characteristic local grid refinement techniques for reservoir flow problems using a sequential time marching approach we describe and implement effective solution strategies for the equations governing immiscible displacement in a porous medium by combining adaptive grid refinement with an operator splitting technique based on the modified method of characteristics the saturation equation is treated in a consistent and accurate way for the pressure equation we construct accurate piecewise linear velocity components from a piecewise linear pressure approximation

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

existence theorems for positive semidefinite and sign indefinite stabilizing solutions of h riccati equations the existence of both positive semidefinite and sign indefinite stabilizing solutions of h infty type riccati equations is determined via joint eigenvalue and hamiltonian tests these tests determine the inertia properties of the solutions they are dependent on a formula derived in this paper that shows how the solution of game riccati equations can be decomposed into a sum of two well known riccati equations namely the standard filter riccati equation and the bounded real riccati equation other properties of stabilizing solutions such as solution partial order spectral radius monotonicity and rank are also discussed

characteristic nonconforming finite element methods for advection dominated diffusion problems in this paper characteristic nonconforming finite element methods are studied for timedependent advection dominated diffusion problems the diffusion term in these problems is discretized using nonconforming finite elements and the temporal differentiation and advection terms are treated by characteristic tracking schemes various nonconforming finite elements and characteristic tracking schemes are studied stability and convergence properties of the characteristic nonconforming methods are obtained unconditionally stable results and error estimates of optimal order are established

stability and convergence of a finite element method for reactive transport in ground water an explicit finite element method is used to solve the linear convection diffusion reaction equations governing contaminant transport in ground water flowing through an adsorbing porous medium the use of discontinuous finite elements for the convective part of the equations combined with mixed finite elements for the diffusive part renders the method for the concentration solution which displays strong gradients trivially conservative and fully parallelizable we carry out a stability and convergence analysis in particular the method is proven to satisfy a maximum principle to be total variation bounded and to converge to the unique weak solution of the equations special attention is paid to the convective part of the equations numerical simulations are presented and discussed

finite volume solutions of convection diffusion test problems the cell vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics more recently one of us has proposed an extension for viscous flows the purpose of the present paper is two fold first we have applied this scheme to a well known convection diffusion model problem involving flow round a 180 degrees bend which highlights some of the issues concerning the application of the boundary conditions in such cell based schemes the results are remarkably good when the boundary conditions are applied in an appropriate manner in our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one dimensional problem our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra convergence phenomena enjoyed by the proposed methods 25 refs 9 figs 1 tab

a gradient method for computing the optimal h sub infinity norm a fast algorithm for computing the optimal h sub infinity norm is presented the algorithm is based on certain explicit computations of gradients examples are given to show that the gradient method is much faster than the bisection method the gradient method can be used without the upper and lower bound computations this could be a substantial additional saving because computing these bounds requires the computation of the h sub infinity norm

a finite element method for an eikonal equation model of myocardial excitation wavefront propagation an efficient finite element method is developed to model the spreading of excitation in ventricular myocardium by treating the thin region of rapidly depolarizing tissue as a propagating wavefront the model is used to investigate excitation propagation in the full canine ventricular myocardium an eikonal curvature equation and an eikonal diffusion equation for excitation time are compared a petrov galerkin finite element method with cubic hermite elements is developed to solve the eikonal diffusion equation on a reasonably coarse mesh the oscillatory errors seen when using the galerkin weighted residual method with high mesh peclet numbers are avoided by supplementing the galerkin weights with c0 functions based on derivatives of the interpolation functions the ratio of the galerkin and supplementary weights is a function of the peclet number such that for one dimensional propagation the error in the solution is within a small constant factor of the optimal error achievable in the trial space an

low lying excitations in the square triangle random tiling model we consider the sub dominant eigenstates of the transfer matrix for the square triangle random tiling model on an infinite strip of width l a numerical algorithm for generation of the corresponding solution of the bethe ansatz is developed numerical finite size scaling analysis of the associated eigenvalues reveals the presence of both integer and non integer critical exponents the analytical value of one of the non integer exponents is found it is also shown numerically that along with the leading correction to the free energy density for some excitations there is a term proportional to

extended variational formulations and f e models for nonlinear beams under nonconservative loading abstract for nonconservative mechanical systems classical variational principles do not hold true hence the principle of incremental virtual work is usually taken as a convenient foundation for finite element solution methods 2 8 9 recent results on the so called inverse problem of calculus of variations however lead to extended variational formulations which can be provided even for nonconservative systems in the present paper this approach is used for the particular case of a nonlinear beam subjected to a co planar follower static loading suitable finite element models are deduced from the extended variational principle and their numerical performance is assessed by means of some meaningful examples

on the numerical solution of non linear reservoir flow models with gravity based on a two phase fluid model for immiscible displacement in a porous medium we develop and analyse numerical solution techniques for certain non linear phenomena two different solution strategies for the treatment of gravity effects which are non trivial to model by existing solution techniques and may be of great influence in many practical flow situations are presented the solution procedures are based on an operator splitting technique combining the modified method of characteristics with finite element techniques and adaptive grid refinement

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

a robust petrov galerkin discretisation of convection diffusion equations a petrov galerkin discretisation is studied of an ultra weak variational formulation of the convection diffusion equation in a mixed form to arrive at an implementable method the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space to prevent that this latter space has to be taken increasingly large for vanishing diffusion a formulation is constructed that is well posed in the limit case of a pure transport problem numerical experiments show approximations that are very close to the best approximations to the solution from the trial space uniformly in the size of the diffusion term

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

on an adaptive monotonic convection diffusion flux discretization scheme abstract the present paper concerns numerical investigation of a two dimensional steady convection diffusion scalar equation specific attention is given to resolving spurious oscillations in a confined region of high gradient in smooth regions a high order accurate solution is desired while in regions containing a sharp gradient the strategy of applying a monotonic capturing scheme is preferred in this paper we model fluxes by means of a finite element model which has spaces spanned by legendre polynomials the propensity to yield an irreducible diagonal dominant stiffness matrix is an attribute of this finite element flux discretization scheme fundamental analyses are extensively conducted in this paper in two areas first by conducting a modified equation analysis we are led to confirm the consistency of the scheme both the stability and monotonicity of the solutions are also addressed a guide for judging whether the stiffness matrix can yield a monotonic solution is rooted in the theory of the m matrix this monotonicity study provides us with greater numerical insight into the importance of the chosen peclet number beyond its critical value oscillatory solutions are present in the flow this monotonic region is however fairly restricted it is this shortcoming which limits the finite element practioner s choices for fairly small grid size the scope of application using the proposed upwind scheme is thus rather small because monotonic solutions are prohibitively expensive to compute circumvention of such deficiency is accomplished by making modifications to the structured grid several test cases have been employed to examine the grid adaption technique for the treatment of advection terms in flow regions having sharp gradients considerable savings in computer storage and execution time are achieved by the employed unstructured grid setting

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

multidisciplinary interactions in energy and environmental modeling abstract mathematical models have been widely used to understand predict or optimize many complex physical processes from semiconductor or pharmaceutical design to large scale applications such as global weather models to astrophysics in particular simulation of environmental effects of air polution is extensive here we address the need for using similar models to understand the fate and transport of groundwater contaminants and to design in situ remediation strategies three basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination one must first obtain an effective model to describe the complex fluid fluid and fluid rock interactions that control the transport of contaminants in groundwater this includes the problem of obtaining accurate reservoir descriptions at various length scales and modeling the effects of this heterogeneity in the reservoir simulators next one must develop accurate discretization techniques that retain the important physical properties of the continuous models finally one should develop efficient numerical solution algorithms that utilize the potential of the emerging computing architectures we will discuss recent advances in each of these three areas

quadratic streamline upwinding for finite element method solutions to 2 d convective transport problems abstract during the past decade finite element methods fems have been recognized to be more powerful tools in the solution of various flow problems as opposed to their predecessors finite difference methods on a fundamental level the fem involves approximations to the solution over a given trial space on a discrete mesh rather than finite difference approximations to the differential operators on a discrete mesh in this way the mechanics operators of the fem remain closer to the physical principles associated with a given problem in addition to this the fem incorporates boundary conditions very efficiently into the numerical formulation much current research is aimed at making these fems viable options for convection dominated flows in 1985 mizukami et al published two papers advancing the approach of streamline upwinding within linear elements for fems the purpose of this work is to demonstrate that streamline upwinding of higher ordered elements is a more accurate option the work goes on to show exactly how streamline upwinding effects the stability of iteratively solving the global algebraic system resulting from the fem

littlewood richardson coefficients for grothendieck polynomials from integrability abstract we study the littlewood richardson coefficients of double grothendieck polynomials indexed by grassmannian permutations geometrically these are the structure constants of the equivariant k theory ring of grassmannians representing the double grothendieck polynomials as partition functions of an integrable vertex model we use its yang baxter equation to derive a series of product rules for the former polynomials and their duals the littlewood richardson coefficients that arise can all be expressed in terms of puzzles without gashes which generalize previous puzzles obtained by knutson tao and vakil

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

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

integrability of the square triangle random tiling model instituut voor theoretische fysica universiteit van amsterdam valckenierstraat 65 1018 xe amsterdam the netherlands received 4 november 1996 it is shown that the square triangle random tiling model is equivalent to an asymmetric limit of the threecoloring model on the honeycomb lattice the latter model is known to be the o n model at t50 andcorresponds to the integrable model connected to the af nea

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

numerical results from large deflection beam and frame problems analysed by means of elliptic integrals numerical evaluations of elliptic integral solutions of some large deflection beam and frame problems are presented the values are given in tabular form with up to six significant figures the numerical technique used for evaluating the elliptic integrals is described

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

digital filters as absolute norm regularizers a variational formulation of the digital filtering problem based on a certain regularizing functional is presented in particular the special case where the regularizing functional is defined by means of l sub 1 norm is considered and a characterization of its minima is presented it is shown that for a certain class of signals one solution to the regularization problem can be obtained by means of a suitable recursive median filter some examples and a possible extension of the variational approach are also discussed

multiblock hybrid grid finite volume method to solve flow in irregular geometries in this work a finite volume based finite element method is suitably developed for solving incompressible flow and heat transfer on collocated hybrid grid topologies the method is generally applicable to arbitrarily shaped elements and orientations and thus challenges the potential to unify many of the different grid topologies into a single formulation the key point in this formulation is the correct estimation of the convective and diffusive fluxes at the cell faces using a novel physical influence scheme this scheme remarkably enhances the achieved solution accuracy it is shown that the extended formulation is robust enough to treat any combination of multiblock meshes with dual element shape employments in this regard the solution domain is broken up into a number of different multiblock arrangements in which each block is filled with only one type of finite element shape the combined grid can decrease the computational time and memory requirements and increase the numerical accuracy the results of the extended formulation are validated against different benchmark and other solutions the current results are in excellent agreement with the other solutions without exhibiting any disturbances around the block boundaries

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

the influence of shear deformations on the stability of thin walled beams under non conservative loading abstract this paper analyses the influence of shear deformations on the stability of thin walled beams of open section subjected to follower type loads the warping of the cross sections is assumed to depend on the shear forces as well as on both primary and secondary torsion the internal stresses are consistently deduced from the kinematic field an algorithm is presented which works in the framework of the least squares method making use of a vector iteration technique an extensive comparison is performed with the results offered by the classical models

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

multiscale stabilization for convection dominated diffusion in heterogeneous media abstract we develop a petrov galerkin stabilization method for multiscale convection diffusion transport systems existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion which may not be sufficient to stabilize multiscale systems we seek a local reduced order model for this kind of multiscale transport problems and thus develop a systematic approach for finding reduced order approximations of the solution we start from a petrov galerkin framework using optimal weighting functions we introduce an auxiliary variable to a mixed formulation of the problem the auxiliary variable stands for the optimal weighting function the problem reduces to finding a test space a dimensionally reduced space for this auxiliary variable which guarantees that the error in the primal variable representing the solution is close to the projection error of the full solution on the dimensionally reduced space that approximates the solution to find the test space we reformulate some recent mixed generalized multiscale finite element methods we introduce snapshots and local spectral problems that appropriately define local weight and trial spaces in particular we use energy minimizing snapshots and local spectral decompositions in the natural norm associated with the auxiliary variable the resulting spectral decomposition adaptively identifies and builds the optimal multiscale space to stabilize the system we discuss the stability and its relation to the approximation property of the test space we design online basis functions which accelerate convergence in the test space and consequently improve stability we present several numerical examples and show that one needs a few test functions to achieve an error similar to the projection error in the primal variable irrespective of the peclet number

the corrected operator splitting approach applied to a nonlinear advection diffusion problem so called corrected operator splitting methods are applied to a 1 d scalar advection diffusion equation of buckley leverett type with general initial data front tracking and a 2nd order godunov method are used to advance the solution in time diffusion is modelled by piece wise linear finite elements at each new time level to obtain correct structure of shock fronts independently of the size of the time step a dynamically defined residual flux term is grouped with diffusion different test problems are considered and the methods are compared with respect to accuracy and runtime finally we extend the corrected operator splitting to 2 d equations by means of dimensional splitting and we apply it to a buckley leverett type problem including gravitational effects

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

adaptive anisotropic petrov galerkin methods for first order transport equations abstract this paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of petrov galerkin type the key issues can be summarized as follows the variational formulations allow us to employ meshes with cells of arbitrary aspect ratios we develop a refinement scheme generating highly anisotropic partitions that is inspired by shearlet systems we establish l 2 approximation rates for n term approximations from corresponding piecewise polynomials for certain compact cartoon classes of functions in contrast to earlier results in a curvelet or shearlet context the cartoon classes are concisely defined through certain characteristic parameters and the dependence of the approximation rates on these parameters is made explicit here the approximation rate results serve then as a benchmark for subsequent applications to adaptive galerkin solvers for transport equations we outline a new class of directionally adaptive petrov galerkin discretizations for such equations in numerical experiments the new algorithms track c 2 discontinuity curves stably and accurately and realize essentially optimal rates finally we treat parameter dependent transport problems which arise in kinetic models as well as in radiative transfer in heterogeneous media these problems feature propagation of singularities along curved characteristics precluding in particular fast marching methods based on ray tracing since now the solutions are functions of spatial variables and parameters one has to address the curse of dimensionality as for non adaptive schemes considered in grella and schwab 2011 and grella 2014 we show computationally for a model parametric transport problem in heterogeneous media in 2 1 dimension that sparse tensorization of the presently proposed directionally adaptive variational discretization scheme in physical space combined with hierarchic collocation in ordinate space can overcome the curse of dimensionality when approximating averaged bulk quantities

subspace correction methods for nonsymmetric parabolic problems abstract a convergence proof for general subspace correction techniques applied to an abstract nonsymmetric parabolic equation is given one of the main concerns is to give a unified convergence analysis for domain decomposition and multigrid methods for parabolic problems the analysis is also valid for nonsymmetric problems for second order parabolic problems the convection can dominate the diffusion the algorithms can be applied to domain decomposition methods with or without the coarse mesh for applications to multigrid methods the coarsest mesh does not need to be very coarse a relation between the coarse mesh size and the time step is needed to get a convergence rate independent of the mesh the number of iterations at each time step for the algorithms is also estimated some numerical experiments are presented for domain decomposition methods with minimum overlap which support the theoretical predictions the algorithms are able to capture the sharp traveling shocks for convection dominated problems

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

extended variational formulations and f e models for nonlinear non conservative mechanical problems abstract for non conservative mechanical systems classical variational principles do not hold true however recent results on the so called inverse problem of calculus of variations lead to extended variational formulations which can be provided even for non conservative systems in this contribution f e models derived from extended variational principles are discussed in a significant case of an illconditioned balance equation it appears that extended mixed variational formulations could improve the numerical performance of f e methods for this class of problems

optimal h p finite element methods abstract a theory of optimal petrov galerkin h p version finite element approximations is presented the optimal scheme is denned relative to a fine mesh solution space and relative to an arbitrary symmetric bilinear form the optimal method leads to a symmetric positive definite stiffness matrix which is independent of the coefficients of the given problem exhibits extra superconvergence properties and has a relative error that can be calculated exactly at each point in the problem domain various generalizations are also discussed including the connection of these methods with certain preconditioning schemes

on the game riccati equations arising in h_ infty control problems in the state space approach to h infty optimal control feasible closed loop gains gamma are characterized via a pair of game riccati equations depending on gamma this paper is concerned with the properties of these equations as gamma varies the most general problem is considered d 11 neq 0 and the variations of the riccati solutions are thoroughly analyzed insight is gained into the behavior near the optimum and into the dependence on gamma of the suboptimality conditions in addition concavity is established for a criterion that synthesizes the three conditions x geq 0 y geq 0 and rho xy gamma 2 this suggests a numerically reliable newton scheme for the computation of the optimal gamma most results presented here are extensions of earlier contributions the main concern is to provide a complete and synthetic overview as well as results and formulas tailored to the development of numerically sound algorithms

discontinuous petrov galerkin method with optimal test functions for thin body problems in solid mechanics we study the applicability of the discontinuous petrov galerkin dpg variational framework for thin body problems in structural mechanics our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate local computation of the corresponding optimal test functions in the timoshenko beam problem the proposed method is shown to provide the best approximation in an energy type norm which is equivalent to the l2 norm for all the unknowns uniformly with respect to the thickness parameter the same formulation remains valid also for the asymptotic euler bernoulli solution as another one dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations in particular we derive a special norm for the test space which leads to a robust method in terms of the shell thickness finally we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp mesh for resolving the boundary layer

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

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

an adaptive characteristic petrov galerkin finite element method for convection dominated linear and nonlinear parabolic problems in one space variable a new adaptive finite element method for convection dominated problems is presented a special feature of the method is that it is based on a petrov galerkin scheme for spatial approximation at a typical time step which employs test functions chosen so that the approximate solution coincides with the exact solution at the nodes of finite element grid this procedure makes possible the derivation of truly local a posteriori error estimates and a very effective solver numerical examples are discussed which illustrate the efficiency and effectively of the method

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

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

a space time accurate method for solving solute transport problems solute transport studies frequently rely on numerical solutions of the classical advection diffusion equation unfortunately solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates recently developed variants of these techniques such as the finite analytic method chen and li 1979 chen and chen 1984 and the optimal test function method celia et al 1989a b c perform well for steady state problems extensions of these methods to the transient case have however not been successful primarily because of inadequate approximations of the temporal derivative the new numerical method proposed in this paper avoids this difficulty by taking the laplace transform of the transient equation the transformed expression behaves like a steady state advection diffusion equation with a first order decay term this expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse laplace transform algorithm the result is an accurate and robust transient solution which performs well over a very wide range of peclet numbers we demonstrate this approach by applying the finite analytic method to a laplace transformed one dimensional model problem a comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection dominated problems a good space approximation combined with a poor temporal approximation or vice versa does not give satisfactory results the method we propose provides a balanced space time approximation which works very well for one dimensional problems extensions to multiple dimensions are conceptually straightforward and briefly discussed

breaking spaces and forms for the dpg method and applications including maxwell equations discontinuous petrov galerkin dpg methods are made easily implementable using broken test spaces i e spaces of functions with no continuity constraints across mesh element interfaces broken spaces derivable from a standard exact sequence of first order unbroken sobolev spaces are of particular interest a characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces this technique is used to provide a complete error analysis of dpg methods for maxwell equations with perfect electric boundary conditions the technique also permits considerable simplifications of previous analyses of dpg methods for other equations reliability and efficiency estimates for an error indicator also follow finally the equivalence of stability for various formulations of the same maxwell problem is proved including the strong form the ultraweak form and various forms in between

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

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

tiles and colors tiling models are classical statistical models in which different geometric shapes the tiles are packed together such that they cover space completely in this paper we discuss a class of two dimensional tiling models in which the tiles are rectangles and isosceles triangles some of these models have been solved recently by means of bethe ansatz we discuss the question why only these models in a larger family are solvable and we search for the yang baxter structure behind their integrability in this quest we find the bethe ansatz solution of the problem of coloring the edges of the square lattice in four colors such that edges of the same color never meet in the same vertex

on the computation of the optimal h norms for two feedback control problems abstract numerical computation of the optimal h norms for the state feedback and the output feedback control problems is considered in 1989 doyle et al derived for a given parameter r 0 necessary and sufficient conditions under which an admissible output feedback controller exists as well as conditions for the existence of a state feedback controller we explore numerical methods to find the infimum of parameter r that satisfies those conditions

a quasi optimal petrov galerkin method for helmholtz problem a petrov galerkin finite element formulation is introduced for helmholtz problem in two dimensions using polynomial weighting functions at each node of the mesh a global basis function for the weighting space is obtained adding to the bilinear c 0 lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node quasi optimal weighting functions with the same support of the corresponding global test functions are obtained after computing the coefficients of these linear combinations attending to optimality criteria this is done numerically through a simple preprocessing technique naturally applied to non uniform and unstructured meshes in particular for uniform mesh a quasi optimal interior stencil of the same order of the quasi stabilized finite element method stencil derived by babuska et al comput methods appl mech engrg 1995 128 325 359 is obtained numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non uniform meshes

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

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

non local and semi local optimal weighting functions for symmetric problems involving a small parameter the method of optimal weighting functions for symmetric problems is described in a general form it is based on a petrov galerkin formulation in which the best approximation property and other mathematical features are achieved for a chosen norm different from the original energy norm of the problem the nonlocality of the weighting functions is shown to have only a minor effect on the efficiency of the method although a localization scheme is also suggested the method is applied to a one and two dimensional singular perturbation problems as well as to a cylindrical shell problem

eulerian lagrangian localized adjoint methods for a nonlinear advection diffusion equation eulerian lagrangian localized adjoint methods ellam are developed for the non linear buckley leverett equation which is characterized by degenerate diffusion and sharpening near shock solutions the ellam methodology employs space time finite elements with edges oriented along flow paths and space time test functions that satisfy a local adjoint condition this combination extends eulerian lagrangian concepts in a systematic mass conservative fashion to problems with general boundary conditions various kinds of boundary conditions are considered and a local time stepping procedure is developed for a no flow outlet condition that leads to a boundary layer numerical experiments illustrate the potential of these methods

on a penalty finite element cfd algorithm for high speed flow abstract the expanding interest in finite element discrete approximation techniques applied to high speed flow prediction has engendered derivation of many distinct theoretical statements quantization of algorithm robustness has focused on accuracy assessment for non smooth solutions i e those containing shocks and or contact discontinuities to reduced forms of the navier stokes equations specifically the euler equation system and the transonic potential flow equation this paper presents derivation of a penalty finite element algorithm applicable to both problem statements with penalty functional derived as a reduced form of a taylor galerkin weak statement numerical results for shocked transonic and euler solutions are cited to document aspects of accuracy convergence and stability

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

multiscale stabilization for convection diffusion equations with heterogeneous velocity and diffusion coefficients abstract we present a new stabilization technique for multiscale convection diffusion problems stabilization for these problems has been a challenging task especially for the case with high peclet numbers our method is based on a constraint energy minimization idea and the discontinuous petrov galerkin formulation in particular the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions and are related to the trial space the resulting test functions have a localization property and can therefore be computed locally we will prove the stability and present several numerical results our numerical results confirm that our test space gives a good stability in the sense that the solution error is close to the best approximation error

finite element solution of the stability problem for nonlinear undamped and damped systems under nonconservative loading a nonlinear finite element analysis of elastic structures which can be studied by a 3d beam theory subjected to conservative as well as to nonconservative forces is presented the stability behaviour of the system is studied by means of an eigenvalue analysis the stiffness matrix of the eigenvalue problem is asymmetric i e non self adjoint system the flutter and divergence modes of instability as well as the values of the critical load are identified for a number of numerical examples belonging to the benchmark tests proposed by nafems 1990 the results demonstrate the reliability of this finite element formulation in particular the effect of damping on the stability behaviour of such structures is investigated and the destabilizing effect of small damping is underlined finally the need to define a number of benchmark tests for nonlinear nonconservative analyses in presence of damping is included

a fast algorithm to computer the h norm of a transfer function matrix abstract a fast algorithm is presented to compute the h norm of a transfer function matrix based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related hamiltonian matrix the norm is computed with guaranteed accuracy a very simple method to compute a rather close lower bound on the h norm is given

spatial filter averaging approach of probabilistic method to linear second order partial differential equations of the parabolic type a backward in time probabilistic method with spatial filter averaging is presented to solve linear second order partial differential equations of the parabolic type an advantage of this methodology is that while forward methods are subject to region with loss of density of particles and hence loss of spatial resolution of the solution the solution given by backward methods is given on any desired grid however traditional backward time probabilistic method using monte carlo averaging are computationally expensive we prove a convergence result and present several examples the method leads to important improvement in computational efficiency and is expected to perform well to solve high dimensional problems where a solution is needed on a large grid

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

analytical treatment of equilibrium configurations of cantilever under terminal loads using jacobi elliptical functions abstract the paper provides an exact analytical solution for the equilibrium configurations of a cantilever rod subject to inclined force and tip moment acting on its free end the solution is given in terms of jacobi s elliptical functions and illustrated by several numerical examples and several graphical presentations of shapes of deformed cantilevers possible forms of the underlying elastica of a cantilever are discussed in detail and various simple formulas are given for calculating the characteristic dimensions of the elastica for the case when a cantilever is subject only to applied force three load conditions are discussed the follower load problem the load determination problem and the conservative load problem for all cases either a formula or an effective procedure for determining the solution is provided in particular a new efficient procedure is given to determine all possible equilibrium shapes in the case of the conservative load problem