stabilized finite element methods with shock capturing for advection diffusion problems abstract stabilized fem of streamline diffusion type for advection diffusion problems may exhibit local oscillations in crosswind direction s as a remedy a shock capturing variant of such stabilized schemes is considered as an additional consistent but nonlinear stabilization we prove existence of discrete solutions then we present some a priori and a posteriori estimates finally we address the efficient solution of the arising nonlinear discrete problems

on stabilized finite element methods for linear systems of convection diffusion reaction equations abstract a stabilized finite element method for solving systems of convection diffusion reaction equations is studied in this paper the method is based on the subgrid scale approach and an algebraic approximation to the subscales after presenting the formulation of the method it is analyzed how it behaves under changes of variables showing that it relies on the law of change of the matrix of stabilization parameters associated to the method an expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a one dimensional 1d model problem applications of the stabilization technique to the stokes problem with convection and to the bending of reissner mindlin plates are discussed next the design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard galerkin method applied to these two problems

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

non linear petrov galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods a new petrov galerkin approach for dealing with sharp or abrupt field changes in discontinuous galerkin dg reduced order modelling rom is outlined in this paper this method presents a natural and easy way to introduce a diffusion term into rom without tuning optimising and provides appropriate modelling and stablisation for the numerical solution of high order nonlinear pdes the approach is based on the use of the cosine rule between the advection direction in cartesian space time and the direction of the gradient of the solution the stabilization of the proper orthogonal decomposition pod model using the new petrov galerkin approach is demonstrated in 1d and 2d advection and 1d shock wave cases error estimation is carried out for evaluating the accuracy of the petrov galerkin pod model numerical results show the new nonlinear petrov galerkin method is a promising approach for stablisation of reduced order modelling

jump number maximization for proper interval graphs and series parallel graphs in this paper the order theoretic problem of finding linear extensions of a partially ordered set poset with certain properties is mapped to the corresponding graphtheorectic problem given a poset a jump exists between two consecutive symbols in a topological sequence if there is no relation between these two symbols in the poset we want to find a topological sequence that minimizes or maximizes the total number of jumps in graph theoretic terms define the jump number optimization problem as finding a linear ordering of vertex sequence of a graph such that the number of pairs of consecutive vertices that are adjacent is maximized jump number minimization problem or minimized jump number maximization problem the complexities of both problems with respect to some special graphs are investigated first in this paper we then study the jump maximization problem for proper interval graphs and series parallel graphs from this study we hope that more researches can be derived from these interdisciplinary knowledge domains

residual based stabilized higher order fem for advection dominated problems abstract we reconsider the numerical solution of linear ized advection diffusion reaction problems using higher order finite elements together with stabilized galerkin methods of streamline diffusion type supg and with shock capturing stabilization the analysis improves the a priori analysis in our previous paper t knopp g lube g rapin stabilized finite element methods with shock capturing for advection diffusion problems comput methods appl mech engrg 191 2002 2997 3013 the theoretical results are supported by some numerical experiments

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

supg reduced order models for convection dominated convection diffusion reaction equations abstract this paper presents a streamline upwind petrov galerkin supg reduced order model rom based on proper orthogonal decomposition pod this rom is investigated theoretically and numerically for convection dominated convection diffusion reaction problems the supg finite element method was used on realistic meshes for computing the snapshots leading to some noise in the pod data numerical analysis is used to propose the scaling of the stabilization parameter for the supg rom two approaches are used one based on the underlying finite element discretization and the other one based on the pod truncation the resulting supg roms and the standard galerkin rom g rom are studied numerically for many settings the results obtained with the supg roms are more accurate finally one of the choices for the stabilization parameter is recommended

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

on the robust adaptive stabilization of a class of nominally first order hybrid systems this paper presents a robust adaptive stabilization scheme for a class of nominally first order and controllable hybrid time invariant linear systems involving both continuous and discrete parametrizations and signals the usual assumptions of inverse stability of the plant and knowledge of the high frequency gain are not required the design philosophy relies on the separation of the contributions of continuous and discrete dynamics to the output by generating two associate additive terms to build the control action the controllability of the nominally estimated plant model is maintained by using a hysteresis switching function this allows relaxing the assumption of stability of the plant inverse

polynomial algorithms for hamiltonian cycle in cocomparability graphs finding a hamiltonian cycle in a graph is one of the classical np complete problems complexity of the hamiltonian problem in permutation graphs has been a well known open problem in this paper the authors settle the complexity of the hamiltonian problem in the more general class of cocomparability graphs it is shown that the hamiltonian cycle existence problem for cocomparability graphs is in p a polynomial time algorithm for constructing a hamiltonian path and cycle is also presented the approach is based on exploiting the relationship between the hamiltonian problem in a cocomparability graph and the bump number problem in a partial order corresponding to the transitive orientation of its complementary graph

the interval merging problem abstract a closed interval is an ordered pair of real numbers x y with x y the interval x y represents the set i r x i y given a set of closed intervals i a 1 b 1 a 2 b 2 a k b k the interval merging problem is to find a minimum cardinality set of intervals m i x 1 y 1 x 2 y 2 x j y j j k such that the real numbers represented by i i 1 k a i b i equal those represented by m i i 1 j x i y i in this paper we show the problem can be solved in o d log d sequential time and in o log d parallel time using o d processors on an erew pram where d is the number of the endpoints of i moreover if the input is given as a set of sorted endpoints then the problem can be solved in o d sequential time and in o log d parallel time using o d log d processors on an erew pram

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

stability and robustness properties of universal adaptive controllers for first order linear systems quite recently examples of non linear adaptive controllers have been found that adaptively stabilize any scalar first order linear system in this paper we describe two general classes of such universal adaptive stabilizers that include the previously proposed controllers as special cases different types of perturbations of these controllers are studied and we report on extensive simulation experiments our results demonstrate considerable qualitative differences in the dynamical behaviour of various adaptive universal stabilizers while the previously proposed controllers all exhibit the bursting phenomena we give an explicit example of a stabilizing controller that does not show bursting behaviour

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

on exact solutions of differential equations arising in universal adaptive stabilization in this paper we analyse non linear willems byrnes adaptive stabilization of first order linear systems x t ax t bu t x 0 r the willems byrnes control law is u h k x k x2 k 0 r and for each x 0 x0 k 0 k0 r x t 0 k t k x0 as t there are three cases according to whether h is smooth piecewise smooth or piecewise constant for each x0 r k x0 defines a limit pole a bh k x0 r we find that in all cases x0 r a bh k x0 0 is open and dense in the piecewise smooth constant case we solve explicitly the closed loop dynamical equation for x t and k t we find that a bh k x0 0 corresponds to solutions which decay only algebraically

stabilized reduced order models for the advection diffusion reaction equation using operator splitting reduced order modeling rom coupled with finite element methods has been used effectively in many disciplines to efficiently solve complex problems however for advection dominated flows numerical simulations often contain spurious nonphysical oscillations which will also be apparent in the rom simulations in this work we consider stabilization methods for rom for the advection diffusion reaction adr equation when it is solved both with and without operator splitting specifically we consider the streamline upwind petrov galerkin supg stabilization method and the spurious oscillations at layers diminishing sold stabilization method we build on these methods by constructing a coherent framework which successfully integrates these model reduction stabilization and operator splitting approaches and we provide numerical examples detailing the application of this framework in the adr setting the stabilized rom results are compared numerically with their corresponding full finite element simulations

on finite element methods for 3d time dependent convection diffusion reaction equations with small diffusion the paper studies finite element methods for the simulation of time dependent convection diffusion reaction equations with small diffusion the supg method a sold method and two types of fem fct methods the methods are assessed in particular with respect to the size of the spurious oscillations in the computed solutions at a 3d example with nonhomogeneous dirichlet boundary conditions and homogeneous neumann boundary conditions

optimal covering of cacti by vertex disjoint paths moran s and y wolfstahl optimal covering of cacti by vertex disjoint paths theoretical computer science 84 1991 179 197 a path couer or in short cover of a graph g is a set of vertex disjoint paths which cover all the vertices of g an optimal couer of g is a cover of the smallest possible cardinality notable applications of graph covering are code optimization and mapping parallel programs to parallel architectures the optimal covering problem is known to be np complete even for cubic 3connected planar graphs motivated by the intractability of this problem we develop an efficient optimal covering algorithm for cacti i e graphs where no edge lies on more than one cycle in doing so we generalize the results of boesch gimpel mchugh 1974 and of pinter wolfstahl 1987 where optimal covering algorithms for trees and graphs where no two cycles share a vertex were presented

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

approximation of functions of large matrices with kronecker structure we consider the numerical approximation of f mathcal a b f a b where b in mathbb r n b rn and mathcal a a is the sum of kronecker products that is mathcal a m 2 otimes i i otimes m 1 in mathbb r n times n a m2 i i m1 rn n here f is a regular function such that f mathcal a f a is well defined we derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations with special emphasis on the exponential and on completely monotonic functions for which the new procedure becomes particularly advantageous our findings are illustrated by numerical experiments with typical functions used in applications

error analysis of the supg finite element discretization of evolutionary convection diffusion reaction equations conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline upwind petrov galerkin supg finite element stabilization of time dependent convection diffusion reaction equations exemplarily it is shown for the supg method combined with the backward euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step the stabilization vanishes in the time continuous limit however based on numerical experience this seems not to be the correct behavior for this reason the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step it is shown that such estimates can be obtained in the case of time independent convection and reaction an error estimate for the time continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady state problem analogous estimates are obtained for the fully discrete case using the backward euler method and the crank nicolson scheme numerical studies support the analytical results

an example of a globally stabilizing adaptive controller with a generically destabilizing parameter estimate in this note we consider the question of whether an adaptive controller can converge to a nonadaptive stabilizing controller specifically we show for a class of back stepping controllers with adaptive tuning functions that the set of initial conditions in a state and estimation parameter for which the estimation parameter converges to a parameter which produces a destabilizing controller can have a nonempty interior and consequently a nonzero lebesgue measure this surprising result is proved by way of a simple example with a quadratic nonlinearity

isogeometric analysis and proper orthogonal decomposition for parabolic problems we investigate the combination of isogeometric analysis iga and proper orthogonal decomposition pod based on the galerkin method for model order reduction of linear parabolic partial differential equations for the proposed fully discrete scheme the associated numerical error features three components due to spatial discretization by iga time discretization with the theta scheme and eigenvalue truncation by pod first we prove a priori error estimates of the spatial iga semi discrete scheme then we show stability and prove a priori error estimates of the space time discrete scheme and the fully discrete iga theta pod galerkin scheme numerical tests are provided to show efficiency and accuracy of nurbs based iga for model order reduction in comparison with standard finite element based pod techniques

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

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

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

isogeometric analysis in electromagnetics b splines approximation abstract we introduce a new discretization scheme for maxwell equations in two space dimension inspired by the new paradigm of isogeometric analysis introduced in hughes et al 2005 16 we propose an algorithm based on the use of bivariate b splines spaces suitably adapted to electromagnetics we construct b splines spaces of variable interelement regularity on the parametric domain these spaces and their push forwards on the physical domain form a de rham diagram and we use them to solve the maxwell source and eigen problem our scheme has the following features i is adapted to treat complex geometries ii is spectral correct iii provides regular e g globally c 0 discrete solutions of maxwell equations

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

an o n 2 log n algorithm for the hamiltonian cycle problem on circular arc graphs a circular arc family f is a collection of arcs on a circle a circular arc graph is the intersection graph of an arc family a hamiltonian cycle hc in a graph is a cycle that passes through every vertex exactly once this paper presents an o n 2 log n algorithm to determine whether a given circular arc graph contains an hc this algorithm is based on two subroutines for interval graphs i a linear time greedy algorithm for the node disjoint path cover problem and ii a linear time hc algorithm if the given graph does not contain an hc this paper can produce a proof either through the deletion of an appropriate cutset or through the failure to obtain a specific type of hc

proper orthogonal decomposition with supg stabilized isogeometric analysis for reduced order modelling of unsteady convection dominated convection diffusion reaction problems abstract we consider reduced order modelling of unsteady convection dominated convection diffusion reaction problems with proper orthogonal decomposition pod in combination with isogeometric analysis isogeometric analysis has potential advantages in exact geometry representations efficient mesh generation different h p and k refinements and smooth bspline nurbs basis functions in order to compensate the oscillations caused by the convection dominated effect the streamline upwind petrov galerkin supg stabilization method is used both in generation of snapshots and pod galerkin method based on the recent novel and promising discretization method isogeometric analysis we propose a new fully discrete supg stabilized scheme the associated numerical error features three components due to spatial discretization by isogeometric analysis with supg stabilization time discretization with the crank nicolson scheme and modes truncation by pod we show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically a variety of two and three dimensional benchmark tests and numerical examples are provided to show the effectiveness accuracy and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis

remarks on adaptive stabilization of first order non linear systems abstract adaptive stabilization of first order non linear systems of the form x x f x ug x are studied for the case where g x 0 one necessary and two sufficient conditions for adaptive stabilization are given for the case of unknown sign of the gain a sufficient condition is givem simulation results are presented

remarks on the adaptive control of linear plants with unknown high frequency gain abstract adaptive control of first order linear systems of the form x ax bu with the sign of b unknown is considered it is shown that stabilizing adaptive controllers exist that have globally bounded gain and prespecified terminal dynamics