Showing papers on "Piecewise linear function published in 2013"

Studies in astronomical time series analysis. vi. bayesian block representations

Abstract: This paper addresses the problem of detecting and characterizing local variability in time series and other forms of sequential data. The goal is to identify and characterize statistically significant variations, at the same time suppressing the inevitable corrupting observational errors. We present a simple nonparametric modeling technique and an algorithm implementing it—an improved and generalized version of Bayesian Blocks [Scargle 1998]—that finds the optimal segmentation of the data in the observation interval. The structure of the algorithm allows it to be used in either a real-time trigger mode, or a retrospective mode. Maximum likelihood or marginal posterior functions to measure model fitness are presented for events, binned counts, and measurements at arbitrary times with known error distributions. Problems addressed include those connected with data gaps, variable exposure, extension to piecewise linear and piecewise exponential representations, multi-variate time series data, analysis of variance, data on the circle, other data modes, and dispersed data. Simulations provide evidence that the detection efficiency for weak signals is close to a theoretical asymptotic limit derived by [Arias-Castro, Donoho and Huo 2003]. In the spirit of Reproducible Research [Donoho et al. (2008)] all of the code and data necessary to reproduce all of the figures in this paper are included as auxiliary material.

Spatial spline regression models

Abstract: Summary We describe a model for the analysis of data distributed over irregularly shaped spatial domains with complex boundaries, strong concavities and interior holes Adopting an approach that is typical of functional data analysis, we propose a spatial spline regression model that is computationally efficient, allows for spatially distributed covariate information and can impose various conditions over the boundaries of the domain Accurate surface estimation is achieved by the use of piecewise linear and quadratic finite elements

Piecewise linear perturbations of a linear center

Abstract: This paper is mainly devoted to study the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line Σ and the singular point of the unperturbed system is in Σ. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirm that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For these last systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.

On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Abstract: Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.

A Coupled Level Set-Moment of Fluid Method for Incompressible Two-Phase Flows

Abstract: A coupled level set and moment of fluid method (CLSMOF) is described for computing solutions to incompressible two-phase flows. The local piecewise linear interface reconstruction (the CLSMOF reconstruction) uses information from the level set function, volume of fluid function, and reference centroid, in order to produce a slope and an intercept for the local reconstruction. The level set function is coupled to the volume-of-fluid function and reference centroid by being maintained as the signed distance to the CLSMOF piecewise linear reconstructed interface. The nonlinear terms in the momentum equations are solved using the sharp interface approach recently developed by Raessi and Pitsch (Annual Research Brief, 2009). We have modified the algorithm of Raessi and Pitsch from a staggered grid method to a collocated grid method and we combine their treatment for the nonlinear terms with the variable density, collocated, pressure projection algorithm developed by Kwatra et al. (J. Comput. Phys. 228:4146---4161, 2009). A collocated grid method makes it convenient for using block structured adaptive mesh refinement (AMR) grids. Many 2D and 3D numerical simulations of bubbles, jets, drops, and waves on a block structured adaptive grid are presented in order to demonstrate the capabilities of our new method.

Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane

Abstract: In this paper we study the existence of limit cycles in a one-parameter family of discontinuous piecewise linear differential systems with two zones in the plane. It is characterized for all the parameter values the number of non-sliding limit cycles of the family studied, detecting a rather non-generic bifurcation leading to the simultaneous generation of three limit cycles.

An Optimal Approximate Dynamic Programming Algorithm for Concave, Scalar Storage Problems With Vector-Valued Controls

Abstract: We prove convergence of an approximate dynamic programming algorithm for a class of high-dimensional stochastic control problems linked by a scalar storage device, given a technical condition. Our problem is motivated by the problem of optimizing energy flows for a power grid supported by grid-level storage. The problem is formulated as a stochastic, dynamic program, where we estimate the value of resources in storage using a piecewise linear value function approximation. Given the technical condition, we provide a rigorous convergence proof for an approximate dynamic programming algorithm, which can capture the presence of both the amount of energy held in storage as well as other exogenous variables. Our algorithm exploits the natural concavity of the problem to avoid any need for explicit exploration policies.

Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics

Abstract: The existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with saddle–saddle dynamics are investigated. Using the Lienard-like canonical form with seven parameters, the parametric regions of the existence of limit cycles are given by constructing proper Poincare maps. In particular, the existence of at least two limit cycles is proved and some parameter regions where two nested limit cycles exist are given.

Sparse/robust estimation and Kalman smoothing with nonsmooth log-concave densities: modeling, computation, and theory

Abstract: We introduce a new class of quadratic support (QS) functions, many of which already play a crucial role in a variety of applications, including machine learning, robust statistical inference, sparsity promotion, and inverse problems such as Kalman smoothing. Well known examples of QS penalties include the l2, Huber, l1 and Vapnik losses. We build on a dual representation for QS functions, using it to characterize conditions necessary to interpret these functions as negative logs of true probability densities. This interpretation establishes the foundation for statistical modeling with both known and new QS loss functions, and enables construction of non-smooth multivariate distributions with specified means and variances from simple scalar building blocks. The main contribution of this paper is a flexible statistical modeling framework for a variety of learning applications, together with a toolbox of efficient numerical methods for estimation. In particular, a broad subclass of QS loss functions known as piecewise linear quadratic (PLQ) penalties has a dual representation that can be exploited to design interior point (IP) methods. IP methods solve nonsmooth optimization problems by working directly with smooth systems of equations characterizing their optimality. We provide several numerical examples, along with a code that can be used to solve general PLQ problems. The efficiency of the IP approach depends on the structure of particular applications. We consider the class of dynamic inverse problems using Kalman smoothing. This class comprises a wide variety of applications, where the aim is to reconstruct the state of a dynamical system with known process and measurement models starting from noisy output samples. In the classical case, Gaussian errors are assumed both in the process and measurement models for such problems. We show that the extended framework allows arbitrary PLQ densities to be used, and that the proposed IP approach solves the generalized Kalman smoothing problem while maintaining the linear complexity in the size of the time series, just as in the Gaussian case. This extends the computational efficiency of the Mayne-Fraser and Rauch-Tung-Striebel algorithms to a much broader nonsmooth setting, and includes many recently proposed robust and sparse smoothers as special cases.

Total variation estimates for the TCP process

Abstract: The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in [0, ∞), is ergodic and irreversible. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. The aim of the present paper is to provide quantitative estimates for the exponential convergence to equilibrium, in terms of the total variation and Wasserstein distances.

Sphere-Meshes: shape approximation using spherical quadric error metrics

Abstract: Shape approximation algorithms aim at computing simple geometric descriptions of dense surface meshes. Many such algorithms are based on mesh decimation techniques, generating coarse triangulations while optimizing for a particular metric which models the distance to the original shape. This approximation scheme is very efficient when enough polygons are allowed for the simplified model. However, as coarser approximations are reached, the intrinsic piecewise linear point interpolation which defines the decimated geometry fails at capturing even simple structures. We claim that when reaching such extreme simplification levels, highly instrumental in shape analysis, the approximating representation should explicitly and progressively model the volumetric extent of the original shape. In this paper, we propose Sphere-Meshes, a new shape representation designed for extreme approximations and substituting a sphere interpolation for the classic point interpolation of surface meshes. From a technical point-of-view, we propose a new shape approximation algorithm, generating a sphere-mesh at a prescribed level of detail from a classical polygon mesh. We also introduce a new metric to guide this approximation, the Spherical Quadric Error Metric in R4, whose minimizer finds the sphere that best approximates a set of tangent planes in the input and which is sensitive to surface orientation, thus distinguishing naturally between the inside and the outside of an object. We evaluate the performance of our algorithm on a collection of models covering a wide range of topological and geometric structures and compare it against alternate methods. Lastly, we propose an application to deformation control where a sphere-mesh hierarchy is used as a convenient rig for altering the input shape interactively.

Acceleration of Univariate Global Optimization Algorithms Working with Lipschitz Functions and Lipschitz First Derivatives

Abstract: This paper deals with two kinds of the one-dimensional global optimization problem over a closed finite interval: (i) the objective function $f(x)$ satisfies the Lipschitz condition with a constant $L$; (ii) the first derivative of $f(x)$ satisfies the Lipschitz condition with a constant $M$. In the paper, six algorithms are presented for the case (i) and six algorithms for the case (ii). In both cases, auxiliary functions are constructed and adaptively improved during the search. In the case (i), piecewise linear functions are constructed and in the case (ii) smooth piecewise quadratic functions are used. The constants $L$ and $M$ either are taken as values known a priori or are dynamically estimated during the search. A recent technique that adaptively estimates the local Lipschitz constants over different zones of the search region is used to accelerate the search. A new technique called the local improvement is introduced in order to accelerate the search in both cases (i) and (ii). The algorithms are...

Annealing between distributions by averaging moments

Abstract: Many powerful Monte Carlo techniques for estimating partition functions, such as annealed importance sampling (AIS), are based on sampling from a sequence of intermediate distributions which interpolate between a tractable initial distribution and the intractable target distribution. The near-universal practice is to use geometric averages of the initial and target distributions, but alternative paths can perform substantially better. We present a novel sequence of intermediate distributions for exponential families defined by averaging the moments of the initial and target distributions. We analyze the asymptotic performance of both the geometric and moment averages paths and derive an asymptotically optimal piecewise linear schedule. AIS with moment averaging performs well empirically at estimating partition functions of restricted Boltzmann machines (RBMs), which form the building blocks of many deep learning models.

Near-optimal piecewise linear fits of static pushover capacity curves for equivalent SDOF analysis

Abstract: analysis. Abstract. The piecewise linear ("multilinear") approximation of realistic force-deformation capacity curves is investigated for structural syst ems incorporating generalized plastic, hard- ening, and negative stiffness behaviors. This fitti ng process factually links capacity and de- mand and lies at the core of nonlinear static asses sment procedures. Despite codification, the various fitting rules used can produce highly heter ogeneous results for the same capacity curve, especially for the highly-curved backbones rfrom the gradual plasticization or the progressive failures of structural elements. To achieve an improved fit, the error intro- duced by the approximation is quantified by studyin g it at the single-degree-of-freedom level, thus avoiding any issues related to multi- versus s ingle-degree-of-freedom realizations. In- cremental Dynamic Analysis is employed to enable a direct comparison of the actual back- bones versus their candidate piecewise linear appro ximations in terms of the spectral acceleration capacity for a continuum of limit-stat es. In all cases, current code-based proce- dures are found to be highly biased wherever widesp read significant stiffness changes occur, generally leading to very conservative estimates of performance. The practical rules deter- mined allow, instead, the definition of standardize d low-bias bilinear, trilinear, or quadrilin- ear approximations, regardless of the details of th e capacity curve shape.

Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation

Abstract: In this paper, we provide a lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line. Here, we only consider nonsliding limit cycles. For those systems, the interior of any limit cycle only contains a unique equilibrium point or a unique sliding segment. Moreover, the linear differential systems that we consider in every half-plane can have either a focus (F), or a node (N), or a saddle (S), these equilibrium points can be real or virtual. Then, we can consider six kinds of planar discontinuous piecewise linear differential systems: FF, FN, FS, NN, NS, SS. We provide for each of these types of discontinuous differential systems examples with two limit cycles.

Nonsmooth Bifurcations, Transient Hyperchaos and Hyperchaotic Beats in a Memristive Murali-Lakshmanan-Chua Circuit

Abstract: In this paper, a memristive Murali-Lakshmanan-Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three segment piecewise linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuty boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity induced non-smooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyper chaotic nature. In addition the circuit admits a codimension-5 bifurcation and transient hyper chaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0-1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincare Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.

Computing Reeb Graphs as a Union of Contour Trees

Abstract: The Reeb graph of a scalar function tracks the evolution of the topology of its level sets. This paper describes a fast algorithm to compute the Reeb graph of a piecewise-linear (PL) function defined over manifolds and non-manifolds. The key idea in the proposed approach is to maximally leverage the efficient contour tree algorithm to compute the Reeb graph. The algorithm proceeds by dividing the input into a set of subvolumes that have loop-free Reeb graphs using the join tree of the scalar function and computes the Reeb graph by combining the contour trees of all the subvolumes. Since the key ingredient of this method is a series of union-find operations, the algorithm is fast in practice. Experimental results demonstrate that it outperforms current generic algorithms by a factor of up to two orders of magnitude, and has a performance on par with algorithms that are catered to restricted classes of input. The algorithm also extends to handle large data that do not fit in memory.

Model reference adaptive control of discrete‐time piecewise linear systems

Abstract: SUMMARY This article presents a switched model reference adaptive controller for discrete-time piecewise linear systems. In the spirit of the work by Landau in the late seventies, proof of asymptotic stability of the closed-loop error system is obtained, recasting its dynamics as a feedback system and showing the feedforward and the feedback paths are both passive. The challenge is that both paths can be piecewise linear. Numerical results show excellent performance of the proposed controller even in the face of sudden variations of the plant parameters. Copyright © 2012 John Wiley & Sons, Ltd.

Human Detection in Images via Piecewise Linear Support Vector Machines

Abstract: Human detection in images is challenged by the view and posture variation problem. In this paper, we propose a piecewise linear support vector machine (PL-SVM) method to tackle this problem. The motivation is to exploit the piecewise discriminative function to construct a nonlinear classification boundary that can discriminate multiview and multiposture human bodies from the backgrounds in a high-dimensional feature space. A PL-SVM training is designed as an iterative procedure of feature space division and linear SVM training, aiming at the margin maximization of local linear SVMs. Each piecewise SVM model is responsible for a subspace, corresponding to a human cluster of a special view or posture. In the PL-SVM, a cascaded detector is proposed with block orientation features and a histogram of oriented gradient features. Extensive experiments show that compared with several recent SVM methods, our method reaches the state of the art in both detection accuracy and computational efficiency, and it performs best when dealing with low-resolution human regions in clutter backgrounds.

Residual-based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

Abstract: This work develops finite element methods with high order stabilization, and robust and efficient adaptive algorithms for Large Eddy Simulation of turbulent compressible flows. The equations are approximated by continuous piecewise linear functions in space, and the time discretization is done in implicit/explicit fashion: the second order Crank-Nicholson method and third/fourth order explicit Runge-Kutta methods. The full residual of the system and the entropy residual, are used in the construction of the stabilization terms. These methods are consistent for the exact solution, conserves all the quantities, such as mass, momentum and energy, is accurate and very simple to implement. We prove convergence of the method for scalar conservation laws in the case of an implicit scheme. The convergence analysis is based on showing that the approximation is uniformly bounded, weakly consistent with all entropy inequalities, and strongly consistent with the initial data. The convergence of the explicit schemes is tested in numerical examples in 1D, 2D and 3D. To resolve the small scales of the flow, such as turbulence fluctuations, shocks, discontinuities and acoustic waves, the simulation needs very fine meshes. In this thesis, a robust adjoint based adaptive algorithm is developed for the time-dependent compressible Euler/Navier-Stokes equations. The adaptation is driven by the minimization of the error in quantities of interest such as stresses, drag and lift forces, or the mean value of some quantity. The implementation and analysis are validated in computational tests, both with respect to the stabilization and the duality based adaptation.

An Efficient Algorithm for Computing the HHSVM and Its Generalizations

Abstract: The hybrid Huberized support vector machine (HHSVM) has proved its advantages over the l1 support vector machine (SVM) in terms of classification and variable selection. Similar to the l1 SVM, the HHSVM enjoys a piecewise linear path property and can be computed by a least-angle regression (LARS)-type piecewise linear solution path algorithm. In this article, we propose a generalized coordinate descent (GCD) algorithm for computing the solution path of the HHSVM. The GCD algorithm takes advantage of a majorization–minimization trick to make each coordinatewise update simple and efficient. Extensive numerical experiments show that the GCD algorithm is much faster than the LARS-type path algorithm. We further extend the GCD algorithm to solve a class of elastic net penalized large margin classifiers, demonstrating the generality of the GCD algorithm. We have implemented the GCD algorithm in a publicly available R package gcdnet.

Continuous curvature path-smoothing algorithm using cubic B zier spiral curves for non-holonomic robots

Abstract: This paper presents a path-smoothing algorithm over the piecewise linear path for non-holonomic robots. Based on the upper-bounded continuous curvature path-smoothing algorithm, three algorithms are proposed to enhance the path smoothing performance. First, an interactive algorithm, which fully utilizes extra distance margins of linear path, is suggested. Second, a bisection algorithm is proposed to relieve the violation of the maximum curvature constraints. Finally, an interpolating path-smoothing algorithm which passes intermediate points is suggested. Simulation results show the validity of the suggested algorithms.