stretching based diagnostics and reduction of chemical kinetic models with diffusion a new method for diagnostics and reduction of dynamical systems and chemical kinetic models is proposed the method makes use of the local structure of the normal stretching rates by projecting the dynamics onto the local directions of maximal stretching the approach is computationally very simple as it implies the spectral analysis of a symmetric matrix notwithstanding its simplicity stretching based analysis derives from a geometric basis grounded on the pointwise applications of concepts of normal hyperbolicity theory as a byproduct a simple reduction method is derived equivalent to a local embedding algorithm which is based on the local projection of the dynamics onto the most unstable and or slow modes compared to the time scale dictated by the local tangential dynamics this method provides excellent results in the analysis and reduction of dynamical systems displaying relaxation towards an equilibrium point limit cycles and chaotic attractors several numerical examples deriving from typical models of reaction diffusion kinetics exhibiting complex dynamics are thoroughly addressed the application to typical combustion models is also analyzed

interval analysis of vibrating systems interval calculus is a tool to evaluate a mathematical expression for ranges of values of its parameters the basic mathematical operations are defined in the interval algebra vibrating systems having system parameters initial conditions or forcing functions defined as intervals rather than single valued quantities have been modelled using interval calculus the eigenvalue problem for determining natural frequencies and vibrating modes of a system in the case of system matrix elements given as a range of values cannot be solved by exhaustion due to the prohibitively large number of solutions of the point number eigenvalue problem eigenvalue solution with interval evaluation of the commonly used numerical techniques is not feasible because if they are applied directly the solution intervals diverge an optimization technique was used to obtain the minimum radius intervals of the solution for the eigenvalue sensitivity problem to assure monotonicity and absolute inclusion necessary for convergence to the exact interval a converging interval halving sequence was developed for finite width interval matrices for numerical tests of the method a monte carlo solution was developed the results showed that interval analysis can predict the range of the eigenvalues with sufficient accuracy the response of a linear system to general excitation for interval matrices of the system parameters cannot be found with interval evaluation of the commonly used numerical techniques because if they are applied directly the solution intervals diverge interval modal analysis and the interval solution of the eigenvalue problem were developed an application is presented for the dynamic response of a rotor with interval bearing properties

sufficient regularity conditions for complex interval matrices and approximations of eigenvalues sets abstract in this paper two approaches are described to establish verifiable sufficient regularity conditions of complex interval matrices in the first approach a complex interval matrix is mapped to a real block interval matrix and then its sufficient regularity conditions are obtained in the second approach a necessary condition for the singularity of a complex interval matrix is derived and used to get its sufficient regularity conditions as an application the above derived sufficient regularity conditions are used to investigate the location of the outer approximations of individual eigenvalue sets of complex interval matrices two algorithms are proposed and results obtained are compared with those obtained by earlier methods and monte carlo simulations the advantages of these algorithms are that they can detect gaps in between the approximations of the whole eigenvalue sets the second algorithm is very effective compared to the first algorithm from the computational time point of view several numerical examples and statistical experiments are worked out to validate and demonstrate the efficacy of our work

characterizing and approximating eigenvalue sets of symmetric interval matrices we consider the eigenvalue problem for the case where the input matrix is symmetric and its entries are perturbed with perturbations belonging to some given intervals we present a characterization of some of the exact boundary points which allows us to introduce an inner approximation algorithm that in many case estimates exact bounds to our knowledge this is the first algorithm that is able to guarantee exactness we illustrate our approach by several examples and numerical experiments

a filtering method for the interval eigenvalue problem we consider the general problem of computing intervals that contain the real eigenvalues of interval matrices given an outer approximation superset of the real eigenvalue set of an interval matrix we propose a filtering method that iteratively improves the approximation even though our method is based on a sufficient regularity condition it is very efficient in practice and our experimental results suggest that it improves in general significantly the initial outer approximation the proposed method works for general as well as for symmetric interval matrices

the extreme eigenvalues and stability of real symmetric interval matrices a novel khoritonov like algorithm for computing the minimal and maximal eigenvalues of n n dimensional symmetric interval matrices is presented it is proved that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2 sup n 1 symmetric vertex matrices whereas its minimal eigenvalue coincides with the minimal of another special set of 2 sup n 1 symmetric vertex matrices as immediate corollaries of this algorithm weak necessary and sufficient conditions for testing the hurwitz and schur stability of symmetric interval matrices where one has to test the stability of 2 sup n 1 and 2 sup n symmetric vertex matrices respectively are obtained

exact maximum singular value calculation of an interval matrix in this note we present a method for calculating the maximum singular value of an interval matrix first we provide an algorithm for calculating the maximum singular value of a square interval matrix then based on this algorithm we extend the result to non square interval matrix case and to the case of computing the minimum singular value through numerical examples the validity of the suggested methods is illustrated particularly we compare the newly proposed method with an existing method to show that the new method finds the correct bound of the maximum singular value with no exception

solving interval linear systems with linear programming techniques abstract in this paper we show how it is possible to use convex polyhedra for solving linear interval systems without using preconditioning we first show how to derive from an enclosure of a b a polyhedron which contains the convex hull of the solution set then a simplex like method enables us to find a new outer inclusion moreover the constraints obtained may be used to compute an inner inclusion of a b

higher order corrections in the approximation of low dimensional manifolds and the construction of simplified problems with the csp method in systems of stiff ordinary differential equations odes both fast and slow time scales are encountered the fast time scales are responsible for the development of low dimensional manifolds on which the solution moves according to the slow time scales in this paper methodologies for constructing highly accurate i expressions describing the manifold and ii simplified non stiff equations governing the slow evolution of the solution on the manifold are developed according to an iterative procedure proposed in the computational singular perturbation csp method it is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the csp vectors spanning the manifold along the solution trajectory here an algorithm is presented which implements these calculations and is validated on the basis of two simple examples

the quasi steady state assumption a case study in perturbation the quasi steady state assumption qssa of biochemistry is studied as an approximation that is important in itself and also that exemplifies an approach to ode systems with an initial fast transient simple estimates of the two relevant time scales of the underlying problem are made these estimates lead to a dimensionless formulation containing a different small parameter from that which is customary earlier results on the qssa are surveyed in the context of the new nondimensionalization it is shown that more comprehensive and simpler error estimates can now be made some general methodological hints are drawn from this example

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

an evolution strategy method for computing eigenvalue bounds of interval matrices in this paper a modified method by means of an evolution strategy es is introduced for computing eigenvalue bounds of interval matrices then a sufficient condition theorem of the evolutionary strategy is presented which guarantees the convergence in probability of the method the numerical examples show the method can effectively yield accurate bounds for interval eigenvalues

explicit time scale splitting algorithm for stiff problems auto ignition of gaseous mixtures behind a steady shock a new explicit algorithm based on the computational singular perturbation csp method is presented this algorithm is specifically designed to solve stiff problems and its performance increases with stiffness the key concept in its structure is the splitting of the fast from the slow time scales in the problem realized by embedding csp concepts into an explicit scheme in simple terms the algorithm marches in time with only the terms producing the slow time scales while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction the new algorithm is designed for the integration of stiff systems of pdes by means of explicit schemes for simplicity in the presentation and discussion of the different features of the new algorithm a simple test case is considered involving the auto ignition of a methane air mixture behind a normal shock wave which is described by a system of odes the performance of the new algorithm accuracy and computational efficiency is then compared with the well known lsode package its merits when used for the solution of systems of pdes are discussed although when dealing with a stiff system of odes the new algorithm is shown to provide equal accuracy with that delivered by lsode at the cost of higher execution time the results indicate that its performance could be superior when facing a stiff system of pdes

approximation of slow and fast dynamics in multiscale dynamical systems by the linearized relaxation redistribution method in this paper we introduce a fictitious dynamics for describing the only fast relaxation of a stiff ordinary differential equation ode system towards a stable low dimensional invariant manifold in the phase space slow invariant manifold sim as a result the demanding problem of constructing sim of any dimensions is recast into the remarkably simpler task of solving a properly devised ode system by stiff numerical schemes available in the literature in the same spirit a set of equations is elaborated for local construction of the fast subspace and possible initialization procedures for the above equations are discussed the implementation to a detailed mechanism for combustion of hydrogen and air has been carried out while a model with the exact chapman enskog solution of the invariance equation is utilized as a benchmark

on the invariance of the sign pattern of matrix eigenvectors under perturbation abstract we study how much perturbation a in a real matrix a is allowed for the i th real eigenvector not to change sign

interval tensors and their application in solving multi linear systems of equations abstract in this paper we introduce interval tensors and present some results about their eigenvalues positive definiteness and application in solving multi linear systems it is proved that the set of maximum z eigenvalues of a symmetric interval tensor is a compact interval also several bounds for eigenvalues of an interval tensor are proposed in addition necessary and sufficient conditions for having a positive definite interval tensor are presented and investigated furthermore solving tensor equations using interval methods is presented and the interval jacobi and gauss seidel algorithms are extended for interval multi linear systems finally some numerical experiments are carried out to illustrate the methods

the g scheme a framework for multi scale adaptive model reduction the numerical solution of mathematical models for reaction systems in general and reacting flows in particular is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics however the dynamics can develop very slow and very fast time scales separated by a range of active time scales an opportunity to reduce the complexity of the problem arises when the fast active and slow active time scales gaps becomes large we propose a numerical technique consisting of an algorithmic framework named the g scheme to achieve multi scale adaptive model reduction along with the integration of the differential equations des the method is directly applicable to initial value odes and by using the method of lines pdes we assume that the dynamics is decomposed into active slow fast and when applicable invariant subspaces the g scheme introduces locally a curvilinear frame of reference defined by a set of orthonormal basis vectors with corresponding coordinates attached to this decomposition the evolution of the curvilinear coordinates associated with the active subspace is described by non stiff des whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem adjusting the active des dynamically during the time integration is the most significant feature of the g scheme since the numerical integration is accomplished by solving a number of des typically much smaller than the dimension of the original problem with corresponding saving in computational work to demonstrate the effectiveness of the g scheme we present results from illustrative as well as from relevant problems

solution theorems for the standard eigenvalue problem of structures with uncertain but bounded parameters generalized eigenvalue problems from the modal analysis are often converted to the standard eigenvalue problems in this paper it evaluates the upper and lower bounds on the eigenvalues of the standard eigenvalue problem of structures subject to severely deficient information about the structural parameters here we focus on non probabilistic interval analysis models of uncertainty which are adapted to the case of severe lack of information on uncertainty non probabilistic interval analysis method in which uncertainties are defined by interval numbers appears as an alternative to the classical probabilistic models for the standard eigenvalue problem of structures with uncertain but bounded parameters the vertex solution theorem the positive semi definite solution theorem and the parameter decomposition solution theorem for the standard eigenvalue problem are presented and compared with deif s solution theorem in numerical examples it is shown that for the upper and lower bounds on the eigenvalues of the standard eigenvalue problem with uncertain but bounded parameters the presented vertex solution theorem is unconditional and the positive semi definite solution theorem and the parameter decomposition solution theorem have less limitary conditions compared with deif s solution theorem the effectiveness of the vertex solution theorem the positive semi definite solution theorem and the parameter decomposition solution theorem are illustrated by numerical examples

an algorithm for addressing the real interval eigenvalue problem in this paper we present an algorithm for approximating the range of the real eigenvalues of interval matrices such matrices could be used to model real life problems where data sets suffer from bounded variations such as uncertainties e g tolerances on parameters measurement errors or to study problems for given states the algorithm that we propose is a subdivision algorithm that exploits sophisticated techniques from interval analysis the quality of the computed approximation and the running time of the algorithm depend on a given input accuracy we also present an efficient c implementation and illustrate its efficiency on various data sets in most of the cases we manage to compute efficiently the exact boundary points limited by floating point representation

natural tangent dynamics with recurrent biorthonormalizations a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic chaotic limit sets abstract this article develops a geometric method referred to as natural tangent dynamics with recurrent biorthonormalizations ntdrb that unifies into a single computational technique the analysis of dynamical systems possessing either a slow manifold associated with an equilibrium point or exhibiting nontrivial limit sets limit cycles and chaotic attractors for the first class of systems the main focus relies on the identification of the slow fast timescales while for the latter dynamics within nontrivial attractors the central issue is the temporal dichotomy between unstable stable components as described by the structure of the lyapunov spectrum the method is based on vector and covector dynamics in the tangent and cotangent bundles induced by the phase space flow and on the periodic application of a biorthonormalization procedure which contrasts the natural tendencies of vectors and covectors to align towards the most unstable slow subspaces the ntdrb method provides the most convenient and natural vector and covector bases for i the characterization of the fast timescales controlling the dynamic relaxation towards a slow manifold ii the identification of the stable and unstable components of the dynamics on a limit set iii the construction of a low dimensional reduced model reproducing the asymptotic behavior of the original systems

efficient inference on fractionally integrated panel data models with fixed effects a dynamic panel data model is considered that contains possibly stochastic individual com ponents and a common fractional stochastic time trend we propose four dierent ways of coping with the individual eects so as to estimate the fractional parameter like models with autoregressive dynamics ours nests a unit root but unlike the nonstandard asymptotics in the autoregressive case estimates of the fractional parameter can be asymptotically normal establishing this property is made di cult due to bias caused by the individual eects or by the consequences of eliminating them and requires the number of time series observations t to increase while the cross sectional size n can either remain xed or increase with t the biases in the central limit theorem are asymptotically negligible only under stringent conditions on the growth of n relative to t but these can be relaxed by bias correction for three of the estimates the biases depend only on the fractional parameter in hypothesis testing bias correc tion of the estimates is readily carried out we evaluate the biases numerically for a range of t and parameter values develop and justify feasible bias corrected estimates and brie y discuss simpli ed but less eective corrections a monte carlo study of nite sample performance is included

real eigenvalue bounds of standard and generalized real interval eigenvalue problems abstract in this paper we study the real eigenvalue bounds of standard and generalized real interval eigenvalue problems based on some known sufficient conditions for the regularity of interval matrices a new algorithm for computing real eigenvalue bounds for real interval matrices is proposed it can easily be extended for applying to generalized real interval eigenvalue problems one advantage of the method is that the computation procedure takes less time than former methods we have proposed therefore it can be applied for solving larger interval eigenvalue problems finally some numerical examples are presented which show the effectiveness of the method

singular values of an interval matrix abstract a simple method is outlined for calculating the singular values of an interval matrix with some restricted width it uses a sensitivity analysis approach to choose among the possible values of the matrix entries to extremize the singular values

eigenvalue bounds for symmetric matrices with entries in one interval in this paper we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval based on some matrix computation theories a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results furthermore the ideas are spread to general symmetric interval matrices finally numerical examples are presented which show the validity of the proposed methods

an algorithm for checking stability of symmetric interval matrices a branch and bound algorithm for checking hurwitz and schur stability of symmetric interval matrices is proposed the algorithm in a finite number of steps either verifies stability or finds a symmetric matrix which is not stable it can also be used for checking positive definiteness of asymmetric interval matrices