Showing papers on "Piecewise linear function published in 1999"

Smooth p-adic analytic spaces are locally contractible

Abstract: 0 Introduction 293 1 Piecewise RS -linear spaces 298 2 R-colored polysimplicial sets 308 3 R-colored polysimplicial sets of length l 313 4 The skeleton of a nondegenerate pluri-stable formal scheme 327 5 A colored polysimplicial set associated with a nondegenerate poly-stable fibration 336 6 p-Adic analytic and piecewise linear spaces 346 7 Strong local contractibility of smooth analytic spaces 355 8 Cohomology with coefficients in the sheaf of constant functions 362

High-level canonical piecewise linear representation using a simplicial partition

Abstract: In this work, we propose a set of high-level canonical piecewise linear (HL-CPWL) functions to form a representation basis for the set of piecewise linear functions f: D/spl rarr/R/sup 1/ defined over a simplicial partition of a rectangular compact set D in R/sup n/. In consequence, the representation proposed uses the minimum number of parameters. The basis functions are obtained recursively by multiple compositions of a unique generating function /spl gamma/, resulting in several types of nested absolute-value functions. It is shown that the representation in a domain in R/sup n/ requires functions up to nesting level n. As a consequence of the choice of the basis functions, an efficient numerical method for the resolution of the parameters of the high-level (HL) canonical representation results. Finally, an application to the approximation of continuous functions is shown.

Boundary integral methods : numerical and mathematical aspects

Abstract: Time-dependent reaction-diffusion problems and the LTDRM approach A generalized BEM for steady and transient heat conduction in media with spatially varying thermal conductivity Numerical solution of the Helmholtz equation The method of fundamental solutions for potential, Helmholtz and diffusion problems On the stability of piecewise linear wavelet collocation and the solution of the double layer equation over polygonal curves Local theory of projection methods for Cauchy singular equations on an interval Numerical solutions of Hammerstein equations Numerical exploitation of symmetric structures in BEM Some recent developments in the convergence analysis discrete projection methods.

The control parameterization enhancing transform for constrained optimal control problems

Abstract: Consider a general class of constrained optimal control problems in canonical form Using the classical control parameterization technique, the time (planning) horizon is partitioned into several subintervals The control functions are approximated by piecewise constant or piecewise linear functions with pre-fixed switching times However, if the optimal control functions to be obtained are piecewise continuous, the accuracy of this approximation process greatly depends on how fine the partition is On the other hand, the performance of any optimization algorithm used is limited by the number of decision variables of the problem Thus, the time horizon cannot be partitioned into arbitrarily many subintervals to reach the desired accuracy To overcome this difficulty, the switching points should also be taken as decision variables This is the main motivation of the paper A novel transform, to be referred to as the control parameterization enhancing transform, is introduced to convert approximate optimal control problems with variable switching times into equivalent standard optimal control problems involving piecewise constant or piecewise linear control functions with pre-fixed switching times The transformed problems are essentially optimal parameter selection problems and hence are solvable by various existing algorithms For illustration, two non-trivial numerical examples are solved using the proposed method

Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

Abstract: We consider the Cahn-Hilliard equation with a logarithmic free energy and non-degenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented.

The slantlet transform

Abstract: The discrete wavelet transform (DWT) is usually carried out by filterbank iteration; however, for a fixed number of zero moments, this does not yield a discrete-time basis that is optimal with respect to time localization. This paper discusses the implementation and properties of an orthogonal DWT, with two zero moments and with improved time localization. The basis is not based on filterbank iteration; instead, different filters are used for each scale. For coarse scales, the support of the discrete-time basis functions approaches two thirds that of the corresponding functions obtained by filterbank iteration. This basis, which is a special case of a class of bases described by Alpert (1992, 1993), retains the octave-band characteristic and is piecewise linear (but discontinuous). Closed-form expressions for the filters are given, an efficient implementation of the transform is described, and improvement in a denoising example is shown. This basis, being piecewise linear, is reminiscent of the slant transform, to which it is compared.

Stability of a three-station fluid network

Abstract: This paper studies the stability of a three-station fluid network. We show that, unlike the two-station networks in Dai and Vande Vate l18r, the global stability region of our three-station network is not the intersection of its stability regions under static buffer priority disciplines. Thus, the “worst” or extremal disciplines are not static buffer priority disciplines. We also prove that the global stability region of our three-station network is not monotone in the service times and so, we may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes this largest monotone subset of the global stability region for our three-station network. We also show that the LP proposed by Bertsimas et al. l1r does not characterize either the global stability region or even the monotone global stability region of our three-station network. Further, we demonstrate that the LP related to the linear Lyapunov function proposed by Chen and Zhang l11r does not characterize the stability region of our three-station network under a static buffer priority discipline.

Absolutely continuous invariant measures for expanding piecewise linear maps

Abstract: We prove the existence of absolutely continuous invariant measures for arbitrary expanding piecewise linear maps on bounded polyhedral domains in Euclidean spaces ℝ d .

Discretized Lyapunov functional for uncertain systems with multiple time-delay

Abstract: The stability of uncertain systems with multiple time-delay is studied using a quadratic Lyapunov functional. By choosing piecewise linear parameters for the kernel of a quadratic expression, the guaranteed stability condition is formulated in the form of linear matrix inequality. The extent of conservatism depends on the grid size. However, numerical results show that even a very coarse discretization can produce quite satisfactory results. The possibility of non-uniform grid size and more general Lyapunov functional is of interest even for a single delay system. Numerical examples are presented.

A parametrization of piecewise linear Lyapunov functions via linear programming

Abstract: In this paper, we present a parametrization of piecewise linear (PWL) Lyapunov functions. To this end, we consider the class of all continuous PWL functions defined over a simplicial partition. We take advantage of a recently developed high level canonical PWL (HL CPWL) representation, which expresses the PWL function in a compact and closed form. Once the parametrization problem is properly stated, we focus on its application to the stabiilty analysis of dynamic systems. We consider uncertain non-linear systems and extend the sector condition obtained by Ohta et al. In addition, we propose a method of selecting an optimal candidate. One of the main advantages of this approach is that the parametrization and choice of the Lyapunov candidate, as well as the stability analysis, result in linear programming problems.

Numerical analysis of the primal problem of elastoplasticity with hardening

Abstract: The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening.

Tight Cooperation and Its Application in Piecewise Linear Optimization

Abstract: Many cooperative systems merge a linear constraint solver and a domain reduction solver over finite domains or intervals. The latter handles a high level formulation of the problem and passes domain variable information. The former handles a linear formulation of the problem and computes a relaxed optimal solution. This paper proposes an extension to this framework called tight cooperation where the linear formulation of a high level constraint is restated in a way, as domains are reduced. This approach is illustrated on piecewise linear optimization. Experimental results are given. These show that tight cooperation can give better results than classical cooperation and mixed-integer programming techniques.

Piecewise linear constrained control for continuous-time systems

Abstract: In the control of linear systems with nonsymmetrical input constraints, larger domains of initialization can be obtained by allowing slow initial dynamics of the system in closed loop and making it faster during its evolution. The positive invariance concept application leads to a piecewise linear constrained control using nonsymmetrical nested polyhedral sets.

Simple deterministic dynamical systems with fractal diffusion coefficients.

Abstract: We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional array of scatterers with moving point particles. The particles move from one scatterer to the next according to a piecewise linear, expanding, deterministic map on unit intervals. The microscopic chaotic scattering process of the map can be changed by a control parameter. The macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation. Our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter. We provide qualitative and quantitative arguments to explain features of this fractal structure.

Rate control strategy for embedded wavelet video coders

Abstract: An investigation into the use of rate-control strategies for embedded wavelet video encoders is presented. It is shown that the best model for the R-D of these encoders is piecewise linear. Also, an effective iterative procedure is proposed for dealing with the problem of frame dependency, which yields improved rate × distortion results.

A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems

Abstract: We present a discretization theory for a class of nonlinear evolution inequalities that encompasses time dependent monotone operator equations and parabolic variational in- equalities. This discretization theory combines a backward Euler scheme for time discretiza- tion and the Galerkin method for space discretization. We include set convergence of convex subsets in the sense of Glowinski-Mosco-Stummel to allow a nonconforming approximation of unilateral constraints. As an application we treat parabolic Signorini problems involving the p-Laplacian, where we use standard piecewise polynomial finite elements for space dis- cretization. Without imposing any regularity assumption for the solution we establish vari- ous norm convergence results for piecewise linear as well piecewise quadratic trial func- tions, which in the latter case leads to a nonconforming approximation scheme.

The ubiquity of Thompson's group F in groups of piecewise linear homeomorphisms of the unit interval

Abstract: We show that Thompson's group F occurs with great frequency in the group of PL homeomorphisms of the unit interval.

Finite volume schemes for Hamilton-Jacobi equations

Abstract: We introduce two classes of monotone finite volume schemes for Hamilton-Jacobi equations. The corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles. The schemes are shown to converge to the viscosity solution of the Hamilton–Jacobi equation.

Piecewise Linear Prewavelets on Arbitrary Triangulations

Abstract: This paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology. It is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high.

Identification of Hammerstein models using genetic algorithms

Abstract: The application of genetic algorithms to the identification of the Hammerstein model is studied. The method uses piecewise linear, and not a polynomial, to approximate the memoryless nonlinear characterisation. Numerical simulation results provided show the efficiency of the method.

Nonlinear adaptive control using networks of piecewise linear approximators

Abstract: This paper presents a stable nonparametric adaptive control approach using a piecewise local linear approximator. The basic structure of the controller is based on the feedback linearizing control plus sliding control structure to ensure the global stability. As for distinctive features, the continuous piecewise linear approximator is newly introduced and a time varying activation region is defined for efficient self-organization of the approximator during operation. The feasibility of the piecewise linear adaptive control method is demonstrated by a computational simulation.

Estimating initial conditions of noisy chaotic signals generated by piecewise linear Markov maps using itineraries

Abstract: Estimating a one-dimensional (1-D) chaotic signal in noise is an important problem in chaotic communications and information processing. This problem is theoretically equivalent to the estimation of the initial condition of a chaotic signal. A few studies on this initial condition estimation problem have been carried out for certain specific maps such as the tent map and the logistic map. This problem is investigated for the piecewise linear Markov maps as well as maps that are topologically conjugate to piecewise linear Markov maps. By using the one-to-one correspondence between the initial conditions of a chaotic map and its space of itineraries, several algorithms extending the halving method are developed to estimate the initial condition of a 1-D chaotic signal embedded in additive noise. Performance of these estimators is evaluated using Monte Carlo simulations. At high SNR, the variance of these estimators is found to approach the Cramer-Rao bound.

Predicting Traction in Web Handling

Abstract: A simple algorithm has been developed for predicting traction in web handling applications. Minimal traction exists when the minimum air film height between the roller and web is greater than three times the rms roughness of the two surfaces in contact. Classical foil bearing theory modified for permeable surfaces is used to determine the air film height. A piecewise linear solution using squeeze film theory is also used to account for side leakage. The minimum air film height is a function of web tension, web and roller velocity, air viscosity, web width, web permeability and roller radius. The algorithm is applicable for permeable and nonpermeable webs. Values obtained from the algorithm can be used to predict if sufficient traction is available between the web and roller for a given set of physical and operating parameters. Traction values can also be used as input for winding, wrinkling, and spreading models.

Pattern recognition based on piecewise linear probability density function

Abstract: The present invention is a method and apparatus to determine a similarity measure between first and second patterns First and second storages store first and second feature vectors which represent the first and second patterns, respectively A similarity estimator is coupled to the first and second storages to compute a similarity probability of the first and second feature vectors using a piecewise linear probability density function (PDF) The similarity probability corresponds to the similarity measure

Two-level Schwarz method for unilateral variational inequalities

Abstract: The numerical solution of variational inequalities of obstacle type associated with second-order elliptic operators is considered. Iterative methods based on the domain decomposition approach are proposed for discrete obstacle problems arising from the continuous, piecewise linear finite element approximation of the differential problem. A new variant of the Schwarz methodology, called the two-level Schwarz method, is developed offering the possibility of making use of fast linear solvers (e.g., linear multigrid and fictitious domain methods) for the genuinely nonlinear obstacle problems. Namely, by using particular monotonicity results, the computational domain can be partitioned into (mesh) subdomains with linear and nonlinear (obstacle-type) subproblems. By taking advantage of this domain decomposition and fast linear solvers, efficient implementation algorithms for large-scale discrete obstacle problems can be developed. The last part of the paper is devoted to illustrative numerical experiments.