Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

Modified sum-product algorithms for decoding low-density parity-check codes

Abstract: The authors deal with the sum-product algorithm (SPA) based on the hyperbolic tangent (tanh) rule when it is applied for decoding low-density parity-check (LDPC) codes. Motivated by the finding that, because of the large number of multiplications required by the algorithm, an overflow in the decoder may occur, two novel modifications of the tanh function (and its inverse) are proposed. By means of computer simulations, both methods are evaluated using random-based LDPC codes with binary phase shift keying (BPSK) signals transmitted over the additive white Gaussian noise (AWGN) channel. It is shown that the proposed modifications improve the bit error rate (BER) performance up to 1 dB with respect to the conventional SPA. These results have also shown that the error floor is removed at BER lower than 10 -6 . Furthermore, two novel approximations are presented to reduce the computational complexity of the tanh function (and its inverse), based on either a piecewise linear function or a quantisation table. It is shown that the proposed approximations can slightly improve the BER performance (up to 0.13 dB) in the former case, whereas small BER performance degradation is observed (<0.25 dB) in the latter case. In both cases, however, the decoding complexity is reduced significantly

Error Analysis for Convex Separable Programs: The Piecewise Linear Approximation and The Bounds on The Optimal Objective Value

Abstract: The bounds have been established on the possible deviation of the optimal objective value of a separable, convex program from the optimal objective value of a program which is its piecewise linear approximation based on a given subdivision interval. By a separable, convex program is meant a separable program with proper convexity-concavity properties which imply that any local optimum is also a global optimum. It is further shown that these bounds are actually attainable, and therefore, cannot be improved in general. Some examples are provided.Naturally, the inquiry requires some study of the piecewise linear approximation itself. The bounds on the function error are determined based on the assumptions—the boundedness of the first, or the second, or both the derivatives—about the original function. Some results are derived for a given piecewise linear function, to determine the nature of the original function with minimum error and satisfying certain conditions; these results would be applicable in those ...

A two-level domain decomposition method for image restoration

Abstract: Image restoration has drawn much attention in recent years and a surge of research has been done on variational models and their numerical studies. However, there remains an urgent need to develop fast and robust methods for solving the minimization problems and the underlying nonlinear PDEs to process images of moderate to large size. This paper aims to propose a two-level domain decomposition method, which consists of an overlapping domain decomposition technique and a coarse mesh correction, for directly solving the total variational minimization problems. The iterative algorithm leads to a system of small size and better conditioning in each subspace, and is accelerated with a piecewise linear coarse mesh correction. Various numerical experiments and comparisons demonstrate that the proposed method is fast and robust particularly for images of large size.

Wiedemann Revisited: New Trajectory Filtering Technique and Its Implications for Car-Following Modeling

Abstract: A new data-driven stochastic car-following model based on the principles of psychospacing or action-point modeling is presented. It uses empirical or experimental trajectory data and mimics the main microscopic behavioral characteristics present in the data. In the action-point model, regions are defined in the relative speed–distance headway plane, in which the follower is likely to perform an action (increase or decrease acceleration) or not. These regions can be established empirically from vehicle trajectory data and thereby yield a joint cumulative probability distribution function of the action points. Furthermore, the conditional distribution of the actions (the size of the acceleration or deceleration given the current distance headway and relative speed or given the acceleration before the action) can be determined from these data as well. To assess the data correctly, a new filtering technique is proposed. The main hypothesis behind this idea is that the speed profile is a continuous piecewise linear function: accelerations are piecewise constant changing values at nonequidistant discrete time instants. The durations of these constant acceleration periods are not fixed but depend on the state of the follower in relation to its leader. The data analysis illustrates that driving behavior shows nonequidistant constant acceleration periods. The distributions of the action points and the conditional accelerations form the core of the presented data-driven stochastic model. The mathematical formalization that describes how these distributions can be used to simulate car-following behavior is presented. Empirical data collected on a Dutch motorway are used to illustrate the workings of the approach and the simulation results.

An approximately divergence‐free 9‐node velocity element (with variations) for incompressible flows

Abstract: Explicit basis functions are constructed for 9-node biquadratic velocity fields which guarantee that a weak form of the continuity equation is satisfied. The corresponding pressure approximations are either piecewise constant, piecewise linear or piecewise bilinear. These results are extended to give bases for bilinear velocity/piecewise constant pressure elements and also to some three-dimensional brick elements.