler equations it is shown that the numerical stability of the rom is intimately tied to the choice of inner product used to define the galerkin projection for the linearized compressible euler equations a symmetry transformation motivates the construction of a weighted l 2 inner product that guarantees certain stability bounds satisfied by the rom sufficient conditions for well posedness and stability of the present galerkin projection method applied to a general linear hyperbolic initial boundary value problem ibvp are stated and proven well posed and stable far field and solid wall boundary conditions are formulated for the linearized compressible euler rom using these more general results a convergence analysis employing a stable penalty like formulation of the boundary conditions reveals that the rom solution converges to the exact solution with refinement of both the numerical solution used to generate the rom and of the pod basis an a priori error estimate for the computed rom solution is derived and examined using a numerical test case published in 2010 by john wiley sons ltd

constructively well posed approximation methods with unity inf sup and continuity constants for partial differential equations starting from the generalized lax milgram theorem and from the fact that the approximation error is minimized when the continuity and inf sup constants are unity we develop a theory that provably delivers well posed approximation methods with unity continuity and inf sup constants for numerical solution of linear partial differential equations we demonstrate our single framework theory on scalar hyperbolic equations to constructively derive two different hp finite element methods the first one coincides with a least squares discontinuous galerkin method and the other appears to be new both methods are proven to be trivially well posed with optimal hpconvergence rates the numerical results show that our new discontinuous finite element method namely a discontinuous petrov galerkin method is more accurate has optimal convergence rate and does not seem to have unphysical diffusion compared to the upwind discontinuous galerkin method

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

exact pdf equations and closure approximations for advective reactive transport mathematical models of advection reaction phenomena rely on advective flow velocity and bio chemical reaction rates that are notoriously random by using functional integral methods we derive exact evolution equations for the probability density function pdf of the state variables of the advection reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions these pdf equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly correlated random field our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large eddy diffusivity led closure approximation 1 we find that the solution to the led based pdf equation which is exact for uncorrelated reaction rates is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty

manifold learning for the emulation of spatial fields from computational models repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible for partial differential equation pde models the outputs of interest are often spatial fields leading to high dimensional output spaces although emulators can be used to find faithful and computationally inexpensive approximations of computer models there are few methods for handling high dimensional output spaces for gaussian process gp emulation approximations of the correlation structure and or dimensionality reduction are necessary linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies manifold learning can overcome the limitations of linear methods if an accurate inverse map is available in this paper we use kernel pca and diffusion maps to construct gp emulators for very high dimensional output spaces arising from pde model simulations for diffusion maps we develop a new inverse map approximation several examples are presented to demonstrate the accuracy of our approach

a monotonically convergent algorithm for factals takane young and de leeuw proposed a procedure called factals for the analysis of variables of mixed measurement levels numerical ordinal or nominal mooijaart pointed out that their algorithm does not necessarily converge and nevels proposed a new algorithm for the case of nominal variables in the present paper it is shown that nevels procedure is incorrect and a new procedure for handling nominal variables is proposed in addition a procedure for handling ordinal variables is proposed using these results a monotonically convergent algorithm is constructed for factals of any mixture of variables

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

the nonconvergence of factals a nonmetric common factor analysis factals is a nonmetric common factor analysis model for multivariate data whose variables may be nominal ordinal or interval in factals an alternating least squares algorithm is utilized which is claimed to be monotonically convergent

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

adaptive rational krylov subspaces for large scale dynamical systems the rational krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems however its success has been hindered by the lack of a parameter free procedure which would effectively generate the sequence of shifts used to build the space in this paper we propose an adaptive computation of these shifts the whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix while further information is automatically generated during the process the approach is a full generalization to the nonsymmetric case of the idea first proposed in druskin et al 2010 18 and it is used for two important problems in control the approximation of the transfer function and the numerical solution of large lyapunov equations the procedure can be naturally extended to other related problems such as the solution of the sylvester equation and parametric or higher order systems several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors

efficient characterization of uncertain model parameters with a reduced order ensemble kalman filter spatially variable model parameters are often highly uncertain and difficult to observe this has prompted the widespread use of bayesian characterization methods that can infer parameter values from measurements of related variables while explicitly accounting for uncertainty ensemble versions of bayesian characterization are particularly convenient when uncertain variables have complex spatial structures that do not conform to gaussian descriptions however ensemble methods can be time consuming for high dimensional problems this paper describes a reduced order approach to ensemble characterization that is particularly well suited for subsurface flow and transport problems it uses a truncated discrete cosine transform to reduce the dimensionality of spatially variable time invariant model parameters and a nonlinear extension of principle orthogonal decomposition to reduce the dimensionality of dynamic model states the resulting nonlinear reduced order model can be included in the forecast step of

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

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

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

reduced order subscales for pod models in this work the reduced order subscales for proper orthogonal decomposition models are presented the basic idea consists in splitting the full order solution into the part which can be captured by the reduced order model and the part which cannot the subscales for which a model is required the proposed model for the subscales is defined as a linear function of the solution of the reduced order model the coefficients of this linear function are obtained by comparing the solution of the full order model with the solution of the reduced order model for the same initial conditions which for convenience are evaluated in the snapshots used to train the original reduced order model the difference between both solutions are the subscales for which a model can be built using a least squares procedure the subscales are then introduced as a correction in the reduced order model resulting in an important improvement in accuracy the enhanced reduced order model is tested in several numerical examples these practical cases show that the use of the subscales leads to more accurate solutions successfully corrects errors introduced by hyper reduction and allows to solve complex flow problems using a reduced number of degrees of freedom

reduced basis anova methods for partial differential equations with high dimensional random inputs in this paper we present a reduced basis anova approach for partial deferential equations pdes with random inputs the anova method combined with stochastic collocation methods provides model reduction in high dimensional parameter space through decomposing high dimensional inputs into unions of low dimensional inputs in this work to further reduce the computational cost we investigate spatial low rank structures in the anova collocation method and develop efficient spatial model reduction techniques using hierarchically generated reduced bases we present a general mathematical framework of the methodology validate its accuracy and demonstrate its efficiency with numerical experiments

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

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

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

reduced dimensional gaussian process emulators of parametrized partial differential equations based on isomap in this paper isomap and kernel isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a gaussian process emulator of parametrized partial differential equations the output space consists of spatial or spatio temporal fields that are functions of multiple input variables for such problems standard multi output gaussian process emulation strategies are computationally impractical and or make restrictive assumptions regarding the correlation structure the method we develop can be applied without modification to any problem involving vector valued targets and vector valued inputs it also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction

simulation based optimal bayesian experimental design for nonlinear systems the optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction particularly in situations where experiments are time consuming and expensive to conduct we propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation based models in particular we focus on finding sets of experiments that provide the most information about targeted sets of parameters our framework employs a bayesian statistical setting which provides a foundation for inference from noisy indirect and incomplete data and a natural mechanism for incorporating heterogeneous sources of information an objective function is constructed from information theoretic measures reflecting expected information gain from proposed combinations of experiments polynomial chaos approximations and a two stage monte carlo sampling method are used to evaluate the expected information gain stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high dimensional settings these algorithms are demonstrated on model problems and on nonlinear parameter inference problems arising in detailed combustion kinetics

accelerating pde constrained optimization by the reduced basis method application to batch chromatography summary in this work we show that the reduced basis method accelerates a partial differential equation constrained optimization problem where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization such an optimization problem arises for example from batch chromatography to reduce the computational burden of repeatedly solving the large scale system under parameter variations a parametric reduced order model with a small number of equations is derived by using the reduced basis method as a result the small reduced order model rather than the full system is solved at each step of the optimization process an adaptive technique for selecting the snapshots is proposed so that the complexity and runtime for generating the reduced basis are largely reduced an output oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically an early stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient numerical examples show that the adaptive technique is very efficient in reducing the offline time the optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution copyright 2015 john wiley sons ltd

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

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

a functional limit theorem for dependent sequences with infinite variance stable limits under an appropriate regular variation condition the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non gaussian stable random variable a functional version of this is known to be true as well the limit process being a stable levy process the main result in the paper is that for a stationary regularly varying sequence for which clusters of high threshold excesses can be broken down into asymptotically independent blocks the properly centered partial sum process still converges to a stable levy process due to clustering the levy triple of the limit process can be different from the one in the independent case the convergence takes place in the space of cadlag functions endowed with skorohod s m 1 topology the more usual j 1 topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values collapsing in the limit to a single jump the result rests on a new limit theorem for point processes which is of independent interest the theory is applied to moving average processes squared garch 1 1 processes and stochastic volatility models

an optimal projection method for the reduced order modeling of incompressible flows a new method to build reduced order models roms for incompressible flows is addressed in this paper proper orthogonal decompositions pods are used to efficiently represent both the fluid velocity and pressure fields with a small number of spatial modes a minimum residual projection is then developed for building the reduced models in which the velocity and pressure temporal coefficients are fully coupled comparisons of numerical calculations based on the proposed approach with results arising from the classical galerkin projection are exposed on three fluid flows a transient ventilated cavity a periodic lid driven cavity and a transient mixed convection flow it is shown that both stability and accuracy of the roms are strongly enhance when the minimum residual projection is used

the structure of optimal parameters for image restoration problems in cambridge this project has been supported by king abdullah university of science and technology kaust award no kuk i1 007 43 epsrc grants nr ep j009539 1 sparse higher order image restoration and nr ep m00483x 1 efficient computational tools for inverse imaging problems in quito the project has been supported by the escuela politecnica nacional de quito under award pis 12 14 and the mathamsud project socde sparse optimal control of differential equations when in quito t valkonen was moreover supported by a prometeo scholarship of the senescyt ecuadorian ministry of science technology education and innovation

stable limits for sums of dependent infinite variance random variables the aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution conditions for this convergence to hold are known in the literature however most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process in this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence we will apply our results to some standard time series models including the garch 1 1 process and its squares the stochastic volatility models and solutions to stochastic recurrence equations

towards efficient backward in time adjoint computations using data compression techniques abstract in the context of a posteriori error estimation for nonlinear time dependent partial differential equations the state of the practice is to use adjoint approaches which require the solution of a backward in time problem defined by a linearization of the forward problem one of the major obstacles in the practical application of these approaches is the need to store or recompute the forward solution to define the adjoint problem and to evaluate the error representation this study considers the use of data compression techniques to approximate forward solutions employed in the backward in time integration the development derives an error representation that accounts for the difference between the standard approach and the compressed approximation of the forward solution this representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data compressed reconstructed solution i e scalar quantities that can be evaluated as the forward problem is integrated this approach is then compared with existing techniques such as checkpointing and time averaged adjoints finally we provide numerical results indicating the potential efficiency of our approach on a transient diffusion reaction equation and on the navier stokes equations these results demonstrate memory compression ratios up to 450 while maintaining reasonable accuracy in the error estimates

reduced order modeling of time dependent pdes with multiple parameters in the boundary data abstract the computational approximation of solutions to nonlinear partial differential equations pdes such as the navier stokes equations is often a formidable task for this reason there is significant interest in the development of very low dimensional models that can be used to determine reasonably accurate approximations in simulation and control problems for pdes many concrete tests have been reported on in the literature that show that several classes of reduced order models roms are effective i e an accurate approximate solution can be obtained using very low dimensional roms most of these tests involved problems for which the solution depends on only a single parameter appearing in the boundary data the extension of the techniques used in those tests to the case of multiple parameters is not a straightforward matter here we present test and compare two methods for treating within the context of a class of roms inhomogeneous dirichlet type boundary conditions that contain multiple parameters in this study we focus on the proper orthogonal decomposition pod approach to rom however the issues and results discussed here in the pod context apply equally well to other rom approaches

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments

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

are the snapshot difference quotients needed in the proper orthogonal decomposition this paper presents a theoretical and numerical investigation of the following practical question should the time difference quotients dqs of the snapshots be used to generate the proper orthogonal decomposition pod basis functions the answer to this question is important since some published numerical studies use the time dqs whereas other numerical studies do not the criterion used in this paper to answer this question is the optimality of the convergence rate of the error of the reduced order model with respect to the number of pod basis functions since to the best of our knowledge a definition of the optimality of the convergence rate is not available we propose one in definition 3 1 in this paper two cases are considered the no dq case in which the snapshot dqs are not used and the dq case in which the snapshot dqs are used for each case two types of pod bases are used the l 2 pod basis in which the basis is generated in the l 2 norm and the h 1 pod basis in which the basi

adaptive hessian based nonstationary gaussian process response surface method for probability density approximation with application to bayesian solution of large scale inverse problems we develop an adaptive hessian based non stationary gaussian process gp response surface method for approximating a probability density function pdf that exploits its structure particularly the hessian of its negative logarithm of particular interest to us are pdfs that arise from the bayesian solution of large scale inverse problems which imply very expensive to evaluate pdfs the method can be considered as a piecewise adaptive gaussian approximation in which a gaussian tailored to the local hessian of the negative log probability density is constructed for each subregion in high dimensional parameter space the task of efficiently partitioning the parameter space into subregions is done implicitly through hessian informed membership probability functions the gp machinery is then employed to glue all local gaussian approximations into a global analytical response surface that is far cheaper to evaluate than the original expensive probability density the resulting response surface is also equipp

a weighted reduced basis method for elliptic partial differential equations with random input data in this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations pdes with random input data the pdes are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure a priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems

reduced basis methods for nonlocal diffusion problems with random input data abstract the construction analysis and application of reduced basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random input data is the subject of this work because of the lack of sparsity of discretized nonlocal models relative to analogous local partial differential equation models the need for reduced order modeling is much more acute in the nonlocal setting in this effort we develop reduced basis approximations for nonlocal diffusion equations with affine random coefficients efficiency estimates of the proposed greedy reduced basis methods are provided numerical examples are used to illustrate the effect varying various model parameters have on the efficiency and accuracy of the reduced basis method relative to sparse grid interpolation using the full finite element method it is shown that the proposed reduced basis approach can indeed provide substantial savings over standard sparse grid methods

reduced order modelling of parameter dependent linear and nonlinear dynamic partial differential equation models in this paper we develop reduced order models for dynamic parameter dependent linear and nonlinear partial differential equations using proper orthogonal decomposition pod the main challenges are to accurately and efficiently approximate the pod bases for new parameter values and in the case of nonlinear problems to efficiently handle the nonlinear terms we use a bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values computational efficiency is ensured by using manifold learning to perform the emulation in a low dimensional space the accuracy of the method is demonstrated on a linear and a nonlinear example with comparisons with a global basis approach

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

stochastic multidisciplinary analysis with high dimensional coupling this paper presents a novel approach for efficient uncertainty quantification and propagation in multidisciplinary analysis with a large number of coupling variables the proposed methodology has three elements bayesian network copula based sampling and principal component analysis the bayesian network represents the joint distribution of multiple variables through marginal distributions and conditional probabilities and it updates the distributions based on new data this paper uses this concept to develop a novel approach for probabilistic multidisciplinary analysis that is inference of distributions of the coupling variables by enforcing the interdisciplinary compatibility condition which is treated similar to data for updating the bayesian network is built using only a few iterations of the coupled multidisciplinary analysis without iterating until convergence a copula based sampling technique is employed for efficient sampling from the joint and conditional distributions further savings

numerical solution of the parameterized steady state navier stokes equations using empirical interpolation methods abstract reduced order modeling is an efficient approach for solving parameterized discrete partial differential equations when the solution is needed at many parameter values an offline step approximates the solution space and an online step utilizes this approximation the reduced basis to solve a smaller reduced problem at significantly lower cost producing an accurate estimate of the solution for nonlinear problems however standard methods do not achieve the desired cost savings empirical interpolation methods represent a modification of this methodology used for cases of nonlinear operators or nonaffine parameter dependence these methods identify points in the discretization necessary for representing the nonlinear component of the reduced model accurately and they incur online computational costs that are independent of n the number of degrees of freedom of the discrete system we will show that empirical interpolation methods can be used to significantly reduce the costs of solving parameterized versions of the navier stokes equations and that iterative solution methods can be used in place of direct methods to further reduce the costs of solving the algebraic systems arising from reduced order models

an improved solution for factals a nonmetric common factor analysis in factals an alternating least squares algorithm is utilized mooijaart 1984 has shown that this algorithm is based upon an erroneous assumption this paper gives a proper solution for the loss function used in factals

a subspace interior and conjugate gradient method for large scale bound constrained minimization problems a subspace adaptation of the coleman li trust region and interior method is proposed for solving large scale bound constrained minimization problems this method can be implemented with either sparse cholesky factorization or conjugate gradient computation under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full space version computational performance on various large test problems is reported advantages of our approach are demonstrated our experience indicates that our proposed method represents an efficient way to solve large bound constrained minimization problems

model reduction for large scale dynamical systems via equality constrained least squares in this paper we present a new method of model reduction for large scale dynamical systems which belongs to the svd krylov based method category it is a two sided projection where one side reflects the krylov part and the other side reflects the svd observability gramian part the reduced model matches the first r i markov parameters of the full order model and the remaining ones approximate in a least squares sense without being explicitly computed where r is the order of the reduced system and i is a nonnegative integer such that email protected i

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

uncertainty quantification for complex systems with very high dimensional response using grassmann manifold variations abstract this paper addresses uncertainty quantification uq for problems where scalar or low dimensional vector response quantities are insufficient and instead full field very high dimensional responses are of interest to do so an adaptive stochastic simulation based methodology is introduced that refines the probability space based on grassmann manifold variations the proposed method has a multi element character discretizing the probability space into simplex elements using a delaunay triangulation for every simplex the high dimensional solutions corresponding to its vertices sample points are projected onto the grassmann manifold the pairwise distances between these points are calculated using appropriately defined metrics and the elements with large total distance are sub sampled and refined as a result regions of the probability space that produce significant changes in the full field solution are accurately resolved an added benefit is that an approximation of the solution within each element can be obtained by interpolation on the grassmann manifold the method is applied to study the probability of shear band formation in a bulk metallic glass using the shear transformation zone theory

multiparameter analysis of aero icing problems using proper orthogonal decomposition and multidimensional interpolation steady and unsteady three dimensional viscous turbulent aero icing simulations are computationally expensive especially for certification campaigns when broad parametric studies are needed to overcome the computational effort of such investigations a reduced order modeling approach based on proper orthogonal decomposition and kriging interpolation is proposed using a database of high fidelity numerical simulations experimental data or combinations of both the proposed technique allows approximating solutions by linear combination of a limited number of eigenfunctions bayesian kriging a recent variant is used to obtain the scalar coefficients for the expansion the accuracy of the proposed method is assessed against reference solutions from two and three dimensional aero icing simulations as well as against experimental data