Showing papers on "Piecewise linear function published in 2010"

Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

Abstract: A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.

Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions

Abstract: We study the modeling of nonconvex piecewise-linear functions as mixed-integer programming (MIP) problems. We review several new and existing MIP formulations for continuous piecewise-linear functions with special attention paid to multivariate nonseparable functions. We compare these formulations with respect to their theoretical properties and their relative computational performance. In addition, we study the extension of these formulations to lower semicontinuous piecewise-linear functions.

Multistability of Recurrent Neural Networks With Time-varying Delays and the Piecewise Linear Activation Function

Abstract: In this brief, stability of multiple equilibria of recurrent neural networks with time-varying delays and the piecewise linear activation function is studied. A sufficient condition is obtained to ensure that n-neuron recurrent neural networks can have (4k-1)n equilibrium points and (2k)n of them are locally exponentially stable. This condition improves and extends the existing stability results in the literature. Simulation results are also discussed in one illustrative example.

Robust Approximation to Multiperiod Inventory Management

Abstract: We propose a robust optimization approach to address a multiperiod inventory control problem under ambiguous demands, that is, only limited information of the demand distributions such as mean, support, and some measures of deviations. Our framework extends to correlated demands and is developed around a factor-based model, which has the ability to incorporate business factors as well as time-series forecast effects of trend, seasonality, and cyclic variations. We can obtain the parameters of the replenishment policies by solving a tractable deterministic optimization problem in the form of a second-order cone optimization problem (SOCP), with solution time; unlike dynamic programming approaches, it is polynomial and independent on parameters such as replenishment lead time, demand variability, and correlations. The proposed truncated linear replenishment policy (TLRP), which is piecewise linear with respect to demand history, improves upon static and linear policies, and achieves objective values that are reasonably close to optimal.

A contour line of the continuum Gaussian free field

Abstract: Consider an instance $h$ of the Gaussian free field on a simply connected planar domain with boundary conditions $-\lambda$ on one boundary arc and $\lambda$ on the complementary arc, where $\lambda$ is the special constant $\sqrt{\pi/8}$. We argue that even though $h$ is defined only as a random distribution, and not as a function, it has a well-defined zero contour line connecting the endpoints of these arcs, whose law is SLE(4). We construct this contour line in two ways: as the limit of the chordal zero contour lines of the projections of $h$ onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property.

A POMDP Extension with Belief-dependent Rewards

Abstract: Partially Observable Markov Decision Processes (POMDPs) model sequential decision-making problems under uncertainty and partial observability. Unfortunately, some problems cannot be modeled with state-dependent reward functions, e.g., problems whose objective explicitly implies reducing the uncertainty on the state. To that end, we introduce ρPOMDPs, an extension of POMDPs where the reward function ρ depends on the belief state. We show that, under the common assumption that ρ is convex, the value function is also convex, what makes it possible to (1) approximate ρ arbitrarily well with a piecewise linear and convex (PWLC) function, and (2) use state-of-the-art exact or approximate solving algorithms with limited changes.

Stochastic resonance in a trapping overdamped monostable system

Abstract: The response of a trapping overdamped monostable system to a harmonic perturbation is analyzed, in the context of stochastic resonance phenomenon. We consider the dynamics of a Brownian particle moving in a piecewise linear potential with a white Gaussian noise source. Based on linear-response theory and Laplace transform technique, analytical expressions of signal-to-noise ratio (SNR) and signal power amplification (SPA) are obtained. We find that the SNR is a nonmonotonic function of the noise intensity, while the SPA is monotonic. Theoretical results are compared with numerical simulations.

A Residual-Based A Posteriori Error Estimator for the Stokes-Darcy Coupled Problem

Abstract: In this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Stokes and Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, and local approximation properties of the Clement interpolant and Raviart-Thomas operator. On the other hand, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

Algorithm 905: SHEPPACK: Modified Shepard Algorithm for Interpolation of Scattered Multivariate Data

Abstract: Scattered data interpolation problems arise in many applications. Shepard’s method for constructing a global interpolant by blending local interpolants using local-support weight functions usually creates reasonable approximations. SHEPPACK is a Fortran 95 package containing five versions of the modified Shepard algorithm: quadratic (Fortran 95 translations of Algorithms 660, 661, and 798), cubic (Fortran 95 translation of Algorithm 791), and linear variations of the original Shepard algorithm. An option to the linear Shepard code is a statistically robust fit, intended to be used when the data is known to contain outliers. SHEPPACK also includes a hybrid robust piecewise linear estimation algorithm RIPPLE (residual initiated polynomial-time piecewise linear estimation) intended for data from piecewise linear functions in arbitrary dimension m. The main goal of SHEPPACK is to provide users with a single consistent package containing most existing polynomial variations of Shepard’s algorithm. The algorithms target data of different dimensions. The linear Shepard algorithm, robust linear Shepard algorithm, and RIPPLE are the only algorithms in the package that are applicable to arbitrary dimensional data.

Optimal convergence analysis of an immersed interface finite element method

Abstract: We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H 1-norm and L 2-norm.

A lattice Boltzmann approach for free-surface-flow simulations on non-uniform block-structured grids

Abstract: In this paper, we present a hybrid volume-of-fluid-based algorithm for the simulation of free-surface-flow problems. For the solution of the flow field, the lattice Boltzmann method is used. The additional advection equation for the volume-of-fluid (VOF) fill level is discretized with a classical finite volume method. For the interface reconstruction, a piecewise linear interface reconstruction in 3D has been implemented. The free-surface-tracking algorithm is embedded into the 3D, non-uniform, lattice-Boltzmann-based solver VirtualFluids; Freudiger et al. (2009) [1], Freudiger (2009) [2]. The advection algorithm is verified and validated with well-known advection test cases. For the validation of the free-surface algorithm, we present simulations of a breaking-dam benchmark.

A Cell-based Merchant-Nemhauser Model for the System Optimum Dynamic Traffic Assignment Problem

Abstract: Using a simple piecewise linear exit-flow function, a cell-based variant of the Merchant-Nemhauser (M-N) model is proposed for the system optimum (SO) dynamic traffic assignment (DTA) problem. Once augmented with additional constraints to capture cross-cell interactions, the model becomes a linear program that embeds a relaxed cell transmission model (CTM) for traffic propagation. The proposed cell-based M-N model is different from the existing CTM-based SO-DTA models in that (1) it does not require intersections to be standard merge and diverge; and 2) it does not directly solve for cell-to-cell flows. Generally, the proposed model has a simpler constraint structure and is easier to construct. Path marginal costs are defined using a recursive formula that involves a subset of multipliers from the linear program. This definition is then employed to interpret the necessary condition, which is a dynamic extension of the Wardrop's second principle. An algorithm is presented to solve the flow holding back problem that is known to exist in many existing SO-DTA models. A numerical experiment is conducted to verify the proposed model and algorithm.

A posteriori error analysis for a class of integral equations and variational inequalities

Abstract: We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order $${2s \in (0,2]}$$. Our main motivation is the pricing of European or American options under Levy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H s -norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms.

On Instantaneous Relaying

Abstract: The Gaussian, three-node relay channel with orthogonal receive components (i.e., the transmitted signals from the source and the relay do not interfere with each other) is investigated. For such channels, linear relaying is a suboptimal strategy in general. This is because a linear scheme merely repeats the received noisy signal and does not utilize the available degrees of freedom efficiently. At this background, nonlinear, symbol-wise (as opposed to block-wise) relaying strategies are developed to compensate for the shortcomings of the linear strategy. Optimal strategies are presented for two special cases of the general scenario, and it is shown that memoryless relaying can achieve the capacity. Furthermore, for the general Gaussian relay channel, a parametric piecewise linear (PL) mapping is proposed and analyzed. The achievable rates obtained by the PL mapping are computed numerically and optimized for a certain number of design parameters. It is concluded that optimized PL relaying always outperforms conventional instantaneous linear relaying (amplify-and-forward). It is also illustrated that the proposed PL relaying scheme can improve on sophisticated block Markov encoding (i.e., decode-and-forward) when the source-relay link is ill-conditioned (relative to other links). Furthermore, PL relaying can work at rates close to those achieved by side-information encoding (i.e., compress-and-forward), but at a much lower complexity.

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.

Analysis and Estimation of Error Constants for P0 and P1 Interpolations over Triangular Finite Elements

Abstract: We give some fundamental results on the error con- stants for the piecewise constant interpolation function and the piece- wise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but the present results also appear to be available for the non-conforming case. We obtain explicit relations for the upper bounds of the constants, and analyze dependence of such constants on the geometric parameters of triangles. In particular, we explicitly determine some special constants including the Babuska- Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The obtained re- sults are expected to be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of nu- merical solutions based on the triangular finite elements. We also give some numerical results for the error constants and for a posteriori estimates of some eigenvalues related to the error constants.

Stabilization of continuous-time switched linear positive systems

Abstract: In this paper we considered a few problems related to linear positive switched systems. First, we provide a result on state-feedback stabilization of autonomous linear positive switched systems through piecewise linear co-positive Lyapunov functions. This is accompanied by a side result on the existence of a switching law guaranteeing an upper bound to the optimal L 1 cost. Then, the induced L 1 guaranteed cost cost is tackled, through constrained piecewise linear co-positive Lyapunov functions. The optimal L 1 cost control is finally studied via Hamiltonian function analysis.

Optimization-based Fluid Simulation on Unstructured Meshes

Abstract: We present a novel approach to fluid simulation, allowing us to take into account the surface energy in a precise manner. This new approach combines a novel, topology-adaptive approach to deformable interface tracking, called the deformable simplicial complexes method (DSC) with an optimization-based, linear finite element method for solving the incompressible Euler equations. The deformable simplicial complexes track the surface of the fluid: the fluid-air interface is represented explicitly as a piecewise linear surface which is a subset of tetrahedralization of the space, such that the interface can be also represented implicitly as a set of faces separating tetrahedra marked as inside from the ones marked as outside. This representation introduces insignificant and controllable numerical diffusion, allows robust topological adaptivity and provides both a volumetric finite element mesh for solving the fluid dynamics equations as well as direct access to the interface geometry data, making inclusion of a new surface energy term feasible. Furthermore, using an unstructured mesh makes it straightforward to handle curved solid boundaries and gives us a possibility to explore several fluid-solid interaction scenarios.

Overlaying Classifiers: A Practical Approach to Optimal Scoring

Abstract: The Receiver Operating Characteristic (ROC) curve is one of the most widely used visual tools to evaluate performance of scoring functions regarding their capacities to discriminate between two populations. It is the goal of this paper to propose a statistical learning method for constructing a scoring function with nearly optimal ROC curve. In this bipartite setup, the target is known to be the regression function up to an increasing transform, and solving the optimization problem boils down to recovering the collection of level sets of the latter, which we interpret here as a continuum of imbricated classification problems. We propose a discretization approach, consisting of building a finite sequence of N classifiers by constrained empirical risk minimization and then constructing a piecewise constant scoring function s N (x) by overlaying the resulting classifiers. Given the functional nature of the ROC criterion, the accuracy of the ranking induced by s N (x) can be conceived in a variety of ways, depending on the distance chosen for measuring closeness to the optimal curve in the ROC space. By relating the ROC curve of the resulting scoring function to piecewise linear approximates of the optimal ROC curve, we establish the consistency of the method as well as rate bounds to control its generalization ability in sup -norm. Eventually, we also highlight the fact that, as a byproduct, the algorithm proposed provides an accurate estimate of the optimal ROC curve.

Noise and Disturbance Reduction for Heart Sounds in Cycle-Frequency Domain Based on Nonlinear Time Scaling

Abstract: Through an investigation of various clinical cases, heart sounds are found to be quasi-cyclostationary. Nonlinear time scaling from cycle-to-cycle is proposed to enhance cyclic stationarity, where nonlinear time scaling is approximated by a piecewise linear function. The techniques of cyclostationary signal processing are employed in this paper to reduce noise and disturbance in the cycle-frequency domain. Heart sounds can be theoretically recovered in the presence of additive, zero mean noise, and disturbance (perhaps non-Gaussian, nonstationary, or colored). The experimental tests in various conditions confirm the theoretical results.

Spatiotemporal dynamics of two generic predator–prey models

Abstract: We present the analysis of two reaction-diffusion systems modelling predator-prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L(∞)-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L(p)-estimate, uniform in time, for all p ≥ 1, implies L(∞)-uniform bounds, given any nonnegative L(∞)-initial data. The applicability of the L(∞)-estimate to general reaction-diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be 'trapped' in an invariant region of phase space.

A novel AR-based robot programming and path planning methodology

Abstract: This paper discusses the benefits of applying Augmented Reality (AR) to facilitate intuitive robot programming, and presents a novel methodology for planning collision-free paths for an n degree-of-freedom (DOF) manipulator in an unknown environment. The targeted applications are where the end-effector is constrained to move along a visible 3D path/curve, which position is unknown, at a particular orientation with respect to the path, such as arc welding and laser cutting. The methodology is interactive as the human is involved in obtaining the 3D data points of the desired curve to be followed through performing a number of demonstrations, defining the free space relevant to the task, and planning the orientations of the end-effector along the curve. A Piecewise Linear Parameterization (PLP) algorithm is used to parameterize the data points using an interactively generated piecewise linear approximation of the desired curve. A curve learning method based on Bayesian neural networks and reparameterization is used to learn and generate 3D parametric curves from the parameterized data points. Finally, the orientation of the end-effector along the learnt curve is planned with the aid of AR. Two case studies are presented and discussed.

Decomposing inventory routing problems with approximate value functions

Abstract: We present a time decomposition for inventory routing problems. The methodology is based on valuing inventory with a concave piecewise linear function and then combining solutions to single-period subproblems using dynamic programming techniques. Computational experiments show that the resulting value function accurately captures the inventory's value, and solving the multiperiod problem as a sequence of single-period subproblems drastically decreases computational time without sacrificing solution quality. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010