Showing papers on "Piecewise linear function published in 2009"

$\ell_1$ Trend Filtering

Abstract: The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on Hodrick-Prescott (H-P) filtering, a widely used method for trend estimation. The proposed $\ell_1$ trend filtering method substitutes a sum of absolute values (i.e., $\ell_1$ norm) for the sum of squares used in H-P filtering to penalize variations in the estimated trend. The $\ell_1$ trend filtering method produces trend estimates that are piecewise linear, and therefore it is well suited to analyzing time series with an underlying piecewise linear trend. The kinks, knots, or changes in slope of the estimated trend can be interpreted as abrupt changes or events in the underlying dynamics of the time series. Using specialized interior-point methods, $\ell_1$ trend filtering can be carried out with not much more effort than H-P filtering; in particular, the number of arithmetic operations required grows linearly with the number of data points. We describe the method and some of its basic properties and give some illustrative examples. We show how the method is related to $\ell_1$ regularization-based methods in sparse signal recovery and feature selection, and we list some extensions of the basic method.

Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces

Abstract: We define higher-order analogues to the piecewise linear surface finite element method studied in [G. Dziuk, “Finite elements for the Beltrami operator on arbitrary surfaces,” in Partial Differential Equations and Calculus of Variations, Springer-Verlag, Berlin, 1988, pp. 142-155] and prove error estimates in both pointwise and $L_2$-based norms. Using the Laplace-Beltrami problem on an implicitly defined surface $\Gamma$ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to $\Gamma$ which likewise are of arbitrary degree. Then we prove a priori error estimates in the $L_2$, $H^1$, and corresponding pointwise norms that demonstrate the interaction between the “PDE error” that arises from employing a finite-dimensional finite element space and the “geometric error” that results from approximating $\Gamma$. We also consider parametric finite element approximations that are defined on $\Gamma$ and thus induce no geometric error. Computational examples confirm the sharpness of our error estimates.

Tighter Approximated MILP Formulations for Unit Commitment Problems

Abstract: The short-term unit commitment (UC) problem in hydrothermal power generation is a large-scale, mixed-integer nonlinear program, which is difficult to solve efficiently, especially for large-scale instances. It is possible to approximate the nonlinear objective function of the problem by means of piecewise-linear functions, so that UC can be approximated by an mixed-integer linear program (MILP); applying the available efficient general-purpose MILP solvers to the resulting formulations, good quality solutions can be obtained in a relatively short amount of time. We build on this approach, presenting a novel way to approximating the nonlinear objective function based on a recently developed class of valid inequalities for the problem, called ldquoperspective cuts.rdquo At least for many realistic instances of a general basic formulation of UC, an MILP-based heuristic obtains comparable or slightly better solutions in less time when employing the new approach rather than the standard piecewise linearizations, while being not more difficult to implement and use. Furthermore, ldquodynamicrdquo formulations, whereby the approximation is iteratively improved, provide even better results if the approximation is appropriately controlled.

Target tracking with binary proximity sensors

Abstract: We explore fundamental performance limits of tracking a target in a two-dimensional field of binary proximity sensors, and design algorithms that attain those limits while providing minimal descriptions of the estimated target trajectory. Using geometric and probabilistic analysis of an idealized model, we prove that the achievable spatial resolution in localizing a target's trajectory is of the order of 1/ρR, where R is the sensing radius and ρ is the sensor density per unit area. We provide a geometric algorithm for computing an economical (in descriptive complexity) piecewise linear path that approximates the trajectory within this fundamental limit of accuracy. We employ analogies between binary sensing and sampling theory to contend that only a “lowpass” approximation of the trajectory is attainable, and explore the implications of this observation for estimating the target's velocity. We also consider nonideal sensing, employing particle filters to average over noisy sensor observations, and geometric geometric postprocessing of the particle filter output to provide an economical piecewise linear description of the trajectory. In addition to simulation results validating our approaches for both idealized and nonideal sensing, we report on lab-scale experiments using motes with acoustic sensors.

B-Spline Modeling of Road Surfaces With an Application to Free-Space Estimation

Abstract: We propose a general technique for modeling the visible road surface in front of a vehicle. The common assumption of a planar road surface is often violated in reality. A workaround proposed in the literature is the use of a piecewise linear or quadratic function to approximate the road surface. Our approach is based on representing the road surface as a general parametric B-spline curve. The surface parameters are tracked over time using a Kalman filter. The surface parameters are estimated from stereo measurements in the free space. To this end, we adopt a recently proposed road-obstacle segmentation algorithm to include disparity measurements and the B-spline road-surface representation. Experimental results in planar and undulating terrain verify the increase in free-space availability and accuracy using a flexible B-spline for road-surface modeling.

Computational Comparison of Piecewise−Linear Relaxations for Pooling Problems

Abstract: This work discusses alternative relaxation schemes for the pooling problem, a theoretically and practically interesting optimization problem. The problem nonconvexities appear in the form of bilinear terms and can be addressed with the relaxation technique based on the bilinear convex and concave envelopes. We explore ways to improve the relaxation tightness, and thus the efficiency of a global optimization algorithm, by employing a piecewise linearization scheme that partitions the original domain of the variables involved and applies the principles of bilinear relaxation for each one of the resulting subdomains. We employ 15 different piecewise relaxation schemes with mixed-integer representations and conduct a comprehensive computational comparison study over a collection of benchmark pooling problems. For each case, various partitioning variants can be envisioned, cumulatively accounting for a total of 56 700 relaxations. The results demonstrate that some of the schemes are clearly superior to their c...

Online piece-wise linear approximation of numerical streams with precision guarantees

Abstract: Continuous "always-on" monitoring is beneficial for a number of applications, but potentially imposes a high load in terms of communication, storage and power consumption when a large number of variables need to be monitored. We introduce two new filtering techniques, swing filters and slide filters, that represent within a prescribed precision a time-varying numerical signal by a piecewise linear function, consisting of connected line segments for swing filters and (mostly) disconnected line segments for slide filters. We demonstrate the effectiveness of swing and slide filters in terms of their compression power by applying them to a real-life data set plus a variety of synthetic data sets. For nearly all combinations of signal behavior and precision requirements, the proposed techniques outperform the earlier approaches for online filtering in terms of data reduction. The slide filter, in particular, consistently dominates all other filters, with up to twofold improvement over the best of the previous techniques.

Simple and Efficient Algorithms for Computing Smooth, Collision-free Feedback Laws Over Given Cell Decompositions

Abstract: This paper presents a novel approach to computing feedback laws in the presence of obstacles. Instead of computing a trajectory between a pair of initial and goal states, our algorithms compute a vector field over the entire state space; all trajectories obtained from following this vector field are guaranteed to asymptotically reach the goal state. As a result, the vector field globally solves the navigation problem and provides robustness to disturbances in sensing and control. The vector field's integral curves (system trajectories) are guaranteed to avoid obstacles and are C∞ smooth. We construct a vector field with these properties by partitioning the space into simple cells, defining local vector fields for each cell, and smoothly interpolating between them to obtain a global vector field. We present an algorithm that computes these feedback controls for a kinematic point robot in an arbitrary dimensional space with piecewise linear boundary; the algorithm requires minimal preprocessing of the environment and is extremely fast during execution. For many practical applications in two-dimensional environments, full computation can be done in milliseconds. We also present an algorithm for computing feedback laws over cylindrical algebraic decompositions, thereby solving a smooth feedback version of the generalized piano movers' problem.

Efficient algorithms for computing Reeb graphs

Abstract: The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)^3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)^3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n^2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

Piecewise linear relaxation of bilinear programs using bivariate partitioning

Abstract: Several operational and synthesis problems of practical interest involve bilinear terms. Commercial global solvers such as BARON appear ineffective at solving some of these problems. Although recent literature has shown the potential of piecewise linear relaxation via ab initio partitioning of variables for such problems, several issues such as how many and which variables to partition, which partitioning scheme(s) and relaxation model(s) to use, placement of grid points, etc., need detailed investigation. To this end, we present a detailed numerical comparison of univariate and bivariate partitioning schemes. We compare several models for the two schemes based on different formulations such as incremental cost (IC), convex combination (CC), and special ordered sets (SOS). Our evaluation using four process synthesis problems shows a formulation using SOS1 variables to perform the best for both partitioning schemes. It also points to the potential usefulness of a 2-segment bivariate partitioning scheme for the global optimization of bilinear programs. We also prove some simple results on the number and selection of partitioned variables and the advantage of uniform placement of grid points (identical segment lengths for partitioning). © 2009 American Institute of Chemical Engineers AIChE J, 2010

Convergence of Time-Stepping Schemes for Passive and Extended Linear Complementarity Systems

Abstract: Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of “least-norm solutions” in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established.

An Optimal Approximate Dynamic Programming Algorithm for the Lagged Asset Acquisition Problem

Abstract: We consider a multistage asset acquisition problem where assets are purchased now, at a price that varies randomly over time, to be used to satisfy a random demand at a particular point in time in the future. We provide a rare proof of convergence for an approximate dynamic programming algorithm using pure exploitation, where the states we visit depend on the decisions produced by solving the approximate problem. The resulting algorithm does not require knowing the probability distribution of prices or demands, nor does it require any assumptions about its functional form. The algorithm and its proof rely on the fact that the true value function is a family of piecewise linear concave functions.

Iterative Solution of Piecewise Linear Systems and Applications to Flows in Porous Media

Abstract: The correct numerical modeling of free-surface hydrodynamic problems often requires to have the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In doing so, one may fulfill relevant physical constraints. The existence, the uniqueness, and two constructive iterative methods to solve a piecewise linear system of the form $\max[\boldsymbol{l},\min(\boldsymbol{u},\mathbf{x})]+T\mathbf{x}=\mathbf{b}$ are analyzed. The methods are shown to have a finite termination property; i.e., they converge to an exact solution in a finite number of steps and, actually, they converge very quickly, as confirmed by a few numerical tests, which are derived from the mathematical modeling of flows in porous media.

Adaptive Predistortion With Direct Learning Based on Piecewise Linear Approximation of Amplifier Nonlinearity

Abstract: We propose an efficient Wiener model for a power amplifier (PA) and develop a direct learning predistorter (PD) based on the model. The Wiener model is formed by a linear filter and a memoryless nonlinearity in which AM/AM and AM/PM characteristics are approximated as piecewise linear and piecewise constant functions, respectively. A two-step identification scheme, wherein the linear portion is estimated first and the nonlinear portion is then identified, is developed. The PD is modeled by a polynomial and its coefficients are directly updated using a recursive least squares (RLS) algorithm. To avoid implementing the inverse of the PA's linear portion, the cost function for the RLS algorithm is defined as the sum of differences between the output of the PA's linear portion and the inverse of the PA's nonlinear portion. The proposed direct learning scheme, which is referred to as the piecewise RLS (PWRLS) algorithm, is simpler to implement, yet exhibits comparable performance, as compared with existing direct learning schemes.

An Anisotropic Error Estimator for the Crank-Nicolson Method: Application to a Parabolic Problem

Abstract: In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the Crank-Nicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure. The error due to the time discretization is obtained using two different continuous, piecewise quadratic time reconstructions. The first reconstruction is developed following G. Akrivis, C. Makridakis, and R. H. Nochetto [Math. Comp., 75 (2006), pp. 511-531], while the second one is new. Moreover, in the case of isotropic meshes only, upper and lower bounds are provided as in [R. Verfurth, Calcolo, 40 (2003), pp. 195-212]. An adaptive algorithm is developed. Numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one. In particular, the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm.

Decentralised coordination of continuously valued control parameters using the max-sum algorithm

Abstract: In this paper we address the problem of decentralised coordination for agents that must make coordinated decisions over continuously valued control parameters (as is required in many real world applications). In particular, we tackle the social welfare maximisation problem, and derive a novel continuous version of the max-sum algorithm. In order to do so, we represent the utility function of the agents by multivariate piecewise linear functions, which in turn are encoded as simplexes. We then derive analytical solutions for the fundamental operations required to implement the max-sum algorithm (specifically, addition and marginal maximisation of general n-ary piecewise linear functions). We empirically evaluate our approach on a simulated network of wireless, energy constrained sensors that must coordinate their sense/sleep cycles in order to maximise the system-wide probability of event detection. We compare the conventional discrete max-sum algorithm with our novel continuous version, and show that the continuous approach obtains more accurate solutions (up to a 10% increase) with a lower communication overhead (up to half of the total message size).

Superconvergence for Optimal Control Problems Governed by Semi-linear Elliptic Equations

Abstract: In this paper, we will investigate the superconvergence of the finite element approximation for quadratic optimal control problem governed by semi-linear elliptic equations. The state and co-state variables are approximated by the piecewise linear functions and the control variable is approximated by the piecewise constant functions. We derive the superconvergence properties for both the control variable and the state variables. Finally, some numerical examples are given to demonstrate the theoretical results.

Optimization Method for Highway Horizontal Alignment Design

Abstract: This research presents an optimization heuristic to solve the horizontal alignment of a highway segment. The iterative heuristic works in two stages. The first stage uses a neighborhood search approach to find a good piecewise linear line that approximates the highway alignment. The second stage further adjusts the alignment so that the external and code requirements are accurately satisfied. Both stages manipulate the piecewise linear line with a neighborhood search heuristic, and use a mixed integer program (MIP) to ensure that the piecewise linear line crosses the control areas and avoids the restricted ones. The optimal objective function value returned by the MIP is used to compare the quality between different piecewise linear lines. The lengths of each line segment are properly constrained in the MIP to ensure that curves can be correctly deployed. Starting from an initial feasible solution the process gradually improves the alignment through iterations. Computational examples are provided.

On optimal and near-optimal turbo decoding using generalized max operator

Abstract: Motivated by a recently published robust geometric programming approximation, a generalized approach for approximating efficiently the max* operator is presented. Using this approach, the max* operator is approximated by means of a generic and yet very simple max operator, instead of using additional correction term as previous approximation methods require. Following that, several turbo decoding algorithms are obtained with optimal and near-optimal bit error rate (BER) performance depending on a single parameter, namely the number of piecewise linear (PWL) approximation terms. It turns out that the known max-log-MAP algorithm can be viewed as special case of this new generalized approach. Furthermore, the decoding complexity of the most popular previously published methods is estimated, for the first time, in a unified way by hardware synthesis results, showing the practical implementation advantages of the proposed approximations against these methods.

Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems

Abstract: We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart- Thomas-Nedelec space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincare, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.

A Priori Error Estimates for Optimal Control Problems Governed by Transient Advection-Diffusion Equations

Abstract: In this paper, we investigate a characteristic finite element approximation of quadratic optimal control problems governed by linear advection-dominated diffusion equations, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by piecewise constant functions. We derive some a priori error estimates for both the control and state approximations. It is proved that these approximations have convergence order $\mathcal{O}(h_{U}+h+k)$ , where h U and h are the spatial mesh-sizes for the control and state discretization, respectively, and k is the time increment. Numerical experiments are presented, which verify the theoretical results.

Initial value problems for lattice equations

Abstract: We describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils. In finitely many directions, we arrive at discrete Goursat problems and in the remaining directions we find Cauchy problems. Next, we consider (s1, s2)-periodic initial value problems. In the Goursat directions, the periodic solutions are generated by correspondences. In the Cauchy directions, assuming the equation to be multi-linear, the periodic solution can be obtained uniquely by iteration of a particularly simple mapping, whose dimension is a piecewise linear function of s1, s2.

Controllability and observability for a class of time-varying impulsive systems☆

Abstract: This paper is concerned with the controllability and observability for a class of piecewise linear time-varying impulsive systems. Several sufficient and necessary conditions for state controllability and observability of such systems are established. Meanwhile, corresponding criteria for time-invariant impulsive systems are also obtained and the criteria are compared with the existing results.

Analysis of Partially Shaded PV Modules Using Piecewise Linear Parallel Branches Model

Abstract: This paper presents an equivalent circuit model based on piecewise linear parallel branches (PLPB) to study solar cell modules which are partially shaded. The PLPB model can easily be used in circuit simulation software such as the ElectroMagnetic Transients Program (EMTP). This PLPB model allows the user to simulate several different configurations of solar cells, the influence of partial shadowing on a single or multiple cells, the influence of the number of solar cells protected by a bypass diode and the effect of the cell connection configuration on partial shadowing. Keywords—Cell Connection Configurations, EMTP, Equivalent Circuit, Partial Shading, Photovoltaic Module.

Digital implementation of the sigmoid function for fpga circuits

Abstract: In this paper, is proposed a method to implement in FPGA (Field Programmable Gate Array) circuits different approximation of the sigmoid function. Three previously published piecewise linear and one piecewise second-order approximation are analyzed from point of view of hardware resources utilization, induced errors caused by the approximation function and bits representation, power consumption and speed processing. The major benefit of the proposed method resides in the possibility to design neural networks by means of predefined block systems created in System Generator environment and the possibility to create a higher level design tools used to implement neural networks in logical circuits.