Showing papers on "Piecewise published in 2000"

Fourth-order partial differential equations for noise removal

Abstract: A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.

High-Order Total Variation-Based Image Restoration

Abstract: The total variation (TV) denoising method is a PDE-based technique that preserves edges well but has the sometimes undesirable staircase effect, namely, the transformation of smooth regions ( ramps) into piecewise constant regions ( stairs). In this paper we present an improved model, constructed by adding a nonlinear fourth order diffusive term to the Euler--Lagrange equations of the variational TV model. Our technique substantially reduces the staircase effect, while preserving sharp jump discontinuities (edges). We show numerical evidence of the power of resolution of this novel model with respect to the TV model in some 1D and 2D numerical examples.

Positive transfer operators and decay of correlations

Abstract: Subshifts of finite type a key symbolic model smooth uniformly expanding dynamics piecewise expanding systems hyperbolic systems.

Scaling up dynamic time warping for datamining applications

Abstract: There has been much recent interest in adapting data mining algorithms to time series databases. Most of these algorithms need to compare time series. Typically some variation of Euclidean distance is used. However, as we demonstrate in this paper, Euclidean distance can be an extremely brittle distance measure. Dynamic time warping (DTW) has been suggested as a technique to allow more robust distance calculations, however it is computationally expensive. In this paper we introduce a modification of DTW which operates on a higher level abstraction of the data, in particular, a Piecewise Aggregate Approximation (PAA). Our approach allows us to outperform DTW by one to two orders of magnitude, with no loss of accuracy.

Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity

Abstract: We study mathematical programs with complementarity constraints. Several stationarity concepts, based on a piecewise smooth formulation, are presented and compared. The concepts are related to stationarity conditions for certain smooth programs as well as to stationarity concepts for a nonsmooth exact penalty function. Further, we present Fiacco-McCormick type second order optimality conditions and an extension of the stability results of Robinson and Kojima to mathematical programs with complementarity constraints.

Piecewise linear quadratic optimal control

Abstract: The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy.

Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints

Abstract: We study the convergence behavior of a sequence of stationary points of a parametric NLP which regularizes a mathematical program with equilibrium constraints (MPEC) in the form of complementarity conditions. Accumulation points are feasible points of the MPEC; they are C-stationary if the MPEC linear independence constraint qualification holds; they are M-stationary if, in addition, an approaching subsequence satisfies second order necessary conditions, and they are B-stationary if, in addition, an upper level strict complementarity condition holds. These results complement recent results of Fukushima and Pang [Convergence of a smoothing continuation method for mathematical programs with equilibrium constraints, in Ill-posed Variational Problems and Regularization Techniques, Springer-Verlag, New York, 1999]. We further show that every local minimizer of the MPEC which satisfies the linear independence, upper level strict complementarity, and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric NLP.

Deadzone compensation in motion control systems using neural networks

Abstract: A compensation scheme is presented for general nonlinear actuator deadzones of unknown width. The compensator uses two neural networks (NNs), one to estimate the unknown deadzone and another to provide adaptive compensation in the feedforward path. The compensator NN has a special augmented form containing extra neurons whose activation functions provide a "jump function basis set" for approximating piecewise continuous functions. Rigorous proofs of closed-loop stability for the deadzone compensator are provided and yield tuning algorithms for the weights of the two NNs. The technique provides a general procedure for using NNs to determine the preinverse of an unknown right-invertible function.

Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits

Abstract: Recent investigations on the bifurcation behavior of power electronic DC-DC converters have revealed that most of the observed bifurcations do not belong to generic classes such as saddle-node, period doubling, or Hopf bifurcations. Since these systems yield piecewise smooth maps under stroboscopic sampling, a new class of bifurcations occur in such systems when a fixed point crosses the border between the smooth regions in the state space. In this paper we present a systematic analysis of such bifurcations through a normal form: the piecewise linear approximation in the neighborhood of the border. We show that there can be many qualitatively different types of border collision bifurcations, depending on the parameters of the normal form. We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. We then use this theoretical framework to explain the bifurcation behavior of the current programmed boost converter.

Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients

Abstract: In this paper we derive global W1,∞ and piecewise C1,α estimates for solutions to divergence form elliptic equations with piecewise Holder continuous coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.

The Explicit-Jump Immersed Interface Method: Finite Difference Methods for PDEs with Piecewise Smooth Solutions

Abstract: Many boundary value problems (BVPs) or initial BVPs have nonsmooth solutions, with jumps along lower-dimensional interfaces. The explicit-jump immersed interface method (EJIIM) was developed following Li's fast iterative immersed interface method (FIIIM), recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic BVPs. EJIIM generalizes the class of problems for which FIIIM is applicable. It handles interfaces between constant and variable coefficients and extends the immersed interface method (IIM) to BVPs on irregular domains with Neumann and Dirichlet boundary conditions. Proofs of second order convergence for a one-dimensional (1D) problem with piecewise constant coefficients and for two-dimensional (2D) problems with singular sources are given. Other problems are reduced to the singular sources case, with additional equations determining the source strengths. The advantages of EJIIM are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries, while still maintaining the efficiency of the FIIIM.

Piecewise smooth subdivision surfaces with normal control

Abstract: In this paper we introduce improved rules for Catmull-Clark and Loop subdivision that overcome several problems with the original schemes, namely, lack of smoothness at extraordinary boundary vertices and folds near concave corners. In addition, our approach to rule modification allows the generation of surfaces with prescribed normals, both on the boundary and in the interior, which considerably improves control of the shape of surfaces.

Stability and stabilization of piecewise affine and hybrid systems: an LMI approach

Abstract: In this paper we present various algorithms both for stability analysis and state-feedback design for discrete-time piecewise affine systems. Our approach hinges on the use of piecewise quadratic Lyapunov functions that can be computed as the solution of a set of linear matrix inequalities. We show that the continuity of the Lyapunov function is not required in the discrete-time case. Moreover, the basic algorithms are made less conservative by exploiting the switching structure of piecewise affine systems and by using relaxation procedures.

A Third-Order Semidiscrete Central Scheme for Conservation Laws and Convection-Diffusion Equations

Abstract: We present a new third-order, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semidiscrete method in [A. Kurgonov and E. Tadmor, J. Comput Phys., 160 (2000) pp. 241--282]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results by focusing on the new third-order central weighted essentially nonoscillatory (CWENO) reconstruction presented in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 21 (1999), pp. 294--322]. The numerical results we present show the desired accuracy, high resolution, and robustness of our method.

Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method

Abstract: The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of the Galerkin method. Error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp-DGFEM gives spectral convergence in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the hp-DGFEM results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms. For the h-version DGFEM algebraically graded time partitions are determined that give the optimal algebraic convergence rates. A fully discrete hp scheme is discussed exemplarily for the heat equation. The use of certain mesh-design principles for the spatial discretizations yields exponential rates of convergence in time and space. Numerical examples confirm the theoretical results.

Convergence of a Difference Scheme for Conservation Laws with a Discontinuous Flux

Abstract: Convergence is established for a scalar finite difference scheme, based on the Godunov or Engquist--Osher (EO) flux, for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. Other works in this direction have established convergence for methods employing the solution of 2 × 2 Riemann problems. The algorithm discussed here uses only scalar Riemann solvers. Satisfaction of a set of Kruzkov-type entropy inequalities is established for the limit solution, from which geometric entropy conditions follow. Assuming a piecewise constant coefficient, it is shown that these conditions imply L1-contractiveness for piecewise C1 solutions, thus extending a well-known theorem.

SRB measures for non-hyperbolic systems with multidimensional expansion

Abstract: We construct ergodic absolutely continuous invariant probability measures for an open class of non-hyperbolic surface maps introduced by Viana (1997), who showed that they exhibit two positive Lyapunov exponents at almost every point. Our approach involves an inducing procedure, based on the notion of hyperbolic time that we introduce here, and contains a theorem of existence of absolutely continuous invariant measures for multidimensional piecewise expanding maps with countably many domains of smoothness.

Optimal controllers for hybrid systems: stability and piecewise linear explicit form

Abstract: We propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical framework, or, equivalently, as piecewise affine systems. A stabilizing controller is obtained by designing a model predictive controller, which is based on the minimization of a weighted 1//spl infin/-norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a mixed-integer linear program (MILP) which depends on the current state. Although efficient branch and bound algorithms exist to solve MILPs, these are known to be NP-hard problems, which may prevent their online solution if the sampling-time is too small for the available computation power. Rather than solving the MILP online, we propose a different approach where all the computation is moved off line, by solving a multiparametric MILP. As the resulting control law is piecewise affine, online computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of a heat exchange system shows the potential of the method.

Continuous time state space modeling of panel data by means of sem

Abstract: Maximum likelihood parameter estimation of the continuous time linear stochastic state space model is considered on the basis of largeN discrete time data using a structural equation modeling (SEM) program. Random subject effects are allowed to be part of the model. The exact discrete model (EDM) is employed which links the discrete time model parameters to the underlying continuous time model parameters by means of nonlinear restrictions. The EDM is generalized to cover not only time-invariant parameters but also the cases of stepwise time-varying (piecewise time-invariant) parameters and parameters varying continuously over time according to a general polynomial scheme. The identification of the continuous time parameters is discussed and an educational example is presented.

Optimal Policies of Yield Management with Multiple Predetermined Prices

Abstract: It is a common practice for industries to price the same products at different levels. For example, airlines charge various fares for a common pool of seats. Seasonal products are sold at full or discount prices during different phases of the season. This article presents a model that reflects this yield management problem. The model assumes that (1) products are offered at multiple predetermined prices over time; (2) demand is price sensitive and obeys the Poisson process; and (3) price is allowed to change monotonically, i.e., either the markup or markdown policy is implemented. To maximize the expected revenue, management needs to determine the optimal times to switch between prices based on the remaining season and inventory. Major results in this research include (1) an exact solution for the continuous-time model; (2) piecewise concavity of the value function with respect to time and inventory; and (3) monotonicity of the optimal policy. The implementation of optimal policies is fairly facile because of the existence of threshold points embedded in the value function. The value function and time thresholds can be solved with a reasonable computation effort. Numerical examples are provided.

Mixing Strategies for Density Estimation

Abstract: General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under Kullback-Leibler and squared $L_2$ losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks. A consequence is that under some mild conditions, an asymptotically minimax-rate adaptive estimator exists for a given countable collection of density classes; that is, a single estimator can be constructed to be simultaneously minimax-rate optimal for all the function classes being considered. A demonstration is given for high-dimensional density estimation on $[0,1]^d$ where the constructed estimator adapts to smoothness and interaction-order over some piecewise Besov classes and is consistent for all the densities with finite entropy.

Applied nonlinear dynamics and chaos of mechanical systems with discontinuities

Abstract: General model of system with discontinuities systems with clearances systems with piecewise variable stiffness impacting systems systems with dry friction systems with multi-mode discontinuities smoothing discontinuities discussion.

Fast Solvers of the Lippmann-Schwinger Equation

Abstract: The electromagnetic and acoustic scattering problems for the Helmholtz equation in two and three dimensions are equivalent to the Lippmann-Schwinger equation which is a weakly singular volume integral equation on the support of the scatterer. We propose for the Lippmann-Schwinger equation two discretizations of the optimal accuracy order, accompanied by fast solvers of corresponding systems of linear equations. The first method is of the second order and based on simplest cubatures; the scatterer is allowed to be only piecewise smooth. The second method is of arbitrary order and is based on a fully discrete version of the collocation method with trigonometric test functions; the scatterer is assumed to be smooth on whole space ℝn and of compact support.

A plane-sweep strategy for the 3d reconstruction of buildings from multiple images

Abstract: A new method is described for automatically reconstructing a 3D piecewise planar model from multiple images of a scene. The novelty of the approach lies in the use of inter-image homographies to validate and best estimate planar facets, and in the minimal initialization requirements — only a single 3D line with a textured neighbourhood is required to generate a plane hypothesis. The planar facets enable line grouping and also the construction of parts of the wireframe which were missed due to the inevitable shortcomings of feature detection and matching. The method allows a piecewise planar model of a scene to be built completely automatically, with no user intervention at any stage, given only the images and camera projection matrices as input. The robustness and reliability of the method are illustrated on several examples, from both aerial and interior views.

Dynamics of piecewise isometries

Abstract: We begin a systematic study of Euclidean piecewise isometric dynamical systems (p.i.d.s.) with a particular focus on the interplay between geometry, symbolic dynamics, and the group of isometries associated with p.i.d.s. We investigate various aspects of the dynamical information contained in the coding: symbolic growth and the periodic behavior of codings and cells. This theoretical investigation is motivated by the many examples of piecewise isometric dynamical systems found recently in the literature. Piecewise isometric dynamical systems are direct generalizations of interval exchange transformations to non-invertible, higher dimensional maps.

Distribution of Lattice Points Visible from the Origin

Abstract: Let Ω be a region in the plane which contains the origin, is star-shaped with respect to the origin and has a piecewise C 1 boundary. For each integer Q≥ 1, we consider the integer lattice points from which are visible from the origin and prove that the 1 st consecutive spacing distribution of the angles formed with the origin exists. This is a probability measure supported on an interval [m Ω,∞), with m Ω >0. Its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel.

Learning Statistical Models of Human Motion

Abstract: Non-linear statistical models of deformation provide methods to learn a priori shape and deformation for an object or class of objects by example. This paper extends these models of deformation to that of motion by augmenting the discrete representation of piecewise nonlinear principle component analysis of shape with a markov chain which represents the temporal dynamics of the model. In this manner, mean trajectories can be learnt and reproduced for either the simulation of movement or for object tracking. This paper demonstrates the use of these techniques in learning human motion from capture data.

A Comparison of Regression Spline Smoothing Procedures

Abstract: Regression spline smoothing involves modelling a regression function as a piecewise polynomial with a high number of pieces relative to the sample size. Because the number of possible models is so large, efficient strategies for choosing among them are required. In this paper we review approaches to this problem and compare them through a simulation study. For simplicity and conciseness we restrict attention to the univariate smoothing setting with Gaussian noise and the truncated polynomial regression spline basis.

Correction of temperature-induced spectral variation by continuous piecewise direct standardization

Abstract: In process analytical applications it is not always possible to keep the measurement conditions constant. However, fluctuations in external variables such as temperature can have a strong influence on measurement results. For example, nonlinear temperature effects on near-infrared (NIR) spectra may lead to a strongly biased prediction result from multivariate calibration models such as PLS. A new method, called Continuous Piecewise Direct Standardization (CPDS) has been developed for the correction of such external influences. It represents a generalization of the discrete PDS calibration transfer method and is able to adjust for continuous nonlinear influences such as the temperature effects on spectra. It was applied to short-wave NIR spectra of ethanol/water/2-propanol mixtures measured at different temperatures in the range 30−70 °C. The method was able to remove, almost completely, the temperature effects on the spectra, and prediction of the mole fractions of the chemical components was close to the...