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

nonlinear equalization of hammerstein ofdm systems a practical orthogonal frequency division multiplexing ofdm system can generally be modelled by the hammerstein system that includes the nonlinear distortion effects of the high power amplifier hpa at transmitter in this contribution we advocate a novel nonlinear equalization scheme for ofdm hammerstein systems we model the nonlinear hpa which represents the static nonlinearity of the ofdm hammerstein channel by a b spline neural network and we develop a highly effective alternating least squares algorithm for estimating the parameters of the ofdm hammerstein channel including channel impulse response coefficients and the parameters of the b spline model moreover we also use another b spline neural network to model the inversion of the hpa s nonlinearity and the parameters of this inverting b spline model can easily be estimated using the standard least squares algorithm based on the pseudo training data obtained as a byproduct of the hammerstein channel identification equalization of the ofdm hammerstein channel can then be accomplished by the usual one tap linear equalization as well as the inverse b spline neural network model obtained the effectiveness of our nonlinear equalization scheme for ofdm hammerstein channels is demonstrated by simulation results

testing time reversibility without moment restrictions in this paper we propose a class of new tests for time reversibility it is shown that this test has an asymptotic normal distribution under the null hypothesis and non trivial power under local alternatives a novel feature of this test is that it does not have any moment restriction in contrast with other time reversibility and linearity tests our simulations also confirm that the proposed test is very robust when data do not possess proper moments an empirical study of stock market indices is also included to illustrate the usefulness of the new test

nonlinear model order reduction via dynamic mode decomposition we propose a new technique for obtaining reduced order models for nonlinear dynamical systems specifically we advocate the use of the recently developed dynamic mode decomposition dmd an equation free method to approximate the nonlinear term dmd is a spatio temporal matrix decomposition of a data matrix that correlates spatial features while simultaneously associating the activity with periodic temporal behavior with this decomposition one can obtain a fully reduced dimensional surrogate model and avoid the evaluation of the nonlinear term in the online stage this allows for a reduction in the computational cost and at the same time accurate approximations of the problem we present a suite of numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches

solution of the implicitly discretised reacting flow equations by operator splitting abstract a non iterative method for handling the coupling of the implicitly discretised time dependent fluid flow equations is described the method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations the main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step with the split sets of equations being amenable to solution by standard techniques at each time step the procedure yields solutions which approximate the exact solution of the difference equations the accuracy of this splitting procedure is assessed for a linearised form of the discretised equations and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time step size by virtue of this it is possible to dispense with iteration thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods this is verified in a companion paper which presents results of computations carried out using the method

on the multivariate horner scheme we present an efficient version of the horner scheme for the evaluation of multivariate polynomials and study its stability properties in particular we show that the backward error of evaluation is bounded by a quantity that is linear in the total degree of the polynomial which itself is usually significantly smaller than the number of operations involved in the evaluation process moreover we show that the same type of estimates holds also for the forward error provided that the polynomial basis is chosen suitably

a note on the optimal stability of bases of univariate functions this note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes

data based stochastic model reduction for the kuramoto sivashinsky equation national science foundation dms 1217065 dms 1418775 dms 1419044 u s department of energy de ac02 05ch11231 national science foundation

estimating dimension from small samples abstract details of a new algorithm to estimate the dimension of a measure from point samples are described the new algorithm emphasizes that dimension need not be the same at all scales that is there need not be a scaling region the algorithm is verified with gaussian measures and is found to be reliable and have less demanding data requirements than conventional estimators using certain convergence properties of gaussian measures it is shown that good estimates can be made from very small samples

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

testing linearity against smooth transition autoregressive models summary we study a general univariate smooth transition autoregressive star model it contains as a special case the self exciting threshold autoregressive setar model we present three tests for testing linearity against star models and discuss their properties the power of the tests in small samples is investigated by simulation when the alternative is the logistic star model one of the tests is identical to tsay s 1986 test statistic and is recommended only in a special case of the two remaining tests with wider applicability one seems superior to the other in small samples it is also more powerful than the cusum test recently proposed for testing linearity against setar models

a tukey nonadditivity type test for time series nonlinearity the past few years have seen a revived interest in the study of nonlinear stationary time series the most general form of a nonlinear stationary process is that referred to as a volterra expansion this is to a linear process what a higher order polynomial is to a linear function the so called bilinear model is a special case which is quite broad and is defined by a small number of parameters volterra expansions involve more than secondorder theory and require higher order cumulant spectra various tests for linearity have been proposed although none is totally satisfactory because of the similarity of volterra expansions to polynomials an analogue of tukey s 1949 one degree of freedom for nonadditivity test seems quite reasonable as a diagnostic for linearity versus a second order volterra expansion such a test would be time domain based and computationally less complex than the frequency domain based alternatives directly generalizable to higher order than second and possibly suggestive of a power transformation toward linearity

evaluation of the derivative of a polynomial in bernstein form in geometric modelling and computer aided design a polynomial is usually represented in bernstein form forward error analysis for the computation of the derivative of a polynomial in bernstein form is performed the corresponding running error is also analyzed we have compared our results with those corresponding to a polynomial represented in power basis and using legendre and chebyshev forms since the conversions between the bernstein and the power bases and between bernstein and chebyshev polynomials were already studied in the literature for the sake of completeness the conversion between bernstein and chebyshev polynomials of the first kind is analyzed in an appendix

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

generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes abstract in this paper we study the error in the approximation of a convex function obtained via a one parameter family of approximation schemes which we refer to as barycentric approximation schemes for a given finite set of pairwise distinct points x n x i i 0 n in r d the barycentric approximation of a convex function f is of the form b f x i 0 n i x f x i where i i 0 n is a set of barycentric coordinates with respect to the point set x n the main content of this paper is two fold the first goal is to derive sharp upper and lower bounds on all barycentric coordinates over the convex polytope conv x n the second objective of the paper is to exploit the convexity assumption heavily and establish a number of upper and lower pointwise bounds on the approximation error for approximating arbitrary convex functions these bounds depend solely on computable quantities related to the data values of the function the largest and smallest barycentric coordinates for convex twice continuously differentiable functions we derive an optimal error estimate we show that the delaunay triangulation gives access to efficient algorithms for computing optimal barycentric approximation finally numerical examples are used to show the success of the method

optimal bases for a class of mixed spaces and their associated spline spaces we consider spaces of the form span where the functions u i i 1 4 are algebraic polynomials or trigonometric or hyperbolic functions we find intervals 0 a where we can guarantee that the spaces possess normalized totally positive bases and so shape preserving representations we construct their normalized b bases with optimal shape preserving properties we also present a unified approach to deal with the associated spline spaces a recursive procedure to obtain bases with optimal shape preserving and stability properties is presented

certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints in this paper we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations we consider the standard linear quadratic problem setting with distributed control and unilateral control constraints for this problem class we propose two different reduced basis approximations and associated error estimation procedures in our first approach we directly consider the resulting optimality system introduce suitable reduced basis approximations for the state adjoint control and lagrange multipliers and use a projection approach to bound the error in the reduced optimal control for our second approach we first reformulate the optimal control problem using a slack variable then develop a reduced basis approximation for the slack problem by suitably restricting the solution space and finally derive error bounds for the slack based optimal control we discuss benefits and drawbacks of b

nonlinearity tests for time series summary this paper considers two nonlinearity tests for stationary time series the idea of tukey s one degree of freedom for nonadditivity test is generalized to the time series setting the case of concurrent nonlinearity is discussed in detail simulation results show that the proposed tests are more powerful than that of keenan 1985

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

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

deriving the continuity of maximum entropy basis functions via variational analysis in this paper we prove the continuity of maximum entropy basis functions using variational analysis techniques the use of information theoretic variational principles to derive basis functions is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints for a set of distinct nodes x i i 1 n in mathbb r d the convex approximation of a function u x is u h x sum i 1 n p i x u i where p i i 1 n are nonnegative basis functions and u h x must reproduce affine functions sum i 1 n p i x 1 sum i 1 n p i x x i x given these constraints we compute p i x by minimizing the relative entropy functional kullback leibler distance d p m sum i 1 n p i x ln p i x m i x where m i x is a known prior weight function distribution to prove the continuity of the basis functions we appeal to the theory of epiconvergence

robust score and portmanteau tests of volatility spillover abstract this paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by large errors or garch type feedback the tests are couched in a general conditional heteroskedasticity framework with idiosyncratic shocks that are only required to have a finite variance if they are independent we negligibly trim test equations or components of the equations and construct heavy tail robust score and portmanteau statistics trimming is either simple based on an indicator function or smoothed in particular we develop the tail trimmed sample correlation coefficient for robust inference and prove that its gaussian limit under the null hypothesis of no spillover has the same standardization irrespective of tail thickness further if spillover occurs within a specified horizon our test statistics obtain power of one asymptotically we discuss the choice of trimming portion including a smoothed p value over a window of extreme observations a monte carlo study shows our tests provide significant improvements over extant garch based tests of spillover and we apply the tests to financial returns data finally based on ideas in patton 2011 we construct a heavy tail robust forecast improvement statistic which allows us to demonstrate that our spillover test can be used as a model specification pre test to improve volatility forecasting

goal adaptive isogeometric analysis with hierarchical splines abstract in this work a method of goal adaptive isogeometric analysis is proposed we combine goal oriented error estimation and adaptivity with hierarchical b splines for local h refinement the goal oriented error estimator is computed with a p refined discrete dual space which is adaptively refined alongside the primal space this discrete dual space is proven to be a strict superset of the primal space hierarchical refinements are introduced in marked regions that are formed as the union of chosen coarse level spline supports from the primal basis we present two ways of extracting localized refinement indicators suitable for the hierarchical refinement procedure one based on a partitioning of the dual weighted residual into contributions of basis function supports and one based on the combination of element indicators within a basis function support the proposed goal oriented adaptive strategy is exemplified for the poisson problem and a free surface flow problem numerical experiments on these problems show convergence of the adaptive method with optimal rates furthermore the corresponding goal oriented error estimators are shown to be accurate with effectivity indices in the range of 0 7 1 1

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

pod and cvt based reduced order modeling of navier stokes flows abstract a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the construction and application of reduced order bases reviews of the pod proper orthogonal decomposition and cvt centroidal voronoi tessellation approaches to reduced order modeling are provided including descriptions of pod and cvt reduced order bases their construction from snapshot sets and their application to the low cost simulation of the navier stokes system some concrete incompressible flow examples are used to illustrate the construction and use of pod and cvt reduced order bases and to compare and contrast the two approaches to reduced order modeling

pod galerkin method for finite volume approximation of navier stokes and rans equations numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications reduced order model rom can be employed whenever fast simulations are required or in general whenever a trade off between computational cost and solution accuracy is a preeminent issue as in process optimization and control in this work the efforts have been put to develop a rom for computational fluid dynamics cfd application based on finite volume approximation starting from the results available in turbulent reynold averaged navier stokes simulations in order to enlarge the application field of proper orthogonal decomposition reduced order model pod rom technique to more industrial fields the approach is tested in the classic benchmark of the numerical simulation of the 2d lid driven cavity in particular two simulations at re 103 and re 105 have been considered in order to assess both a laminar and a turbulent case some quantities have been compared with the full order model in order to assess the performance of the proposed rom procedure i e the kinetic energy of the system and the reconstructed quantities of interest velocity pressure and turbulent viscosity in addition for the laminar case the comparison between the rom steady state solution and the data available in literature has been presented the results have turned out to be very satisfactory both for the accuracy and the computational times as a major outcome the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent pod galerkin methods

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

least supported bases and local linear independence we introduce the concept of least supported basis which is very useful for numerical purposes we prove that this concept is equivalent to the local linear independence of the basis for any given locally linearly independent basis we characterize all the bases of the space sharing the same property several examples for spline spaces are given

stability and accuracy of periodic flow solutions obtained by a pod penalty method we develop a new penalty method to derive low dimensional galerkin models for fluid flows with time dependent boundary conditions we then outline a procedure based on bifurcation analysis in selecting the proper values of the penalty parameter s that yield asymptotically stable periodic solutions of the highest possible accuracy we illustrate this new approach by studying flow past a circular cylinder using direct numerical simulation dns data and a wave structure interaction problem using particle image velocimetry piv data

radial basis functions for the multivariate interpolation of large scattered data sets an efficient method for the multivariate interpolation of very large scattered data sets is presented it is based on the local use of radial basis functions and represents a further improvement of the well known shepard s method although the latter is simple and well suited for multivariate interpolation it does not share the good reproduction quality of other methods widely used for bivariate interpolation on the other band radial basis functions which have proven to be highly useful for multivariate scattered data interpolation have a severe drawback they are unable to interpolate large sets in an efficient and numerically stable way and maintain a good level of reproduction quality at the same time both problems have been circumvented using radial basis functions to evaluate the nodal function of the modified shepard s method this approach exploits the flexibility of the method and improves its reproduction quality the proposed algorithm has been implemented and numerical results confirm its efficiency

non linearly stable reduced order models for incompressible flow with energy conserving finite volume methods abstract a novel reduced order model rom formulation for incompressible flows is presented with the key property that it exhibits non linearly stability independent of the mesh of the full order model the time step the viscosity and the number of modes the two essential elements to non linear stability are 1 first discretise the full order model and then project the discretised equations and 2 use spatial and temporal discretisation schemes for the full order model that are globally energy conserving in the limit of vanishing viscosity for this purpose as full order model a staggered grid finite volume method in conjunction with an implicit runge kutta method is employed in addition a constrained singular value decomposition is employed which enforces global momentum conservation the resulting velocity only rom is thus globally conserving mass momentum and kinetic energy for non homogeneous boundary conditions a one time poisson equation is solved that accounts for the boundary contribution the stability of the proposed rom is demonstrated in several test cases furthermore it is shown that explicit runge kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

two level discretizations of nonlinear closure models for proper orthogonal decomposition proper orthogonal decomposition has been successfully used in the reduced order modeling of complex systems its original promise of computationally efficient yet accurate approximation of coherent structures in high reynolds number turbulent flows however still remains to be fulfilled to balance the low computational cost required by reduced order modeling and the complexity of the targeted flows appropriate closure modeling strategies need to be employed since modern closure models for turbulent flows are generally nonlinear their efficient numerical discretization within a proper orthogonal decomposition framework is challenging this paper proposes a two level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced order models the two level method computes the nonlinear terms of the reduced order model on a coarse mesh compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step the two level method attains the same level of accuracy while dramatically reducing the computational cost we numerically illustrate these improvements in the two level method by using it in three settings the one dimensional burgers equation with a small diffusion parameter n 10 3 the two dimensional flow past a cylinder at reynolds number re 200 and the three dimensional flow past a cylinder at reynolds number re 1000

approximation by smooth multivariate splines abstract one of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh this report shows this to be untrue in the multivariate setting in particular it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant in an extreme case piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or rho continuous will fail to approximate certain smooth functions at all as the grid goes to zero unless rho is kept below r 3 2 during the analysis of approximation on a certain regular triangular grid a novel kind of bivariate b spline is introduced this b spline in contrast to the established multivariate b spline derived from a simplex can be made to have all its breaklines in a given regular grid this makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation and its properties will be explored further

a parametric solution for simple stress strength model of failure with an application abstract estimation of p r s is considered for the simple stress strength model of failure using the pareto and power distributions together with their combined form a useful parametric solution is obtained and is illustrated numerically it is shown that these models are also applicable when only the tails of distributions for r and s are considered an application to the failure study concerning the fractures is also included

data driven reduced order modeling for time dependent problems abstract a data driven reduced basis rb method for parametrized time dependent problems is proposed this method requires the offline preparation of a database comprising the time history of the full order solutions at parameter locations based on the full order data a reduced basis is constructed by the proper orthogonal decomposition pod and the maps between the time parameter values and the projection coefficients onto the rb are approximated as a regression model with a natural tensor grid between the time and the parameters in the database a singular value decomposition svd is used to extract the principal components in the data of projection coefficients the regression functions are represented as the linear combinations of several tensor products of two gaussian processes one of time and the other of parameters during the online stage the solutions at new time parameter locations in the domain of interest can be recovered rapidly as outputs from the regression models featuring a non intrusive nature and the complete decoupling of the offline and online stages the proposed approach provides a reliable and efficient tool for approximating parametrized time dependent problems and its effectiveness is illustrated by non trivial numerical examples

on the foundation of bases of spline spaces the concept of foundation is a sharper concept than support which allows us to recognize spline bases suitable for design and computational purposes we present new results on least founded bases of univariate spaces in the case of tensor product spaces we find under very general conditions bases of minimally founded functions although we show that in contrast to the univariate case they are no least founded bases we also analyze related questions in a space of splines defined on angular regions of the plane

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

non intrusive reduced order modeling of nonlinear problems using neural networks abstract we develop a non intrusive reduced basis rb method for parametrized steady state partial differential equations pdes the method extracts a reduced basis from a collection of high fidelity solutions via a proper orthogonal decomposition pod and employs artificial neural networks anns particularly multi layer perceptrons mlps to accurately approximate the coefficients of the reduced model the search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine relying upon a joint use of the latin hypercube sampling lhs and the levenberg marquardt lm training algorithm this guarantees a complete offline online decoupling leading to an efficient rb method referred to as pod nn suitable also for general nonlinear problems with a non affine parametric dependence numerical studies are presented for the nonlinear poisson equation and for driven cavity viscous flows modeled through the steady incompressible navier stokes equations both physical and geometrical parametrizations are considered several results confirm the accuracy of the pod nn method and show the substantial speed up enabled at the online stage as compared to a traditional rb strategy

on the optimal stability of bases of univariate functions this paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation the only bases considered are those whose elements have no sign changes among these an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval a uniqueness result and many examples of such optimally stable bases are also provided

comparative performance of complex valued b spline and polynomial models applied to iterative frequency domain decision feedback equalization of hammerstein channels complex valued cv b spline neural network approach offers a highly effective means for identifying and inverting practical hammerstein systems compared with its conventional cv polynomial based counterpart a cv b spline neural network has superior performance in identifying and inverting cv hammerstein systems while imposing a similar complexity this paper reviews the optimality of the cv b spline neural network approach advantages of b spline neural network approach as compared with the polynomial based modeling approach are extensively discussed and the effectiveness of the cv neural network based approach is demonstrated in a real world application more specifically we evaluate the comparative performance of the cv b spline and polynomial based approaches for the nonlinear iterative frequency domain decision feedback equalization nifddfe of single carrier hammerstein channels our results confirm the superior performance of the cv b spline based nifddfe over its cv polynomial based counterpart

data driven pod galerkin reduced order model for turbulent flows abstract in this work we present a reduced order model which is specifically designed to deal with turbulent flows in a finite volume setting the method used to build the reduced order model is based on the idea of merging combining projection based techniques with data driven reduction strategies in particular the work presents a mixed strategy that exploits a data driven reduction method to approximate the eddy viscosity solution manifold and a classical pod galerkin projection approach for the velocity and the pressure fields respectively the newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with reynolds up to r e o 10 5

proper orthogonal decomposition closure models for turbulent flows a numerical comparison this paper puts forth two new closure models for the proper orthogonal decomposition reduced order modeling of structurally dominated turbulent flows the dynamic subgrid scale model and the variational multiscale model these models which are considered state of the art in large eddy simulation together with the mixing length and the smagorinsky closure models are tested in the numerical simulation of 3d turbulent flow past a circular cylinder at re 1000 five criteria are used to judge the performance of the proper orthogonal decomposition reduced order models the kinetic energy spectrum the mean velocity the reynolds stresses the root mean square values of the velocity fluctuations and the time evolution of the pod coefficients all the numerical results are benchmarked against a direct numerical simulation based on these numerical results we conclude that the dynamic subgrid scale and the variational multiscale models are the most accurate

a stabilized pod model for turbulent flows over a range of reynolds numbers optimal parameter sampling and constrained projection abstract we present a reduced basis technique for long time integration of parametrized incompressible turbulent flows the new contributions are threefold first we propose a constrained galerkin formulation that corrects the standard galerkin statement by incorporating prior information about the long time attractor for explicit and semi implicit time discretizations our statement reads as a constrained quadratic programming problem where the objective function is the euclidean norm of the error in the reduced galerkin algebraic formulation while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the n term expansion second we propose an a posteriori error indicator which corresponds to the dual norm of the residual associated with the time averaged momentum equation we demonstrate that the error indicator is highly correlated with the error in mean flow prediction and can be efficiently computed through an offline online strategy third we propose a greedy algorithm for the construction of an approximation space procedure valid over a range of parameters the greedy is informed by the a posteriori error indicator developed in this paper we illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two dimensional turbulent lid driven cavity flow on the reynolds number

galerkin v least squares petrov galerkin projection in nonlinear model reduction least squares petrov galerkin lspg model reduction techniques such as the gauss newton with approximated tensors gnat method have shown promise as they have generated stable accurate solutions for large scale turbulent compressible flow problems where standard galerkin techniques have failed however there has been limited comparative analysis of the two approaches this is due in part to difficulties arising from the fact that galerkin techniques perform optimal projection associated with residual minimization at the time continuous level while lspg techniques do so at the time discrete level this work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators linear multistep schemes and runge kutta schemes we present a number of new findings including conditions under which the lspg rom has a time continuous representation conditions under which the two techniques are equivalent and time discrete error bounds for the two approaches perhaps most surprisingly we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the lspg rom instead the time step should be matched to the spectral content of the reduced basis in numerical experiments carried out on a turbulent compressible flow problem with over one million unknowns we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the lspg reduced order model by an order of magnitude

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

proper general decomposition pgd for the resolution of navier stokes equations in this work the pgd method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions in the first stage the equations of stokes and burgers will be solved then we will solve the navier stokes problem in the case of the lid driven cavity for different reynolds numbers re 100 1000 and 10 000 finally the pgd method will be compared to the standard resolution technique both in terms of cpu time and accuracy

on a certified smagorinsky reduced basis turbulence model in this work we present a reduced basis smagorinsky turbulence model for steady flows we approximate the nonlinear eddy diffusion term using the empirical interpolation method cf m a grepl et al esaim math model numer anal 41 2007 pp 575 605 barrault et al c r acad sci paris ser i math 339 2004 pp 667 672 and the velocity pressure unknowns by an independent reduced basis procedure this model is based upon an a posteriori error estimation for a smagorinsky turbulence model the theoretical development of the a posteriori error estimation is based on s deparis siam j sci comput 46 2008 pp 2039 2067 and a manzoni esaim math model numer anal 48 2014 pp 1199 1226 according to the brezzi rappaz raviart stability theory and adapted for the nonlinear eddy diffusion term we present some numerical tests programmed in freefem cf f hecht j numer math 20 2012 pp 251 265 in which we show a speedup on the computation by factor larger

pod galerkin reduced order methods for combined navier stokes transport equations based on a hybrid fv fe solver abstract the purpose of this work is to introduce a novel pod galerkin strategy for the semi implicit hybrid high order finite volume finite element solver introduced in bermudez et al 2014 and busto et al 2018 the interest is into the incompressible navier stokes equations coupled with an additional transport equation the full order model employed in this article makes use of staggered meshes this feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes the reduced order model presented herein accounts for velocity pressure and a transport related variable the pressure term at both the full order and the reduced order level is reconstructed making use of a projection method more precisely a poisson equation for pressure is considered within the reduced order model results are verified against three dimensional manufactured test cases moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

a corner cutting algorithm for evaluating rational bezier surfaces and the optimal stability of the basis the usual method for evaluating rational bezier surfaces uses the projection operator and the representation provided by the bernstein basis we prove the optimal stability of the basis used in this representation we also propose an alternative method for evaluating rational surfaces through that representation we show the stability properties of this last method and prove that it has better properties than other known methods from the point of view of avoiding underflow and overflow

tail index estimation for heavy tailed models accommodation of bias in weighted log excesses we are interested in the derivation of the distributional properties of a weighted es estimatorof a positive tail index one of the main objectives of such an estilog excesses estimator of a positive tail index one of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias together with the maintenance of the asymptotic variance of the maximum likelihood estimator of under a strict pareto model we consider the external estimation not only of a second order shape parameter p but also of a second order scale parameter this will enable us to reduce the asymptotic variance of the final estimators under consideration compared with second order reduced bias estimators that are already available in the literature the second order reduced bias estimators that are considered are also studied for finite samples through monte carlo techniques as well as applied to real data in the field of finance

estimation and testing of varying coefficients in quantile regression in this article we establish a novel connection between the null hypothesis h0 on the coefficients and a rank reducible form of the varying coefficient model in quantile regression we use b splines to approximate the varying coefficients in the rank reducible model and make use of the fact that the null hypothesis h0 implies a unidimensional structure of a transformed coefficient matrix for the b spline basis functions by evaluating the unidimensional structure we alleviate the difficulty of testing such hypotheses commonly considered in varying coefficient quantile models we demonstrate through numerical studies that the proposed method can be much more powerful than the rank score test which is widely used in the quantile regression literature supplementary materials for this article are available online

on the robustness of nonlinearity tests to moment condition failure abstract this paper investigates the consequences of testing nonlinearities in data that suffer from moment condition failure simulation experiments indicate that tests designed to have maximal power against misspecification of the conditional variance seem to be especially sensitive to the nonexistence of moments the relationship between findings of nonlinearity and moment condition failure is further investigated in a sample of daily stock returns the empirical study shows that evidence of nonlinearity in stock returns cannot all be attributed to the nonrobustness of nonlinearity tests to moment condition failure however it is shown that most tests of nonlinearity are not reliable in testing situations involving heavy tailed data

stable distributions and the mixtures of distributions hypotheses for common stock returns abstract the form of the distribution underlying common stock returns has many implications for financial modeling among the frequently proposed models for stock return distributions is the family of stable distributions numerous studies of stock return distributions have also proposed various types of mixtures of normal distributions this article examines the characteristics of several different combinations of mixtures of normal and stable distributions and compares them with actual stock price distributions the principal tests applied are based on the stability under addition property of stable distributions

finite element approximation of the dirichlet problem using the boundary penalty method summary this paper considers a finite element approximation of the dirichlet problem for a second order self adjoint elliptic equation a u f in a region f2cn n 2 or 3 by the boundary penalty method if the finite element space defined over d h a union of elements has approximation power h in the l 2 norm then i for f2 d h convex polyhedral we show that choosing the penalty parameter e h x with 2 k yields optimal h 1 and l 2 error bounds if u hk i q

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

b splines and optimal stability it is proved that among all nonnegative bases of its space the b spline basis is optimally stable for evaluating spline functions

moment condition tests for heavy tailed time series abstract we develop an asymptotically chi squared statistic for testing moment conditions e m t 0 0 where m t 0 may be weakly dependent scalar components of m t 0 may have an infinite variance and e m t need not exist for any under the alternative score tests are a natural application and in general a variety of tests can be heavy tail robustified by our method including white noise garch affects omitted variables distribution functional form causation volatility spillover and over identification the test statistic is derived from a tail trimmed sample version of the moments evaluated at a consistent plug in t for 0 depending on the test in question and heaviness of tails t may be any consistent estimator including sub t 1 2 convergent and or asymptotically non gaussian ones since t can be assured not to affect the test statistic asymptotically we adapt bootstrap p value occupation time and covariance determinant methods for selecting the trimming fractile in any sample and apply our statistic to tests of white noise omitted variables and volatility spillover we find it obtains sharp empirical size and strong power while conventional tests exhibit size distortions

calibrated reduced order pod galerkin system for fluid flow modelling to improve the behaviour of reduced order proper orthogonal decomposition pod galerkin systems two numerical methods are proposed these methods determine free parameters in the pod galerkin system from flow simulations via a minimization problem they give rise to linear systems and their computational costs are reasonable both methods are assessed for two flow configurations a two dimensional flow around a square cylinder for a reynolds number of 100 and a three dimensional flow past a backward facing step for a reynolds number of 7432 based on the step height and the streamwise velocity at the middle of the inlet for both configurations the methods are effective since accurate calibrated reduced order pod galerkin systems are obtained

a general class of bernstein like bases this paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials bases with optimal shape preserving and stability properties are constructed evaluation and subdivision algorithms for them are provided bases for the corresponding and mixed spline spaces are also constructed some nice properties of these bases and the generated curves are shown

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

supremizer stabilization of pod galerkin approximation of parametrized steady incompressible navier stokes equations in this work we present a stable proper orthogonal decomposition pod galerkin approximation for parametrized steady incompressible navier stokes equations with low reynolds number supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced order system considering in particular the fulfillment of an inf sup condition the stability analysis is first carried out from a theoretical standpoint then confirmed by numerical tests performed on a parametrized two dimensional backward facing step flow

on the estimation of extreme tail probabilities applications of extreme value theory to problems of statistical inference typically involve estimating tail probabilities well beyond the range of the data without the benefit of a concise mathematical model for the sampling distribution the available model is generally only an asymptotic one that is an approximation to probabilities of extreme deviation is supposed which is assumed to become increasingly accurate as one moves further from the range of the data but whose concise accuracy is unknown quantification of the level of accuracy is essential for optimal estimation of tail probabilities in the present paper we suggest a practical device based on a nonstandard application of the bootstrap for determining empirically the accuracy of the approximation and thereby constructing appropriate estimators

estimating the stable index alpha in order to measure tail thickness a critique stable laws are often fit to outlier prone data and if the index alpha is estimated to be much less than two then the normal law is rejected in favor of an infinite variance stable law this paper derives the theoretical properties of such a procedure when the true distribution is stable the distribution of the m l e of alpha is non regular if alpha 2 when the true distribution is not stable the estimate of alpha is not a robust measure of the rate of decrease of the tail probabilities a more robust procedure is developed and a statistic for describing and comparing the tail shapes of arbitrary samples is proposed

richardson method and totally nonnegative linear systems we show that modified richardson method converges for any nonsingular totally nonnegative stochastic matrix for any choice of the parameter between 0 and 2 we present a variant of the modified richardson method that is convergent for any nonsingular totally nonnegative matrix we obtain the optimal parameter value for this method and give a procedure for estimating it numerical experiments are presented