Finite Linear Systems Research Articles

Exact aggregate solutions for M/G/1-type Markov processes

We introduce a new methodology for the exact analysis of M/G/1-type Markov processes. The methodology uses basic, well-known results for Markov chains by exploiting the structure of the repetitive portion of the chain and recasting the overall problem into the computation of the solution of a finite linear system. The methodology allows for the calculation of the aggregate probability of a finite set of classes of states from the state space, appropriately defined. Further, it allows for the computation of a set of measures of interest such as the system queue length or any of its higher moments. The proposed methodology is exact. Detailed experiments illustrate that the methodology is also numerically stable, and in many cases can yield significantly less expensive solutions when compared with other methods, as shown by detailed time and space complexity analysis.

Differential Inequalities Applied to The Stochastic Birth and Death Process

We describe an approach to a countably infinite system of ordinary differential equations belonging to the theory of the stochastic birth and death process. The main novelty in our method is the systematic use of a classical theorem on sub- and supersolutions for finite linear systems of the form y '( t ) = Ay ( t ). It leads in a simple way to the minimal solution and some of its properties. For convenience a proof of the theorem is given at the end.

Infinite Quasimonotone Systems of Ordinary Differential Equations with Applications to Stochastic Processes

We deal with the initial value problem for countably infinite linear systems of ordinary differential equations of the form y '( t ) = A ( t ) y ( t ) where A ( t ) = ( a ij ( t ): i , j S 1) is a measurable, infinite and essentially positive matrix, i.e., a ij ( t ) S 0 for i p j . The main novelty of our approach is the systematic use of a classical comparison theorem for finite linear systems which leads easily to the existence of a nonnegative minimal solution and its properties. Application to generalized stochastic birth and death processes produces criteria for honest and dishonest probability distributions. A short proof of the Kolmogorov and Chapman-Kolmogorov equations for stochastic processes follows. The results hold for L 1 -coefficients. Our method extends to nonlinear infinite systems of quasimonotone type and can be used for numerical procedures that yield exact results; cf. the Addendum.

A polynomial-time algorithm for checking equivalence under certain semiring congruences motivated by the state-space isomorphism problem for hybrid systems

This paper presents a polynomial-time algorithm for equivalence under certain semiring congruences. These congruences arise when studying the isomorphism of state spaces for a class of hybrid systems. The area of hybrid systems concerns issues of modeling, computation, and control for systems which combine discrete and continuous components. The subclass of piecewise linear (PL) systems provides one systematic approach to discrete-time hybrid systems, naturally blending switching mechanisms with classical linear components. PL systems model arbitrary interconnections of finite automata and linear systems. Tools from automata theory, logic, and related areas of computer science and finite mathematics are used in the study of PL systems, in conjunction with linear algebra techniques, all in the context of a “PL algebra” formalism. PL systems are of interest as controllers as well as identification models. Basic questions for any class of systems are those of equivalence, and, in particular, whether state spaces are equivalent under a change of variables. This paper studies this state-space equivalence problem for PL systems. The problem was known to be decidable, but its computational complexity was potentially exponential; here it is shown to be solvable in polynomial time.

A new bounded real lemma representation for the continuous-time case

A differential linear matrix inequality (DLMI) approach is introduced for the solution of various linear continuous-time control problems. The proposed method permits the application of linear matrix inequalities (LMIs) to the solution of control design problems under uncertainty. These problems are solved for finite horizon linear systems while considerably reducing the overdesign inherent in previous methods. The new approach also allows for the solution of the output-feedback control problem for systems belonging to a finite set of uncertain plants with hardly any overdesign. Four examples are given to demonstrate the applicability of the new method.

An iterative solver for the Oseen problem and numerical solution of incompressible Navier-Stokes equations

Incompressible unsteady Navier–Stokes equations in pressure–velocity variables are considered. By use of the implicit and semi-implicit schemes presented the resulting system of linear equations can be solved by a robust and efficient iterative method. This iterative solver is constructed for the system of linearized Navier–Stokes equations. The Schur complement technique is used. We present a new approach of building a non-symmetric preconditioner to solve a non-symmetric problem of convection–diffusion and saddle-point type. It is shown that handling the differential equations properly results in constructing efficient solvers for the corresponding finite linear algebra systems. The method has good performance for various ranges of viscosity and can be used both for 2D and 3D problems. The analysis of the method is still partly heuristic, however, the mathematically rigorous results are proved for certain cases. The proof is based on energy estimates and basic properties of the underlying partial differential equations. Numerical results are provided. Additionally, a multigrid method for the auxiliary convection–diffusion problem is briefly discussed. Copyright © 1999 John Wiley & Sons, Ltd.

An approximate waves-bordering algorithm for adaptive finite elements analysis

In this paper an Approximate Waves-Bordering algorithm (AWB) is presented. It computes the finite elements linear system solution-update after a refinement/unrefinement step. This is done taking into consideration only the equations that correspond to the nodes whose solution is modified above a certain tolerance and it appears to be very efficient. The algorithm considers an increasing set of equations that updates recursively and stops when the norm of the residual has gone under a user-defined threshold.

Locally polyhedral linear inequality systems

Linear systems of an arbitrary number of inequalities provide external representations for the closed convex sets in the Euclidean space. In particular, the locally polyhedral systems introduced in this paper are the natural linear representation for quasipolyhedral sets (those subsets of the Euclidean space whose nonempty intersections with polytopes are polytopes). For these systems the geometrical properties of the solution set are investigated, and their extreme points and edges are characterized. The class of locally polyhedral systems includes the quasipolyhedral systems, introduced by Marchi, Puente, and Vera de Serio in order to generalize the Weyl property of finite linear inequality systems.

Locally polyhedral linear inequality systems

Linear systems of an arbitrary number of inequalities provide external representations for the closed convex sets in the Euclidean space. In particular, the locally polyhedral systems introduced in this paper are the natural linear representation for quasipolyhedral sets (those subsets of the Euclidean space whose nonempty intersections with polytopes are polytopes). For these systems the geometrical properties of the solution set are investigated, and their extreme points and edges are characterized. The class of locally polyhedral systems includes the quasipolyhedral systems, introduced by Marchi, Puente, and Vera de Serio in order to generalize the Weyl property of finite linear inequality systems.

Quasipolyhedral sets in linear semiinfinite inequality systems

This paper provides an extension to linear semiinfinite systems of a well-known property of finite linear inequality systems, the so-called Weyl property, which characterizes the extreme points of the solution set as those solution points such that the gradient vectors of the active constraints form a complete set. A class of linear semiinfinite systems which satisfy this property is identified, the p-systems. It is also shown that any p-system contains an equivalent minimal subsystem.

A connection between scalar conservation laws and infinite linear systems of PDE's

We establish an explicit connection between solutions of the scalar conservation law and infinite hyperbolic systems of linear partial differential equations. As an immediate corollary of this connection combined with the well-known local existence theorem for the scalar conservation law, we obtain the corresponding local existence of smooth solutions for infinite linear systems. This approach also allows a way to seek numerical solutions of scalar conservation laws via solutions of truncated finite linear systems of PDE's.

Finite symmetric systems and their analysis

Symmetry of linear mechanical systems permits one to substantially reduce the effort in their analysis. Classification and structural analysis of the finite linear symmetric systems (models) are studied in this paper. There are two variants of the symmetry approach: mechanical and algebraic. In accordance with the former, a symmetric system is replaced by one or several small nonsymmetric subsystems which are subjected to special loads obtained from the initial set. The total response of the original symmetric system is found by special superposition of partial responses of these subsystems. The algebraic approach is based on the explicit block diagonal decomposition of the matrix equation corresponding to a symmetric system. While both approaches have the same efficiency the latter is easier to implement and describe. It is presented here. The condtions under which the symmetry technique may be utilized do not include symmetry of the applied loads (specifically, symmetry of those loads which form the right side of the associated equations). Nevertheless, if the loads are symmetric, the efficiency of the symmetry approach substantially increases. Group theory, which is widely used in this paper, is the mathematical tool for the study of symmetry, and all necessary notions are introduced.

Finite and infinite dimensional linear systems Green, W.L.: by Richard D. Jarvinen, 54:179–182 (1983)

Finite and infinite dimensional linear systems Green, W.L.: by Richard D. Jarvinen, 54:179–182 (1983)

Correlation effects in the electronic structure of high T c superconductors

We model Cu-O high T c superconductors by a generalized Hubbard Hamiltonian. We obtain the ground state for finite linear chain systems exactly. This allows us to calculate physical quantities to show that: a) those systems can change from a highly ionic to a highly metallic state as holes are added. b) In the metallic state holes tend to form pairs in next nearest neighbour sites.

Deconvolution of elliptical nuclei of galaxies - Application to M81

view Abstract Citations (14) References (24) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Deconvolution of Elliptical Nuclei of Galaxies: Application to M81 Bendinelli, O. ; Parmeggiani, G. ; Zavatti, F. Abstract The convolution integral equation relating the observed azimuthal brightness profiles of an elliptical galaxy is derived with constant ellipticity to the point-spread function (PSF) and the true brightness distribution along the semimajor axis. As an application, the nuclear structure of the low-level active galaxy M81 is studied, taking the brightness and PSF from Illingworth. The solution of the discretizing finite linear algebraic system is reached by means of Tikhonov regularization, since image restoration belongs to the class of numerically unstable inverse problems. The deconvolved structure found consists of a delta-spike of 14.24 + or - 0.07 integrated R mag, superposed on a smooth profile, having a central brightness of 14.02 + or - 0.07 R mag/sq arcsec. The reliability of the deconvolution and its errors (solution variance and bias) are confirmed by perturbation analysis and extensive simulations. Publication: The Astrophysical Journal Pub Date: September 1986 DOI: 10.1086/164531 Bibcode: 1986ApJ...308..611B Keywords: Brightness Distribution; Convolution Integrals; Elliptical Galaxies; Galactic Nuclei; Active Galaxies; Astronomical Photometry; Error Analysis; Galactic Structure; Astrophysics; GALAXIES: INDIVIDUAL MESSIER NUMBER: M81; GALAXIES: NUCLEI; GALAXIES: PHOTOMETRY; GALAXIES: STRUCTURE full text sources ADS | data products SIMBAD (1) NED (1) Related Materials (1) Erratum: 1987ApJ...315..408B

Type Curves for Finite Radial and Linear Gas-Flow Systems: Constant-Terminal-Pressure Case

Abstract This paper presents gas-prpduction-rate results in type curve form for finite radial and linear flow systems produced at a constant terminal (bottomhole) pressure. These results can be used in the analysis of acfual gas and oil rate/time data to estimate reservoir size and to infer reservoir shape. The type curves are based on dimensionless variables that are a generalized form of those presented previously.1,2 In addition, an approximate drawdown parameter is presented. Example applications that demonstrate the applicability of the type curves to a variety of reservoir configurations are given. The Appendix contains derivations of the dimensionless variables and the drawdown parameter.

System identification via state characterization

Arbib and Zeiger's generalization of Ho's algorithm for system identification is presented from an alternative viewpoint called “state characterization”. This is an essentially nonlinear, nonparametric method for designing experiments which will directly determine a state space representation for an unknown black box. This method assumes (1) a finite dimensional state space is possible, (2) the unknown black box can be reset to its initial state at will, so that an arbitrary set of experiments can be performed, and (3) there is no plant noise, so that applying given inputs to the initial state will always yield the same state, so various experiments can be performed on it. In the initial, informal discussion the properties of this method of system identification are compared to the properties of the method of parameter estimation via loss function minimization, e.g. least square error or maximum likelihood, which is the only general approach to system identification presently available. Loss function minimization will utilize any data, but it is computationally difficult and essentially parametric. State characterization eliminates the computational requirement at the expense of requiring specific data, and it is essentially nonparametric. It is proposed that state characterization may have practical application in determining an approximate, low order description of a complex system about which we have little prior information, for instance in the social sciences and medicine. The formal presentation is limited to finite state automata and discrete time linear systems. The situation is considered in which the order of the unknown black box, the minimum size state space required to represent it, is not known a priori; so it is necessary to continue experimenting indefinitely, using the results to obtain successively better descriptions of the unknown black box. Detailed algorithms are presented for obtaining either a Moore model or a Mealy model description. Branching rules for using past data to choose subsequent experiments in order to hasten convergence are presented. Known theorems are used to prove that after some finite time these algorithms will yield a correct description, and new theorems show that the subsequent representations will be invariant after a correct description is obtained—although there is no way for the experimenter to know when he has obtained a correct description.

Two stochastic approximation procedures for identifying linear systems

A Robbins-Monro [1] stochastic approximation procedure for identifying a finite memory time-discrete time-stationary linear system from noisy input-output measurements is developed. Two algorithms are presented to sequentially identify the linear system. The first one is derived, based on the minimization of the mean-square error between the unknown system and a model, and is shown to develop a bias which depends only on the variance of the input signal measurement error. Under the assumption that this variance is known a priori, a second algorithm is developed which, in the limit, identifies the unknown system exactly. The case when the covariance matrix of the random input sequence is not positive definite is also considered.

The Numerical Solution of Fredholm integral Equations of the Second Kind

When the kernel has several continuous derivatives, the method reduces to replacing the integral with a numerical integral and then to solving a finite linear system; see [1], [2], [3], [5], [11], [13]. In the followiing section, a generalized form of numerical inltegration is given for functioins of oine variable. It is applied to (1) in ?3, aild convergence of the resulting method is showii in ?4. Section 5 conitains computational notes and a numerical example.

Continuously infinite matrices and the algebraic theory of Fredholm's equation

In this note the theory of the integral equation is reduced to elementary matrix algebra. The Fredholm theorems are proved without the introduction of the usual parameter λ and without using the properties of the resolvent considered as an analytic function of the complex variable λ. There is an exact correspondence between the properties of the equation (1) and the properties of a finite linear system. Here the proofs of these properties are also in extact correspondence with the same for the elementary case.