Showing papers on "Piecewise linear function published in 2011"
TL;DR: Sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty and a Lyapunov function, in quadratic form, which is non-increasing at the switching instants is assigned to each subsystem.
Abstract: Sufficient conditions are given for the stability of linear switched systems with dwell time and with polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, which is non-increasing at the switching instants is assigned to each subsystem. During the dwell time, this function varies piecewise linearly in time after switching occurs. It becomes time invariant afterwards. This function leads to asymptotic stability conditions for the nominal set of subsystems that can be readily extended to the case where these subsystems suffer from polytopic type parameter uncertainties. The method proposed is then applied to stabilization via state-feedback both for the nominal and the uncertain cases.
333 citations
TL;DR: It is proved that the new formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions have favorable tightness properties and can significantly outperform other mixed integer binary formulations.
Abstract: Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra. Using these conditions we introduce mixed integer binary formulations for SOS1 and SOS2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.
229 citations
TL;DR: In this article, the authors refer to the generalized piecewise linear (GPL) loss function, which nests the asymmetric piece-wise linear loss, and show that the level of the quantile depends on a generic asymmetry parameter which reflects the possibly distinct costs of underprediction and overprediction.
204 citations
TL;DR: In this paper, a multistage planning problem of a distribution network is considered, where the objective function is the net present value of the investment cost to add, reinforce or replace feeders and substations, losses cost, and operation and maintenance cost.
Abstract: This paper presents a model for solving the multistage planning problem of a distribution network. The objective function to be minimized is the net present value of the investment cost to add, reinforce or replace feeders and substations, losses cost, and operation and maintenance cost. The model considers three levels of load in each node and two investment alternatives for each resource to be added, reinforced or replaced. The nonlinear objective function is approximated by a piecewise linear function, resulting in a mixed integer linear model that is solved using standard mathematical programming. The model allows us to find multiple solutions in addition to the optimal one, helping the decision maker to analyze and choose from a pool of solutions. In addition to the optimization problem, reliability indexes and associated costs are computed for each solution, based on the regulation model used in Brazil. Numerical results and discussion are presented for an illustrative 27-node test network.
155 citations
TL;DR: Inspired by recent work, a formulation for the piecewise linear relaxation of bilinear functions with a logarithmic number of binary variables is introduced and computationally compare the performance of this new formulation to the best-performing piecewise relaxations with a linear number ofbinary variables.
134 citations
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15 Sep 2011
TL;DR: This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry and finds problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points.
Abstract: This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry. Among these applications we find problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points. All results are either combinatorial or computational in nature. They are based on the combinatorial analysis in two companion papers [PS] and [E2] and a divide-and-conquer algorithm for computing envelopes described in this paper.
132 citations
TL;DR: Theoretical analysis and computer simulation indicate that the proposed algorithm presents several interesting features, such as high message and key sensitivity, good statistical properties, collision resistance and secure against meet-in-the-middle attacks, which can satisfy the performance requirements of Hash function.
Abstract: An algorithm for constructing a one-way novel Hash function based on two-layer chaotic neural network structure is proposed. The piecewise linear chaotic map (PWLCM) is utilized as transfer function, and the 4-dimensional and one-way coupled map lattices (4D OWCML) is employed as key generator of the chaotic neural network. Theoretical analysis and computer simulation indicate that the proposed algorithm presents several interesting features, such as high message and key sensitivity, good statistical properties, collision resistance and secure against meet-in-the-middle attacks, which can satisfy the performance requirements of Hash function.
121 citations
TL;DR: In this article, it was shown that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBED(d-1)→d is undecidable for each d ≥ 5.
Abstract: Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into Rd? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3 (even if k is not considered fixed).We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBEDd→d and EMBED(d-1)→d are undecidable for each d ≥ 5. Our main result is NP-hardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d → 4 and d → k → (2d - 2)/3.
98 citations
TL;DR: In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost.
Abstract: A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS based on sparse models, where N is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximization-maximization (Max-Max) algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost.
98 citations
TL;DR: A transfer matrix formalism is developed to visualize the framing of discrete piecewise linear curves in three-dimensional space and draws attention to the conceptual similarity between inflection points where the curvature vanishes and topologically stable solitons.
Abstract: We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three-dimensional space. Our approach is based on the concept of an intrinsically discrete cur ...
97 citations
TL;DR: A new divergence-free finite element on 3D Powell–Sabin grids is constructed for Stokes equations, where the velocity is approximating by continuous piecewise quadratic polynomials while the pressure is approximated by discontinuous piecewise linear polynmials on the same grid.
Abstract: Given a tetrahedral grid in 3D, a Powell---Sabin grid can be constructed by refining each original tetrahedron into 12 subtetrahedra. A new divergence-free finite element on 3D Powell---Sabin grids is constructed for Stokes equations, where the velocity is approximated by continuous piecewise quadratic polynomials while the pressure is approximated by discontinuous piecewise linear polynomials on the same grid. To be precise, the finite element space for the pressure is exactly the divergence of the corresponding space for the velocity. Therefore, the resulting finite element solution for the velocity is pointwise divergence-free, including the inter-element boundary. By establishing the inf-sup condition, the finite element is stable and of the optimal order. Numerical tests are provided.
TL;DR: This work analyzes numerically the role played by the asymmetry of a piecewise linear potential, in the presence of both a Gaussian white noise and a dichotomous noise, on the resonant activation phenomenon.
Abstract: This work analyzes numerically the role played by the asymmetry of a piecewise linear potential, in the presence of both a Gaussian white noise and a dichotomous noise, on the resonant activation phenomenon. The features of the asymmetry of the potential barrier arise by investigating the stochastic transitions far behind the potential maximum, from the initial well to the bottom of the adjacent potential well. Because of the asymmetry of the potential profile together with the random external force uniform in space, we find, for the different asymmetries: (1) an inversion of the curves of the mean first passage time in the resonant region of the correlation time τ of the dichotomous noise, for low thermal noise intensities; (2) a maximum of the mean velocity of the Brownian particle as a function of τ; and (3) an inversion of the curves of the mean velocity and a very weak current reversal in the miniratchet system obtained with the asymmetrical potential profiles investigated. An inversion of the mean first passage time curves is also observed by varying the amplitude of the dichotomous noise, behavior confirmed by recent experiments.
TL;DR: This technical note addresses optimal control and scheduling (controlled switching) of discrete-time switched linear systems by relaxed dynamic programming, reducing complexity while relaxing optimality within prespecified bounds.
Abstract: This technical note addresses optimal control and scheduling (controlled switching) of discrete-time switched linear systems. A receding-horizon control and scheduling (RHCS) problem is introduced and solved by dynamic programming, leading to a combinatorial optimization problem with exponential complexity. By relaxed dynamic programming, complexity is reduced while relaxing optimality within prespecified bounds. The resulting RHCS strategy is expressed explicitly as a piecewise linear state feedback control law defined over regions implied by quadratic forms. Closed-loop stability is not guaranteed inherently for the RHCS strategy. Therefore, a posteriori stability criteria based on piecewise quadratic Lyapunov functions are proposed. Finally, a region-reachability criterion is presented.
TL;DR: In this paper, the authors studied the dynamics of a point particle in a periodic array of spherical scatterers and constructed a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (BoltzmannGrad limit).
Abstract: We study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (BoltzmannGrad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.
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15 Sep 2011TL;DR: The overall combinatorial complexity of the boundary ofC is at mostO(nd+1−ɛ(d+1)>0.5 for some fixed constantɚ(d-1) >0.1.
Abstract: Letf1, ...,fmbe (partially defined) piecewise linear functions ofd variables whose graphs consist ofn d-simplices altogether. We show that the maximal number ofd-faces comprising the upper envelope (i.e., the pointwise maximum) of these functions isO(nd?(n)), where?(n) denotes the inverse of the Ackermann function, and that this bound is tight in the worst case. If, instead of the upper envelope, we consider any single connected componentC enclosed byn d-simplices (or, more generally, (d ? 1)-dimensional compact convex sets) in ?d+1, then we show that the overall combinatorial complexity of the boundary ofC is at mostO(nd+1??(d+1)) for some fixed constant?(d+1)>0.
TL;DR: In this article, a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2D is presented, which makes use of the global infsup condition, Helmholtz decompositions in both media, and local approximation properties of the Clement interpolant and Raviart-Thomas operator.
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TL;DR: In this article, the authors characterize optimal tax parameters for the cases in which budget sets are convex and nonconvex respectively, and a numerical analysis of a discrete version of the model shows the circumstances under which each of these cases will hold as a global optimum.
Abstract: Given its significance in practice, piecewise linear taxation has received relatively little attention in the literature. This paper offers a simple and transparent analysis of its main characteristics. We fully characterize optimal tax parameters for the cases in which budget sets are convex and nonconvex respectively. A numerical analysis of a discrete version of the model shows the circumstances under which each of these cases will hold as a global optimum. We find that, given plausible parameter values and wage distributions, the globally optimal tax system is convex, and marginal rate progressivity increases with rising inequality.
TL;DR: In this paper, a systematic approach is developed to identify the bivariate relation of two fundamental traffic variables, traffic volume and density, from single-loop detector data, motivated by the observation of a peculiar feature of traffic fluctuations.
Abstract: A systematic approach is developed to identify the bivariate relation of two fundamental traffic variables, traffic volume and density, from single-loop detector data. The approach is motivated by the observation of a peculiar feature of traffic fluctuations. That is, in a short time, traffic usually experiences fluctuations without a significant change in speed. This fact is used to define equilibrium in a new manner, and a mixed integer programming approach is proposed for constructing a piecewise linear fundamental diagram (FD) accordingly. By construction, the proposed method is data adaptive and optimal in the sense of least absolute deviation. This method is used to perform a case study with data from one section of a multilane freeway. The results indicate that both capacity drop and concave–convex FD shapes abound in practice. Differences in traffic behavior across freeway lanes and along freeway sections revealed through the FD are discussed.
TL;DR: In this paper, the authors provided asymptotic estimates for the transition probabilities of a Markov process with memory two for a point particle in a periodic array of spherical scatterers.
Abstract: The dynamics of a point particle in a periodic array of spherical scatterers converges, in the limit of small scatterer size, to a random flight process, whose paths are piecewise linear curves generated by a Markov process with memory two. The corresponding transport equation is distinctly different from the linear Boltzmann equation observed in the case of a random configuration of scatterers. In the present paper we provide asymptotic estimates for the transition probabilities of this Markov process. Our results in particular sharpen previous upper and lower bounds on the distribution of free path lengths obtained by Bourgain, Golse and Wennberg.
TL;DR: Digital architectures for the circuit realization of multivariate piecewise‐linear (PWL) functions are reviewed and compared and two architectures with different levels of parallelism/complexity are proposed.
Abstract: Digital architectures for the circuit realization of multivariate piecewise-linear (PWL) functions are reviewed and compared. The output of the circuits is a digital word representing the value of the PWL function at the n-dimensional input. In particular, we propose two architectures with different levels of parallelism/complexity. PWL functions with n = 3 inputs are implemented on an FPGA and experimental results are shown. The accuracy in the representation of PWL functions is tested through three benchmark examples, two concerning three-variate static functions and one concerning a dynamical control system defined by a bi-variate PWL function. Copyright © 2009 John Wiley & Sons, Ltd.
20 Jun 2011
TL;DR: This work proposes a deterministic branch and bound method to solve the MaxFS problem with guaranteed global optimality and introduces a piecewise linear relaxation to build very tight under- and over-estimators for square terms by partitioning variable bounds into smaller segments.
Abstract: Many computer vision problems can be accounted for or properly approximated by linearity, and the robust model fitting (parameter estimation) problem in presence of outliers is actually to find the Maximum Feasible Subsystem (MaxFS) of a set of infeasible linear constraints. We propose a deterministic branch and bound method to solve the MaxFS problem with guaranteed global optimality. It can be used in a wide class of computer vision problems, in which the model variables are subject to the unit norm constraint. In contrast to the convex and concave relaxations in existing works, we introduce a piecewise linear relaxation to build very tight under- and over-estimators for square terms by partitioning variable bounds into smaller segments. Based on this novel relaxation technique, our branch and bound method can converge in a few iterations. For homogeneous linear systems, which correspond to some quasi-convex problems based on L ∞ -L ∞ -norm, our method is non-iterative and certainly reaches the globally optimal solution at the root node by partitioning each variable range into two segments with equal length. Throughout this work, we rely on the so-called Big-M method, and successfully avoid potential numerical problems by exploiting proper parametrization and problem structure. Experimental results demonstrate the stability and efficiency of our proposed method.
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TL;DR: In this paper, the authors studied limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin.
Abstract: In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclnic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.
TL;DR: A hierarchical algorithm to compute the normal vector and curvature of an interface line with the height function method that switches automatically between second- and fourth-order approximations and that can be applied also when the local radius of curvature is of the order of the grid spacing.
TL;DR: In this article, a piecewise linear method was used to model and forecast the demand for Macau tourism on a monthly basis using data over the period January 1991-December 2005 and a seasonally adjusted series for tourism demand.
TL;DR: A low-error approximation of the sigmoid function and hyperbolic tangent, which are mainly used to activate the artificial neuron, are proposed based on the piecewise linear method, showing better results than the state-of-the-art.
TL;DR: In this paper, the authors investigated the convergence rate of interpolating splines with respect to sparse grids for Besov spaces of dominating mixed smoothness (Tensor Product Besov Spaces).
Abstract: We investigate the rate of convergence of interpolating splines with respect to sparse grids for Besov spaces of dominating mixed smoothness (tensor product Besov spaces). Main emphasis is given to the approximation by piecewise linear functions.
TL;DR: A recently derived numerical algorithm for one-dimensional time-dependent Stefan problems is extended for the purposes of solving a moving boundary problem for the transient heating of an evaporating spherical droplet using the Keller box finite-difference scheme.
TL;DR: In this paper, an analytical approach that applies to the case of a general waveshape of the channel-base current, based on the piecewise straight-line approximation, in cylindrical coordinates, which is functional when using the TL return-stroke current model and the Agrawal et al. coupling model is presented.
Abstract: The evaluation of the lightning performance of overhead lines requires, mainly for distribution lines, the calculation of the voltages induced by nearby lightning strokes hitting the ground. Generally, when calculating lightning-induced voltages using the time-domain approach and the TEM assumption, the most time-consuming stage is the calculation of the source terms of the coupling equations, which means, in turn, the electromagnetic field radiated by the lightning channel. Hence, simplified approaches are still desirable in order to save computational time. Analytical solution of the lighting-originated electric- and magnetic field is available in literature for the case of an ideal transmission-line (TL) return-stroke model with channel-base current having a stepwise waveshape. This paper describes an analytical approach that applies to the case of a general waveshape of the channel-base current, based on the piecewise straight-line approximation, in cylindrical coordinates, which is functional when using the TL return-stroke current model and the Agrawal et al. coupling model. This paper reports the exact solution for the case of trapezoidal waveshape of the current at the base of the channel and the analytical solution for the case of piecewise linear waveform.
TL;DR: The applicability of the concave-convex approach on a number of standard distributions is demonstrated and an application to the efficient construction of sequential Monte Carlo proposal distributions for inference over genealogical trees is described.
Abstract: We describe a method for generating independent samples from univariate density functions using adaptive rejection sampling without the log-concavity requirement. The method makes use of the fact that many functions can be expressed as a sum of concave and convex functions. Using a concave-convex decomposition, we bound the log-density by separately bounding the concave and convex parts using piecewise linear functions. The upper bound can then be used as the proposal distribution in rejection sampling. We demonstrate the applicability of the concave-convex approach on a number of standard distributions and describe an application to the efficient construction of sequential Monte Carlo proposal distributions for inference over genealogical trees. Computer code for the proposed algorithms is available online.
TL;DR: A bent line quantile regression model is proposed that is theoretically justified and its parameters are of direct biological interests and its parameter estimates are asymptotically valid given the existence of a change-point.
Abstract: Quantile regression, which models the conditional quantiles of the response variable given covariates, usually assumes a linear model. However, this kind of linearity is often unrealistic in real life. One situation where linear quantile regression is not appropriate is when the response variable is piecewise linear but still continuous in covariates. To analyze such data, we propose a bent line quantile regression model. We derive its parameter estimates, prove that they are asymptotically valid given the existence of a change-point, and discuss several methods for testing the existence of a change-point in bent line quantile regression together with a power comparison by simulation. An example of land mammal maximal running speeds is given to illustrate an application of bent line quantile regression in which this model is theoretically justified and its parameters are of direct biological interests.