Showing papers on "Piecewise published in 2001"
TL;DR: It is shown that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function.
Abstract: We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of \dRd It is assumed that function evaluations are expensive and that no additional information is available Radial basis function interpolation is used to define a utility function The maximizer of this function is the next point where the objective function is evaluated We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm A general framework for both methods is presented Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods
793 citations
TL;DR: In this paper, a double-moment parameterization of microphysical processes in warm clouds is derived directly from the stochastic collection equation, which is able to reproduce the results of the spectral reference model within a wide range of initial conditions, while other parameterizations show large errors when assuming continental clouds with small mean radii.
389 citations
01 Jan 2001
TL;DR: The analysis of the application of a chaotic piecewise-linear one-dimensional map as random number generator (RNG) enables us to mathematically find parameter values for which a generating partition is Markov and the RNG behaves as a Markov information source.
Abstract: This paper and its companion (Part II) are devoted to the analysis of the application of a chaotic piecewise-linear one- dimensional (PL1D) map as random number generator (RNG). Piecewise linearity of the map enables us to mathematically find parameter values for which a generating partition is Markov and the RNG behaves as a Markov information source, and then to mathematically analyze the information generation process and the RNG. In the companion paper we discuss practical aspects of our chaos-based RNGs. Index Terms—Chaos, random number generator, symbolic dy- namics.
380 citations
TL;DR: This work considers the problem of interpolating a signal using a linear combination of shifted versions of a compactly-supported basis function phi(x) using an explicit form of the MOMS that maximizes the approximation accuracy when the step-size is small enough and implies that regularity has little to do with approximating performance.
Abstract: We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly-supported basis function /spl phi/(z). We first give the expression for the cases of /spl phi/ that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we call maximal-order-minimal-support functions (MOMS) is made of linear combinations of the B-spline of the same order and of its derivatives. We provide an explicit form of the MOMS that maximizes the approximation accuracy when the step-size is small enough. We compute the sampling gain obtained by using these optimal basis functions over the splines of the same order. We show that it is already substantial for small orders and that it further increases with the approximation order L. When L is large, this sampling gain becomes linear; more specifically, its exact asymptotic expression is 2/(/spl pi/e)L. Since the optimal functions are continuous, but not differentiable, for even orders, and even only piecewise continuous for odd orders, our result implies that regularity has little to do with approximating performance. These theoretical findings are corroborated by experimental evidence that involves compounded rotations of images.
223 citations
TL;DR: In this paper, the authors present a unified framework for performing local analysis of grazing bifurcations in n-dimensional piecewise-smooth systems of ODEs, where a periodic orbit has a point of tangency with a smooth (n−1)-dimensional boundary dividing distinct regions in phase space where the vector field is smooth.
208 citations
TL;DR: It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.
Abstract: We will propose a branch and bound algorithm for calculating a globally optimal solution of a portfolio construction/rebalancing problem under concave transaction costs and minimal transaction unit constraints. We will employ the absolute deviation of the rate of return of the portfolio as the measure of risk and solve linear programming subproblems by introducing (piecewise) linear underestimating function for concave transaction cost functions. It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.
193 citations
TL;DR: This paper discusses Markov random fields problems in the context of a representative application---the image segmentation problem and presents an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear.
Abstract: Problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements, or order constraints. In such problems, the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modify value. In Markov random field problems, there is also a pairwise relationship between the objects. The objective in Markov random field problem is to minimize the sum of the deviation cost function and a penalty function that grows with the distance between the values of related pairs---separation function.We discuss Markov random fields problems in the context of a representative application---the image segmentation problem. In this problem, the goal is to modify color shades assigned to pixels of an image so that the penalty function consisting of one term due to the deviation from the initial color shade and a second term that penalizes differences in assigned values to neighboring pixels is minimized. We present here an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear; and in strongly polynomial time when the deviation cost function is linear, quadratic or piecewise linear convex with few pieces (where “few” means a number exponential in a polynomial function of the number of variables and constraints). The complexity of the algorithm for a problem on n pixels or variables, m adjacency relations or constraints, and range of variable values (colors) U, is O(T(n,m) + n log U) where T(n,m) is the complexity of solving the minimum s, t cut problem on a graph with n nodes and m arcs. Furthermore, other algorithms are shown to solve the problem with convex deviation and convex separation in running time O(mn log n log nU) and the problem with nonconvex deviation and convex separation in running time O(T(nU, mU). The nonconvex separation problem is NP-hard even for fixed value of U.For the family of problems with convex deviation functions and linear separation function, the algorithm described here runs in polynomial time which is demonstrated to be fastest possible.
179 citations
TL;DR: A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes, which links grazings with border-collisions in nonsmooth maps.
Abstract: A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes. This links grazings with border-collisions in nonsmooth maps. Contrary to previous literature, piecewise linear maps correspond only to nonsmooth discontinuity boundaries. All other maps have either square-root or $(3/2)$-type singularities.
164 citations
TL;DR: The present work has been motivated by the desire to accurately model strong hydrodynamic and magnetohydrodynamic shocks, and a key issue has therefore been to achieve a near-optimal representation of the simulated system at all times.
Abstract: Smoothed particle hydrodynamics (SPH) has proven to be a useful numerical tool in studying a number of different astrophysical problems. Still, used on other problems, such as the modeling of low-β MHD systems, the method has so far not performed as well as one might have hoped. The present work has been motivated by the desire to accurately model strong hydrodynamic and magnetohydrodynamic shocks, and a key issue has therefore been to achieve a near-optimal representation of the simulated system at all times. Using SPH, this means combining the Lagrangian nature of the method with a smoothing-length profile that varies in both space and time. In this paper, a scheme containing two novel features is proposed. First, the scheme assumes a piecewise constant smoothing-length profile. To avoid substantial errors near the steps in the profile, alternative forms of the SPH equations of motion are used. Second, a predictive attitude toward optimizing the particle distribution is introduced by activating a mass, momentum, and energy conservation regularization process at intervals. The scheme described has been implemented in a new code called regularized smoothed particle hydrodynamics (RSPH), and test results for a number of standard hydrodynamic and magnetohydrodynamic tests in one and two dimensions using this code are presented.
148 citations
07 Oct 2001
TL;DR: Analysis shows that the proposed method is more aggressive than anisotropic diffusion at enhancing and preserving edges, and is less sensitive to the edge contrast parameter, and empirical results confirm these advantages.
Abstract: This paper describes a partial differential equation for denoising images. The proposed method is demonstrably superior to anisotropic diffusion (and its many variations) for denoising images that are approximately piecewise constant. The method relies on an equation that is the level-set equivalent of the anisotropic diffusion equation proposed by Perona and Malik (1990). This proposed equation has come up in the literature, but has failed to be fully utilized due to a lack of analysis and the need for a stable, accurate numerical implementation. Our analysis shows that the proposed method is more aggressive than anisotropic diffusion at enhancing and preserving edges, and is less sensitive to the edge contrast parameter. Empirical results confirm these advantages, and show that for certain classes of images, one should always prefer the proposed method over anisotropic diffusion.
148 citations
01 Dec 2001
TL;DR: The technique is interactive and updates the model in real time as constraints are added, allowing fast reconstruction of photorealistic scene models, and is shown to yield high quality results on a large variety of images.
Abstract: This paper presents a novel approach for reconstructing free-form, texture-mapped, 3D scene models from a single painting or photograph. Given a sparse set of user-specified constraints on the local shape of the scene, a smooth 3D surface that satisfies the constraints is generated This problem is formulated as a constrained variational optimization problem. In contrast to previous work in single view reconstruction, our technique enables high quality reconstructions of free-form curved surfaces with arbitrary reflectance properties. A key feature of the approach is a novel hierarchical transformation technique for accelerating convergence on a non-uniform, piecewise continuous grid. The technique is interactive and updates the model in real time as constraints are added, allowing fast reconstruction of photorealistic scene models. The approach is shown to yield high quality results on a large variety of images.
02 Dec 2001
TL;DR: It is shown that the global stability of the closed loop fuzzy control systems can be established, and the control laws can be obtained by solving a set of linear matrix inequalities (LMI).
Abstract: This paper presents a controller synthesis method for fuzzy dynamic systems based on a piecewise smooth Lyapunov function. It is shown that the global stability of the closed loop fuzzy control systems can be established, and the control laws can be obtained by solving a set of linear matrix inequalities (LMI).
TL;DR: The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points.
Abstract: In the first part of this paper we study the regularity properties of solutions of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically. The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points.
01 Dec 2001
TL;DR: This paper addresses the problem of extracting depth information of non-rigid dynamic 3D scenes from multiple synchronized video streams by presenting a framework in which the scene is modeled as a collection of 3D piecewise planar surface patches induced by color based image segmentation.
Abstract: This paper addresses the problem of extracting depth information of non-rigid dynamic 3D scenes from multiple synchronized video streams. Three main issues are discussed in this context. (i) temporally consistent depth estimation, (ii) sharp depth discontinuity estimation around object boundaries, and (iii) enforcement of the global visibility constraint. We present a framework in which the scene is modeled as a collection of 3D piecewise planar surface patches induced by color based image segmentation. This representation is continuously estimated using an incremental formulation in which the 3D geometric, motion, and global visibility constraints are enforced over space and time. The proposed algorithm optimizes a cost function that incorporates the spatial color consistency constraint and a smooth scene motion model.
TL;DR: An optimal spline-based algorithm for the enlargement or reduction of digital images with arbitrary (noninteger) scaling factors is presented, and the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal-to-noise ratio.
Abstract: We present an optimal spline-based algorithm for the enlargement or reduction of digital images with arbitrary (noninteger) scaling factors. This projection-based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are B-splines of any degree n. A noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a. For a given choice of basis functions, the results of our method are consistently better than those of the standard interpolation procedure; the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal-to-noise ratio. The method can be generalized to include other classes of piecewise polynomial functions, expressed as linear combinations of B-splines and their derivatives.
TL;DR: In this article, a new analytical model of a gear pair with time varying mesh stiffness, viscous damping and sliding friction parameters is presented, where the excitation consists of three separate terms, namely the unloaded transmission error, time-invariant external torque and the periodically varying sliding friction force.
TL;DR: The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps to exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known.
Abstract: Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives The standard Fourier–Pade approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known Implementation requires just the solution of a linear system, as in standard Pade approximation The new methods compare favorably in experiments with existing techniques
TL;DR: The finite-element (FE) method combines the significant advantages of both real-space-grid and basis-oriented approaches and so promises to be particularly well suited for large, accurate ab initio calculations.
TL;DR: In this paper, a new sliding control scheme for non-linear systems containing time-varying uncertainties with unknown bounds is proposed, where uncertainties are assumed to be piecewise continuous functions of time and satisfy the Dirichlet conditions.
Abstract: The sliding controller is very effective in dealing with system uncertainties defined in compact sets. If the bounds of the uncertainties are not available, the adaptive sliding controller might be designed. One restriction for the adaptive sliding scheme is that the unknown parameter should be constant, which is not always satisfied in practice. For a non-linear system with general uncertainties (i.e. time varying with unknown bounds), both the traditional sliding control and adaptive sliding control do not work properly. This paper proposes a new sliding control scheme for non-linear systems containing time-varying uncertainties with unknown bounds. The uncertainties are assumed to be piecewise continuous functions of time and satisfy the Dirichlet conditions. By representing these uncertainties in finite-term Fourier series, they can be estimated by updating the Fourier coefficients. Since the coefficients are time-invariant, update laws are easily obtained from the Lyapunov approach to guarantee outpu...
TL;DR: In this paper, a new Fourier-von Mises image model is identified, with phase differences between Fouriertransformed images having von Mises distributions, and null set distortion criteria are proposed, with each criterion uniquely minimized by a particular set of polynomial functions.
Abstract: A warping is a function that deforms images by mapping between image domains. The choice of function is formulated statistically as maximum penalized likelihood, where the likelihood measures the similarity between images after warping and the penalty is a measure of distortion of a warping. The paper addresses two issues simultaneously, of how to choose the warping function and how to assess the alignment. A new, Fourier–von Mises image model is identified, with phase differences between Fourier-transformed images having von Mises distributions. Also, new, null set distortion criteria are proposed, with each criterion uniquely minimized by a particular set of polynomial functions. A conjugate gradient algorithm is used to estimate the warping function, which is numerically approximated by a piecewise bilinear function. The method is motivated by, and used to solve, three applied problems: to register a remotely sensed image with a map, to align microscope images obtained by using different optics and to discriminate between species of fish from photographic images.
26 Mar 2001
TL;DR: GDEVS as mentioned in this paper is a generalized discrete event specification, where the trajectories are organized through piecewise polynomial segments, which can be used to model continuous processes as discrete event abstractions.
Abstract: Given a process whose output is a dynamic function of time, the traditional discrete event specification (DEVS) approximates the input, output, and state trajectories through piecewise constant segments, where the segments correspond to discrete time intervals that are not necessarily equal in length. For processes that defy accurate modeling through piecewise constant segments, this paper presents GDEVS, a generalized discrete event specification, wherein the trajectories are organized through piecewise polynomial segments. The utilization of arbitrary polynomial functions for segments promises higher accuracies in modeling continuous processes as discrete event abstractions. In general, discrete event systems including DEVS and GDEVS execute faster on host computers because executions occur corresponding to significant changes in the system unlike in continuous simulations where execution is on a continuous basis. GDEVS' superiority over DEVS lies in its ability to discretize a system characteristic. A key contribution of GDEVS is that it permits the development of a uniform simulation environment for hybrid, i.e. both continuous and discrete, systems. GDEVS is illustrated for a first order system and a hybrid system, with piecewise linear segments. Two representative systems have been modeled under GDEVS and executed on a simulator developed for GDEVS.
TL;DR: The defined donating region (DDR) scheme as mentioned in this paper uses a linear piecewise method of free surface reconstruction, coupled with a fully multi-dimensional method of cell boundary flux integration.
Abstract: This paper presents a new volume of fluid (VOF) advection algorithm, termed the defined donating region (DDR) scheme. The algorithm uses a linear piecewise method of free surface reconstruction, coupled to a fully multi-dimensional method of cell boundary flux integration. The performance of the new scheme has been compared with the performance of a number of alternative schemes using translation, rotation and shear advection tests. The DDR scheme is shown to be generally more accurate than linear constant and flux limited schemes, and comparable with an alternative linear piecewise scheme. The DDR scheme conserves fluid volume rigorously without local redistribution algorithms, and generates no fluid ‘flotsam’ or other debris, making it ideal in applications where stability of the free surface interface is paramount. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: In this article, it was shown that piecewise linear isometries have zero topological entropy in any dimension, i.e. non-necessarily invertible maps defined on a finite union of polytopes and coinciding with an isometry on the interior of each polytope.
Abstract: We show that piecewise isometries, i.e. non-necessarily invertible maps defined on a finite union of polytopes and coinciding with an isometry on the interior of each polytope, have zero topological entropy in any dimension. This had been conjectured by a number of authors. The proof is by an induction on the dimension and uses a device (the differential of a piecewise linear map) introduced by M. Tsujii.
TL;DR: In this paper, an algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations.
Abstract: An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians.
The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.
The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations.
Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.
01 Dec 2001-Compel-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering
TL;DR: In this article, a purely mathematical model of the saturation curve and the hysteresis loop based on the fundamental similarities between the Langevin function the specified T(x) function and the sigmoid shape is presented.
Abstract: This paper starts with the description of a purely mathematical model of the saturation curve and the hysteresis loop based on the fundamental similarities between the Langevin function the specified T(x) function and the sigmoid shape. The T(x) function which is composed of tangent hyperbolic and linear functions with its free parameters can describe the regular anhysteretic magnetisation curve. Developed from this function the model describes not only the regular hysteresis loop but also the biased and other minor loops like the ones produced by the interrupted and reversed magnetisation process and the open “loops” created by a piecewise monotonic magnetising field input of diminishing amplitude. The remanent magnetism as the function of the interrupted field co‐ordinates is predicted by the model in this mathematical form for the first time. The model presented here is based on the principle that all processes follow the shape of the T(x) function describing the shape of the major hysteresis loop of the ferromagnetic specimen under investigation. The model is also applicable to hysteretic processes in other fields.
TL;DR: In this paper, an F-16 nonlinear sixdegree-of-freedom model is considered for tracking causal reference output with a finite number of nonzero time derivatives (piecewise polynomial spline model) in the presence of unmatched disturbances of the same kind.
Abstract: The approximate causal nonminimum phase output tracking problem is considered for an F-16 nonlinear sixdegree-of-freedom modelandaddressed via sliding modecontrol.Asymptoticoutputtracking-errordynamicswith desired eigenvalue placement are provided in case of tracking causal reference output proe le with a e nite number of nonzero time derivatives (piecewise polynomial spline model ) in the presence of unmatched disturbances of the same kind. A complete constructive algorithm for tracking controller design is built for a class of uncertain nonlinear multi-input/multi-output systems with known linear unstable internal dynamics. An analysis is made of the issue that the given nonlinear aircraft model yields to the approach developed.
TL;DR: The use of a state-selecting metamodeling approach that provides an accurate approximation for piecewise continuous responses is investigated and is applied to a desk lamp performance model.
Abstract: Metamodels are effective for providing fast-running surrogate approximations of product or system performance. Because these approximations are generally based on continuous functions, they can provide poor fits of discontinuous response functions. Many engineering models produce functions that are only piecewise continuous, due to changes in modes of behavior or other state variables. The use of a state-selecting metamodeling approach that provides an accurate approximation for piecewise continuous responses is investigated. The proposed approach is applied to a desk lamp performance model. Three types of metamodels, quadratic polynomials, spatial correlation (kriging) models, and radial basis functions, and five types of experimental designs, full factorial designs, D-best Latin hypercube designs, fractional Latin hypercubes, Hammersley sampling sequences, and uniform designs, are compared based on three error metrics computed over the design space. The state-selecting metamodeling approach outperforms a combined metamodeling approach in this example, and radial basis functions perform well for metamodel construction.
TL;DR: In this paper, a modified Zig-Zag technical theory for the analysis of thick composite beams with rectangular cross section, general lay-up and in cylindrical bending is developed and tested.
Abstract: A modified zig-zag technical theory, suitable for the analysis of thick composite beams with rectangular cross section, general lay-up and in cylindrical bending is developed and tested. An equivalent single-layer model and a multiple-layer model are implemented. The displacement field of both these models is postulated as to allow for appropriate jumps in the strains, so that the transverse shear and the transverse normal stress and stress gradient continuity at the interfaces are met. A third-order piecewise approximation for the in-plane displacement and a fourth-order piecewise approximation for the transverse displacement are assumed in the two models. Their predictive capability is investigated in sample cases wherein the exact three-dimensional elasticity and other approximate solutions are available. On the basis of this numerical investigation, they appear to predict accurately and efficiently the displacement and stress fields of composite beams with layers of different materials.
TL;DR: In this article, the stability problem for systems with distributed delay is considered using discretized Lyapunov functional, where coefficients associated with the distributed delay are assumed to be piecewise constant, and the discretization mesh may be non-uniform.
Abstract: The stability problem for systems with distributed delay is considered using discretized Lyapunov functional. The coefficients associated with the distributed delay are assumed to be piecewise constant, and the discretization mesh may be non-uniform. The resulting stability criteria are written in the form of linear matrix inequality. Numerical examples are also provided to illustrate the effectiveness of the method. The basic idea can be extended to a more general setting with more involved formulation.
TL;DR: In this paper, the authors explore the applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps (SOMs).
Abstract: I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps.