Showing papers on "Piecewise linear function published in 1988"

The hierarchical basis multigrid method

Abstract: We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauβ-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

Polygonal Approximations of a Curve — Formulations and Algorithms

Abstract: A general framework for polygonal approximation problems in various situations is given, and solution algorithms are discussed. The problems are classified according to three criteria: (1) whether the curve is the graph of a piecewise linear function y=f(x) of the independent variable x, or it is a general polygonal curve in the plane, (2) which error criterion is adopted (for a piecewise linear function, we consider only one criterion), (3) whether the objective is to minimize the number of vertices of the approximate curve when an error bound is given, or to minimize the approximation error when the number of vertices of the approximate curve is specified. The problems are first formulated in terms of graph theory, and then computationally efficient algorithms are constructed by taking advantage of fundamental algorithms for convex hulls in computational geometry.

Homology of smooth splines: generic triangulations and a conjecture of Strang

Abstract: For A a triangulated d-dimensional region in Rd, let Sr (A) denote the vector space of all cr functions F on A that, restricted to any simplex in A, are given by polynomials of degree at most m. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds A in the plane, getting lower bounds on the dimension of Sr (A) for all r. For r = 1, we prove a conjecture of Strang concerning the generic dimension of the space of Cl splines over a triangulated manifold in R2. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.

PLOP-hashing: A grid file without directory

Abstract: The authors consider the case of nonuniform weakly correlated or independent multidimensional record distributions. After demonstrating the advantages of multidimensional hashing schemes without directory, they suggest using piecewise linear expansions to distribute the load more evenly over the pages of the file. The resulting piecewise linear order preserving hashing scheme (PLOP-hashing) is then compared to the two-level grid file, which turned out to be the most popular scheme in practical applications. >

Why computers like lebesgue measure

Abstract: Let τ: [0,1]→[0,1] be a transformation which has an absolutely continuous invariant measure μ. Let τ be a realistic, deterministic model for τ. We prove that if τ has long computer trajectories, either periodic or non-periodic, then these computer trajectories have histograms which approach the density of μ. For a large class of piecewise linear transformations, we prove the existence of long periodic trajectories.

A simplex algorithm for piecewise-linear programming 11: finiteness, feasibility and degeneracy

Abstract: The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. Part I of this paper has developed a general and direct simplex algorithm for piecewise-linear programming, under convenient assumptions that guarantee a finite number of basic solutions, existence of basic feasible solutions, and nondegeneracy of all such solutions. Part II now shows how these assumptions can be weakened so that they pose no obstacle to effective use of the piecewise-linear simplex algorithm. The theory of piecewise-linear programming is thereby extended, and numerous features of linear programming are generalized or are seen in a new light. An analysis of the algorithm's computational requirements and a survey of applications will be presented in Part III.

A separable piecewise linear upper bound for stochastic linear programs

Abstract: Stochastic linear programs require the evaluation of an integral in which the integrand is itself the value of a linear program. This integration is often approximated by discrete distributions that bound the integral from above or below. A difficulty with previous upper bounds is that they generally require a number of function evaluations that grows exponentially in the number of variables. We give a new upper bound that requires operations that only grow polynomially in the number of random variables. We show that this bound is sharp if the function is linear and give computational results to illustrate its performance.

A sample performance function of closed Jackson queueing networks

Abstract: A stochastic system such as a queueing network can be specified by system parameters and a set of sequences of random variables that represents the randomness in the system. A “sample performance function” is a measure of system performance as a function of system parameters, for each realization of the random sequences. Although the average of N sample performance functions converges to the expected value of the performance with probability one when N goes to infinity, the average of the derivatives of these N sample performance functions with respect to a parameter may not converge to the derivative of the expected value. In this paper, we study a sample performance function of a closed Jackson queueing network; specifically, the time required by a server to serve a finite number of customers. We show that this sample performance function is a continuous, piecewise linear function of the mean service time. We prove that the average of derivatives of this sample performance function with a given initial ...

On the hierarchical design of VLSI processor arrays

Abstract: The author investigates systematic methods for the design of processor arrays. The proposed concept enables the efficient realization of more general classes of algorithms than the systolic concept. In particular, instance-dependent branching, instance-dependent processor configurations, and hierarchical formulations of imperative programs can be taken into account. The concept of a piecewise-regular dependence graph and that of its reduced description is given. The definition of piece-wise regular algorithms leads to their mapping onto piecewise regular systolic arrays using piecewise linear transformations. The hierarchical description of algorithms and dependence graphs and the corresponding transformations such as condensation, unfolding, clustering and unclustering are applied to partitioning problems (assignment, schedule segmentation, multidimensional mapping). >

The Combined Use of a Nonlinear Chernoff Formula with a Regularization Procedure for Two-Phase Stefan Problems

Abstract: The approximation of two-phase Stefan problems in 2-D by a nonlinear Chernoff formula combined with a regularization procedure is analyzed. The first technique allows the associated strongly nonlinear parabolic P.D.E. to be approximated by a sequence of linear elliptic problems. In addition, non-degeneracy properties can be properly exploited through the use of a smoothing process. A fully discrete scheme involving piecewise linear and constant finite elements is proposed. Energy error estimates are proven for both physical variables, namely enthalpy and temperature. These rates of convergence improve previous results.

Automatic Qualitative Analysis of Ordinary Differential Equations Using Piecewise Linear Approximations

Abstract: This paper explores automating the qualitative analysis of physical systems. It describes a program, called PLR, that takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values. PLR approximates intractable nonlinear systems with piecewise linear ones, analyzes the approximations, and draws conclusions about the original systems. It chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes, yet simple enough to be analyzed completely and efficiently. It derives additional properties, such as boundedness or periodicity, by theoretical methods. I demonstrate PLR on several common nonlinear systems and on published examples from mechanical engineering.

Piecewise uniform vector quantizers

Abstract: The companding model for quantizer design and analysis has been widely applied in the scalar quantization case. However, if the signal to be quantized is a vector, then the optimum companding system can be designed for only a limited number of distributions. On the other hand, multidimensional piecewise linear companders can be designed for any signal density, generating quantizers that are uniform on each region of the compander. These systems, while not optimal, can have asymptotic performance arbitrarily close to the optimum. Their analysis and implementation can be simpler than those of optimal systems. Piecewise linear companders for asymptotic multidimensional quantization are analyzed, and a method for their design is suggested. >

A matrix method for estimating the Liapunov exponent of one-dimensional systems

Abstract: Let τ: [0, 1]→[0, 1] be a piecewise monotonie expanding map. Then τ admits an absolutely continuous invariant measureμ. A result of Kosyakin and Sandler shows thatμ can be approximated by a sequence of absolutely continuous measuresμn invariant under piecewise linear Markov maps τitn. Each τitn is constructed on the inverse images of the turning points of τ. The easily computable measuresμn are used to estimate the Liapunov exponent of τ. The idea of using Markov maps for estimating the Liapunov exponent is applied to both expanding and nonexpanding maps.

Piecewise Linear Incentive Schemes

Abstract: General Outlook and Stabilization Theories and Policies of piecewis e linear incentive schemes are studied in a setting of moral hazard and adverse selection The form of the best such function is analyzed Particular attention is given to the loss from using piecewise linear schemes relative to "optimal" schemes, as well as the gain from usin g simpler functional forms such as the linear functions The incentive problem is studied under various informational assumptions, which allow some calculations of the value of information Copyright 1988 by The editors of the Scandinavian Journal of Economics

Normal forms of continuous piecewise linear Vector fields and chaotic attractors Part II: chaotic attractors

Abstract: This paper represents Part II of a 2-part paper which provides the normal forms of piecewise linear vector fields (abbr. PL-systems) under affine conjugacy and the prototype chaotic attractors in the PL-systems. We derive in Part I the general forms of PL-systems and the normal forms of linear systems with a section which play an important role in Part II. The normal forms of 2-region PL-systems and the prototype attractors (Spiral, Double Scroll, Double Screw, Toroidal, Sparrow, Lorenz and Duffing attractors) are provided in Part II. It is also proved in Part II that the affine conjugate classes of proper systems are uniquely determined by the eigen values in each region.

Piecewise linear filtering with small observation noise

Abstract: We consider a piecewise linear filtering problem with small observation noise. It is shown that one can construct an approximate finite dimensional filter which uses a bunch of Kalman filters, together with a test procedure to decide which Kalman filter to follow.

Fundamental study for high accuracy calculation of 3-D electromagnetic field

Abstract: The effect of the improper integral and the discretization in 3-D, which are the main error sources in the boundary element method, is investigated. The accuracy of the double numerical integral with a singularity is improved by using the integral variable transform of a double exponential function and executing the integral synchronously along both integral variables. Trigonometric expressions for computation of the integrand are concisely described. Fundamental characteristics of the discretization error in 3-D are demonstrated. The magnitude of the discretization error for the charge distribution can be decreased by combining the piecewise linear solutions with stepwise solutions according to the sizes of the adjacent elements. >

Convergence results for piecewise linear quadratures for Cauchy principal value integrals

Abstract: Etablissement des conditions de convergence simple et uniforme des quadratures lineaires par morceaux pour les integrales a valeur principale de Cauchy

The ranges of transfer and return maps in three-region piecewise-linear dynamical systems

Abstract: The ranges of the transfer and the return maps that are induced by the flow of a piecewise-linear continuous dynamical system describing Chua's circuit are investigated analytically. the two sets turn out to be intertwined for all values of the parameters. For high enough frequencies of the saddle-focus, located at the origin of state space, the first set forms ‘inclusions’ inside the second one. A charting of parameter space describing the creation and annihilation of the above inclusions quantitatively is indicated.

Normal forms of continuous piecewise linear vector fields and chaotic attractors Part I: Linear vector fields with a section

Abstract: This paper represents Part I of a 2-part paper which provides the normal forms of piecewise linear vector fields (abbr. PL-systems) under affine conjugacy and the prototype chaotic attractors in the PL-systems. We derive in Part I the general forms of PL-systems and the normal forms of linear systems with a section which play an important role in Part II. The normal forms of 2-region PL-systems and the prototype attractors (Spiral, Double Scroll, Double Screw, Troidal, Sparrow, Lorenz and Duffing attractors) are provided in Part II. It is also proved in Part II that the affine conjugate classes of proper systems are uniquely determined by the eigenvalues in each region.

Shape Matching Using Curvature Processes

Abstract: Shape matching is a fundamental problem of vision in general atrd interpretation of deforming shapes in particular. The objective of marching in this instance is to recover the deformation and therefore generalizes the notion of correlation, which aims to only produce a numerical me:rsure of the similarity between two shapes. To address shape matching, we introduce a new representation of a closed 2D shape as a cyclic sequence of the extended circular images of the convex and concave segments of its contour. This representation is then used to establish correspondences between segments of the two contours using dynamic programming. Finally, we compute a recrvery of the differences between two similar contours in terms of the action of curvature process. Computation of convex and concave segments of the contours, given in piecewise linear form, is accomplished using the analytic representation of a local B-spline fit. We show the result of our deformation recovery scheme appliedto dynamis cloud silhouette analysis using hand-traced input from real satellite images. o lese Academic Prcss, Inc.

Response of MDOF Systems to Multiple Support Seismic Excitation

Abstract: The response of general multiple‐degrees‐of‐freedom (MDOF) systems subjected to several components of earthquake at multiple support points is obtained using a random‐vibration formulation. The excitation is modeled as a colored, correlated, vector‐valued, nonstationary random process. A new filter is used, such that the excitation may have multiple predominant frequencies and a wide range of spectral shapes. The time history of the RMS excitation at support points which is the output of the filter is prescribed directly. The corresponding input of the filter, a fictitious piecewise linear strength envelope, is generated accordingly before engaging the filter with the system. This filter also allows the support motion to be prescribed in terms of displacement, velocity, or acceleration, which is not true for other filters that are popular and widely in use. Finally, the correlation between any two components can also be prescribed as a function of time. The concept of “pseudo‐static displacement” used by ...

Latency exploitation in circuit simulation by sparse matrix techniques

Abstract: The most important operations for a circuit simulator are component model linearization, updating the network matrix, performing large unsymmetric decomposition on this matrix, and solving the network variables by forward and backward substitution. Methods are presented to keep all these operations localized to the part of the network that is active at the current time point, thus obtaining a considerable reduction in computational effort. The methods depend upon the sparse matrix structure itself, yielding a very effective fine-grained latency use, contrary to methods based on the large blocks specified by the circuit hierarchy. Results obtained from an implementation of the algorithms in a piecewise linear circuit simulator with an implicit multirate integration scheme are presented. >

Solving Parametric Problems on Trees

Abstract: We consider optimization problems on weighted trees, including weighted versions of the minimum vertex cover and the minimum dominating set problems, where vertex and edge weights vary as linear functions of time. For each of the problems studied here, we derive bounds on the number of breakpoints of the piecewise linear functions describing the cost of the optimum solution and we present efficient methods for constructing these functions. We also consider the problem of determining optimum steady-state solutions and finding the time t∞ at which the steady state is reached.

Piecewise Linear Filtering

Abstract: We consider a nonlinear filtering problem, whose dynamics is “piecewise linear”. We construct a suboptimal filter consisting of a finite number of Kalman filters working in parallel, and exchanging their information at times δ, 2δ, 3δ …. We establish the convergence of that filter towards the optimal filter, as δ → 0.

Parametric maximal flows in generalized networks – complexity and algorithms

Abstract: The problem of determining parametric maximal flows in networks with gains is considered. A worst-case analysis with respect to the number of breakpoints in the optimal objective value function is performed for both parametric flows leaving the source and parametric capacities of the arcs. The result is an exponential growth of the number of breakpoints depending on the number of vertices in the underlying graph. From there, the idea of a horizontal approximation algorithm developed from Hamachee and Foulds [5] is extended to generalized flows. In each iteration the horizontal approach makes an improvement which is a piecewise linear function of the whole parameter interval. This process can be applied up to optimality