Showing papers on "Piecewise linear function published in 1997"

Representation of Topography by Shaved Cells in a Height Coordinate Ocean Model

Abstract: Height coordinate ocean models commonly represent topography as a ‘‘staircase’’ of discontinuous steps that are fitted to the model grid. Here the ramifications of an alternative approach are studied in which ‘‘shaved cells’’ are used to represent irregular topography. The problem is formulated using the finite-volume method and care is taken to ensure that the discrete forms have appropriate conservation properties. Two representations of topography, ‘‘partial step’’ and ‘‘piecewise linear,’’ are considered and compared with the staircase approach in some standard problems such as the topographic b effect and flow over a Gaussian bump. It is shown that shaved cells are clearly more accurate than the conventional staircase representation. The use of partial steps, although not as accurate as the piecewise linear approach, is seen to be superior to the staircase approach. Moreover, partial steps can be readily implemented in existing height coordinate models.

Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h - p Version of the FEM

Abstract: In this paper, which is part II in a series of two, the investigation of the Galerkin finite element solution to the Helmholtz equation is continued. While part I contained results on the $h$ version with piecewise linear approximation, the present part deals with approximation spaces of order $p \ge 1$. As in part I, the results are presented on a one-dimensional model problem with Dirichlet--Robin boundary conditions. In particular, there are proven stability estimates, both with respect to data of higher regularity and data that is bounded in lower norms. The estimates are shown both for the continuous and the discrete spaces under consideration. Further, there is proven a result on the phase difference between the exact and the Galerkin finite element solutions for arbitrary $p$ that had been previously conjectured from numerical experiments. These results and further preparatory statements are then employed to show error estimates for the Galerkin finite element method (FEM). It becomes evident that the error estimate for higher approximation can---with certain assumptions on the data---be written in the same form as the piecewise linear case, namely, as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference. The paper is concluded with a numerical evaluation.

Piecewise Linear Quadratic Optimal Control

Abstract: The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy.

Piecewise linear quadratic optimal control

Abstract: A technique for computation of piecewise quadratic Lyapunov functions is further developed for performance analysis and controller synthesis for nonlinear systems. In this way, degree of observability is estimated, L/sub 2/ induced gain is computed and optimal control problems are solved. The computations are based on convex optimization in terms of linear matrix inequalities.

Inverter harmonic reduction using Walsh function harmonic elimination method

Abstract: A pulse-width-modulated (PWM) inverter using the Walsh function harmonic elimination method is proposed in this paper. By using the Walsh domain waveform analytic technique, the harmonic amplitudes of the inverter output voltage can be expressed as functions of switching angles. Thus, the switching angles are optimized by solving linear algebraic equations instead of solving nonlinear transcendental equations. The local piecewise linear relations between the switching angles and the fundamental amplitude can be obtained under an appropriate initial condition. By searching all feasible initial conditions, the global solutions are obtained. The relations between switching angles and fundamental amplitude can be approximated by straight-line curve fitting. Thus, on-line control of fundamental amplitude and frequency is possible for the microcomputer-based implementation. The developed algorithm can be applied to both bipolar and unipolar switching schemes. The theoretical predictions are confirmed by computer simulations and DSP-based hardware implementation.

Boundary crossing probability for Brownian motion and general boundaries

Abstract: An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

Multiwavelets for Second-Kind Integral Equations

Abstract: We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N2 to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.

The multilevel finite element method for adaptive mesh optimization and visualization of volume data

Abstract: Multilevel representations and mesh reduction techniques have been used for accelerating the processing and the rendering of large datasets representing scalar- or vector-valued functions defined on complex 2D or 3D meshes. We present a method based on finite element approximations which combines these two approaches in a new and unique way that is conceptually simple and theoretically sound. The main idea is to consider mesh reduction as an approximation problem in appropriate finite element spaces. Starting with a very coarse triangulation of the functional domain, a hierarchy of highly non-uniform tetrahedral (or triangular in 2D) meshes is generated adaptively by local refinement. This process is driven by controlling the local error of the piecewise linear finite element approximation of the function on each mesh element. A reliable and efficient computation of the global approximation error and a multilevel preconditioned conjugate gradient solver are the key components of the implementation. In order to analyze the properties and advantages of the adaptively generated tetrahedral meshes, we implemented two volume visualization algorithms: an iso-surface extractor and a ray-caster. Both algorithms, while conceptually simple, show significant speedups over conventional methods delivering comparable rendering quality from adaptively compressed datasets.

Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications

Abstract: We consider the problem of minimizing a separable convex objective function over the linear space given by a system Mx=0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [J. Assoc. Comput. Mach., 36 (1989), pp. 873--886.] and extended to some other problems [T. R. Ervolina and S. T. McCormick, Discrete Appl. Math., 46 (1993), pp. 133--165], [A. Frank and A. V. Karzanov, Technical Report RR 895-M, Laboratoire ARTEMIS IMAG, Universite Joseph Fourier, Grenoble, France, 1992], [T. Ibaraki, A. V. Karzanov, and H. Nagamochi, private communication, 1993], [M. Hadjiat, Technical Report, Groupe Intelligence Artificielle, Faculte des Sciences de Luminy, Marseille, France, 1994] can be generalized to give a combinatorial method with geometric convergence for our problem. We also generalize the computationally more efficient cancel-and-tighten method. We then consider objective functions that are piecewise linear, pure and piecewise quadratic, or piecewise mixed linear and quadratic, and we show how both methods can be implemented to find exact solutions in polynomial time (strongly polynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and cocirculations in a network. We finish by showing how to extend our methods to find optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles.

Planning and control in stochastic domains with imperfect information

Abstract: Partially observable Markov decision processes (POMDPs) can be used to model complex control problems that include both action outcome uncertainty and imperfect observability. A control problem within the POMDP framework is expressed as a dynamic optimization problem with a value function that combines costs or rewards from multiple steps. Although the POMDP framework is more expressive than other simpler frameworks, like Markov decision processes (MDP), its associated optimization methods are more demanding computationally and only very small problems can be solved exactly in practice. The thesis focuses on two possible approaches that can he used to solve larger problems: approximation methods and exploitation of additional problem structure. First, a number of new efficient approximation methods and improvements of existing algorithms are proposed. These include (1) the fast informed bound method based on approximate dynamic programming updates that lead to piecewise linear and convex value functions with a constant number of linear vectors, (2) a grid-based point interpolation method that supports variable grids, (3) an incremental version of the linear vector method that updates value function derivatives, as well as (4) various heuristics for selecting grid-points. The new and existing methods are experimentally tested and compared on a set of three infinite discounted horizon problems of different complexity. The experimental results show that methods that preserve the shape of the value function over updates, such as the newly designed incremental linear vector and fast informed bound methods, tend to outperform other methods on the control performance test. Second, the thesis presents a number of techniques for exploiting additional structure in the model of complex control problems. These are studied as applied to a medical therapy planning problem--the management of patients with chronic ischemic heart disease. The new extensions proposed include factored and hierarchically structured models that combine the advantages of the POMDP and MDP frameworks and cut down the size and complexity of the information state space.

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

Abstract: A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly.

On the computation of piecewise quadratic Lyapunov functions

Abstract: A new parametrization of piecewise quadratic functions is presented, aimed for convex optimization of Lyapunov functions The parametrization ensures continuity of the functions, but is also flexible enough to approximate any continuous function with arbitrary accuracy A converse Lyapunov theorem is given, stating that all exponentially stable systems of a certain class will in principle admit a piecewise quadratic Lyapunov function, to be found by the given optimization procedure

Piecewise linear test functions for stability and instability of queueing networks

Abstract: We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient.

Locally and globally riddled basins in two coupled piecewise-linear maps

Abstract: The chaos synchronization and riddled basins phenomena are discussed for a family of two-dimensional piecewise linear endomorphisms that consist of two linearly coupled one-dimensional maps. Rigorous conditions for the occurrence of both phenomena are given. Different scenarios for the transition from locally to globally riddled basins and blowout bifurcation have been identified and described.

Visualization of higher order singularities in vector fields

Abstract: Presents an algorithm for the visualization of vector field topology based on Clifford algebra. It allows the detection of higher-order singularities. This is accomplished by first analysing the possible critical points and then choosing a suitable polynomial approximation, because conventional methods based on piecewise linear or bilinear approximation do not allow higher-order critical points and destroy the topology in such cases. The algorithm is still very fast, because of using linear approximation outside the areas with several critical points.

Generation of broadband speech from narrowband speech using piecewise linear mapping.

Abstract: This paper proposes a recovery method of broadband speech form narrowband speech based on piecewise linear mapping. In this method, narrowband spectrum envelope of input speech is transformed to broadband spectrum envelope using linearly transformed matrices which are associated with several spectrum spaces. These matrices were estimated by speech training data, so as to minimize the mean square error between the transformed and the original spectra. This algorithm is compared the following other methods, (1)the codebook mapping, (2)the neural network. Through the evaluation by the spectral distance measure, it was found that the proposed method achieved a lower spectral distortion than the other methods. Perceptual experiments indicates a good performance for the reconstructed broadband speech.

New formulation and relaxation to solve a concave-cost network flow problem

Abstract: In this paper we study a minimum cost, multicommodity network flow problem in which the total cost is piecewise linear, concave of the total flow along the arcs. Specifically, the problem can be defined as follows. Given a directed network, a set of pairs of communicating nodes and a set of available capacity ranges and their corresponding variable and fixed cost components for each arc, the problem is to select for each arc a range and identify a path for each commodity between its source and destination nodes so as to minimize the total costs. We also extend the problem to the case of piecewise nonlinear, concave cost function. New mathematical programming formulations of the problems are presented. Efficient solution procedures based on Lagrangean relaxations of the problems are developed. Extensive computational results across a variety of networks are reported. These results indicate that the solution procedures are effective for a wide range of traffic loads and different cost structures. They also show that this work represents an improvement over previous work made by other authors. This improvement is the result of the introduction of the new formulations of the problems and their relaxations.

Secure communication with chaotic systems of difference equations

Abstract: The paper presents chaotic systems of difference equations that can effectively encrypt information. Two classes of systems are presented: The first one (Class 1) is optimized for secure communications over reliable channels, while the second (Class 2) tolerates transmission noise at the expense of reduced parameter space size. The nonlinearity of these systems is achieved by designing proper piecewise linear functions and by using module operations. The utilization of additional nonlinear terms can improve the enciphering efficiency. The encrypting performance of the algorithms is evaluated analytically and by simulation experiments. Also, the case of an imperfect transmission channel that inserts noise in the transmitted signal is addressed and the design is modified in order to offer reliable secure transmission over channels with very small signal to noise ratios.

Hopf-like bifurcations in planar piecewise linear systems

Abstract: Continuous planar piecewise linear systems with two linear zones are considered. Due to their low differentiability specific techniques of analysis must be developed. Several bifurcations giving rise to limit cycles are pointed out.

Robust stability of piecewise linear discrete time systems

Abstract: This paper presents efficient algorithms for verifying the stability of uncertain discrete time piecewise linear (PL) systems. While PL systems are intuitively simple, they are computationally hard. Two approaches to verifying stability are presented. For each approach, separate necessary and sufficient conditions are given. The first approach requires the solution of a linear matrix inequality. This method is only applicable to a restricted class of PL systems, and is generally very conservative. It is demonstrated that for most PL systems, these conditions yield no information. The second, more general, approach is based upon robust simulation. This method is useful for all PL systems, and will always yield a definitive answer. If a system initially satisfies necessity, but fails sufficiency, these algorithms can be systematically refined and after a finite number of refinements, a definitive answer is guaranteed. The algorithms are illustrated on four examples.

Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a convection–diffusion problem

Abstract: We analyze a streamline diffusion scheme on a special piecewise unform mesh for a model time-dependent convection-diffusion problem The method with piecewise linear elements is shown to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside the boundary layer and almost order 3/4 inside the boundary layer Numerical results are also given

Effects of membership function parameters on the performance of a fuzzy signal detector

Abstract: This paper describes a signal-detection algorithm based on fuzzy logic. The detector combines evidence provided by two waveform features and explicitly considers uncertainty in the detection decision. The detector classifies waveforms including a signal, not including a signal, or being uncertain, in which case no conclusion regarding presence or absence of a signal is drawn. Piecewise linear membership functions are used, and a method to describe the membership functions in terms of two parameters is developed. The performance of the detector is compared to a Bayesian maximum likelihood detector, using brainstem auditory evoked potential signals in simulated noise, and the effects of the steepness (slope) and overlap of the membership functions on detector performance are evaluated. By varying the membership function steepness and overlap, the fuzzy detector can almost completely eliminate classification errors at the cost of a large number of uncertain classifications or it can be made to perform similarly to the Bayesian detector.

Wage determination under non-linear taxes: estimation and an application to panel data

Abstract: This paper studies wage determination under piecewise linear taxes in a unionized labor market. The purpose of the paper is to model how piecewise linear taxation affects the choice set of the union and to take this information into account in the estimation. The empirical application is based on panel data. Piecewise linear taxes necessitates formal assumptions about the sources of randomness, and the authors find that both unobserved heterogeneity and measurement errors in the wage rate are important to consider in the estimation. The authors also find that taxes are likely to have a nonnegligible impact on the (pre-tax) wage rate. Copyright 1997 by Royal Economic Society.