Abstract: We examine bifurcation phenomena for continuous one-dimensional maps that are piecewise smooth and depend on a parameter μ. In the simplest case, there is a point c at which the map has no derivative (it has two one-sided derivatives). The point c is the border of two intervals in which the map is smooth. As the parameter μ is varied, a fixed point (or periodic point) Eμ may cross the point c, and we may assume that this crossing occurs at μ=0. The investigation of what bifurcations occur at μ=0 reduces to a study of a map fμ depending linearly on μ and two other parameters a and b. A variety of bifurcations occur frequently in such situations. In particular, Eμ may cross the point c, and for μ 0 there may be a period-3 attractor or even a three-piece chaotic attractor which shrinks to E0 as μ→0. More generally, for every integer m≥2, bifurcations from a fixed point attractor to a period-m attractor, a 2m-piece chaotic attractor, an m-piece chaotic attractor, or a one-piece chaotic attractor can occur for piecewise smooth one-dimensional maps. These bifurcations are called border-collision bifurcations. For almost every point in the region of interest in the (a, b)-space, we state explicitly which border-collision bifurcation actually does occur. We believe this phenomenon will be seen in many applications.

Abstract: Conewise linear elastic (CLE) materials are proposed as the proper generalization to two and three dimensions of one-dimensionalbimodular models. The basic elements of classical smooth elasticity are extended tononsmooth (or piecewise smooth) elasticity. Firstly, a necessary and sufficient condition for a stress-strain law to becontinous across the interface of the tension and compression subdomains is established. Secondly, a sufficient condition for the strain energy function to be strictlyconvex is derived. Thirdly, the representations of the energy function, stress-strain law and elasticity tensor are obtained fororthotropic, transverse isotropic andisotropic CLE materials. Finally, the previous results are specialized to apiecewise linear stress-strain law and it is found out that the pieces must be polyhedral convex cones, thus the CLE name.

Abstract: Segmentation of range images has long been considered in computer vision as an important but extremely difficult problem. In this paper we present a new paradigm for the segmentation of range images into piecewise continuous surfaces. Data aggregation is performed via model recovery in terms of variable-order bi-variate polynomials using iterative regression. Model recovery is initiated independently in regularly placed seed regions in the image. All the recovered models are potential candidates for the final description of the data. Selection of the models is defined as a quadratic Boolean problem, and the solution is sought by the WTA (winner-takes-all) technique, which turns out to be a good compromise between the speed of computation and the accuracy of the solution. The overall efficiency of the method is achieved by combining model recovery and model selection in an iterative way. Partial recovery of the models is followed by the selection (optimization) procedure and only the “best” models are allowed to develop further. The major novelty of the approach lies in an effective combination of simple component algorithms, which stands in contrast to methods which attempt to solve the problem in a single processing step using sophisticated means. We present the results on several real range images.

Abstract: We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators. We show that, unlike the analogous situation for kernel density estimators, this MISE formula is relatively unaffected by assumptions of continuity. In particular, it is available for densities which are smooth in only a piecewise sense. Another difference is that in the wavelet case the classical MISE formula is valid only for sufficiently small values of the bandwidth. For larger bandwidths MISE assumes a very different form and hardly varies at all with changing bandwidth. This remarkable property guarantees a high level of robustness against oversmoothing, not encountered in the context of kernel methods. We also use the MISE formula to describe an asymptotically optimal empirical bandwidth selection rule.

Abstract: While the ML-EM algorithm for reconstruction for emission tomography is unstable due to the ill-posed nature of the problem. Bayesian reconstruction methods overcome this instability by introducing prior information, often in the form of a spatial smoothness regularizer. More elaborate forms of smoothness constraints may be used to extend the role of the prior beyond that of a stabilizer in order to capture actual spatial information about the object. Previously proposed forms of such prior distributions were based on the assumption of a piecewise constant source distribution. Here, the authors propose an extension to a piecewise linear model-the weak plate-which is more expressive than the piecewise constant model. The weak plate prior not only preserves edges but also allows for piecewise ramplike regions in the reconstruction. Indeed, for the authors' application in SPECT, such ramplike regions are observed in ground-truth source distributions in the form of primate autoradiographs of rCBF radionuclides. To incorporate the weak plate prior in a MAP approach, the authors model the prior as a Gibbs distribution and use a GEM formulation for the optimization. They compare quantitative performance of the ML-EM algorithm, a GEM algorithm with a prior favoring piecewise constant regions, and a GEM algorithm with their weak plate prior. Pointwise and regional bias and variance of ensemble image reconstructions are used as indications of image quality. The authors' results show that the weak plate and membrane priors exhibit improved bias and variance relative to ML-EM techniques.

Abstract: The paper presents a discussion on the application of the Kirchhoff transformation to the thermal analysis of semiconductor devices with temperature-dependent and piecewise inhomogeneous thermal conductivity. The Kirchhoff transformation is shown to generally reduce the problem to the solution of the linear heat equation with nonlinear jump conditions on the apparent temperature across subdomains, unless the ratio of the thermal conductivities is temperature independent, in which case the apparent temperature is continuous. In many practical cases, the temperature dependence of the thermal conductivity can be approximated in all subdomains so as to enforce this condition; one and two-dimensional examples are discussed to show that in realistic configurations (devices with metal heat sinks, multilayered structures made of different semiconductors) the error thereby introduced is acceptably low.

Abstract: A class of linear-quadratic piecewise deterministic soft-constrained zero-sum differential games is formulated and solved, where the minimizing player has access to perfect or imperfect (continuous) state measurements. Such systems are also known as jump linear-quadratic systems, and the underlying game problem can also be viewed as an H∞ optimal control problem, where the system and cost matrices depend on the outcome of a Markov chain. Both finite- and infinite-horizon cases are considered, and a set of sufficient, as well as a set of necessary, conditions are obtained for the upper value of the game to be bounded. Policies for the minimizing player that achieve this upper value (which is zero) are piecewise linear on each sample path of the stochastic process, and are obtained from solutions of linearly coupled generalized Riccati equations. For the associated H∞-optimal control problem, these policies guarantee an L 2 gain type inequality on the closed-loop system.

Abstract: The NIRVANA project is concerned with the development of a nonoscillatory, integrally reconstructed, volume‐averaged numerical advection scheme. The conservative, flux‐based finite‐volume algorithm is built on an explicit, single‐step, forward‐in‐time update of the cell‐average variable, without restrictions on the size of the time‐step. There are similarities with semi‐Lagrangian schemes; a major difference is the introduction of a discrete integral variable, guaranteeing conservation. The crucial step is the interpolation of this variable, which is used in the calculation of the fluxes; the (analytic) derivative of the interpolant then gives sub‐cell behaviour of the advected variable. In this paper, basic principles are described, using the simplest possible conditions: pure one‐dimensional advection at constant velocity on a uniform grid. Piecewise Nth‐degree polynomial interpolation of the discrete integral variable leads to an Nth‐order advection scheme, in both space and time. Nonoscillatory result...

Abstract: The construction of quadratic $C^1 $ surfaces from B-spline control points is generalized to a wider class of control meshes capable of outlining arbitrary free-form surfaces in space. Irregular meshes with nonquadrilateral cells and more or less than four cells meeting at a point are allowed so that arbitrary free-form surfaces with or without boundary can be modeled in the same conceptual frame work as tensor-product B-splines. That is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local, evaluates by averaging, and obeys the convex hull property. For a regular region of the input mesh, the representation reduces to the standard quadratic spline. In general, a surface spline is represented by Bernstein–Bezier patches of degree two and three with derivatives matching across boundaries after local reparametrization. According to the user’s choice, these patches can be polynomial or rational, and three-sided, four-sided, or a combination thereof.

Abstract: We present efficient algorithms for interference detection between geometric models described by linear or curved boundaries and undergoing rigid motion. The set of models include surfaces described by rational spline patches or piecewise algebraic functions. In contrast to previous approaches, we first describe an efficient algorithm for interference detection between convex polytopes using coherence and local features. Then an extension using hierarchical representation to concave polytopes is presented. We apply these algorithms along with properties of input models, local and global algebraic methods for solving polynomial equations, and the geometric formulation of the problem to devise efficient algorithms for convex and nonconvex curved objects. Finally, a scheduling scheme to reduce the frequency of interference detection in large environments is described. These algorithms have been successfully implemented and we discuss their performance in various environments.

Abstract: We consider singularly perturbed high-order elliptic two-point boundary value problems of convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear form is proved and a representation result for the solutions of such problems is given. A family of Galerkin finite-element methods based on piecewise polynomial test/trial functions on a Shishkin mesh is constructed and proved to be convergent, uniformly in the perturbation parameter, in energy and W∞ k norms. Numerical results are presented for a second-order problem and fourth-order problems

Abstract: In this paper we define and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations. We prove that these estimators are equivalent to an appropriate norm of the error. For the case of piecewise linear elements we define two estimators. Both of them are easy to compute, but the second is simpler because it can be computed using only the right-hand side and the approximate velocity. We show how the first estimator can be generalized to higher-order elements. Finally, we present several numerical examples in which one of our estimators is used for adaptive refinement.

Abstract: Electric utilities are becoming increasingly interested in using synchronized phasor measurements from around power systems to enhance their protection and remedial action control strategies. Accordingly, the task of predicting future behavior of the power system before it actually occurs has become an important area of research. This paper presents and analyses several approaches for solving the real-time prediction problem. In order to solve power systems with detailed load models fast enough for real-time prediction, the authors present a new piecewise constant current load model approximation technique that can solve a model as complex as the New England 39 bus system with composite voltage dependent loads much faster than in real-time. If the reduced order model is too large for real-time solution, then a pattern recognition tool such as decision trees can be trained off line to associate the post-fault phasor measurements with the outcome of future behavior. In this case also, the piecewise constant current technique would be needed to perform the offline training set generation with sufficient speed and accuracy. >

Abstract: The finite-element method possesses many advantages over more traditional numerical techniques used to solve systems of differential equations. These advantages include a number of conservation properties and a natural treatment of boundary conditions. The method's piecewise nature makes it useful when dealing with irregular domains and similarly when using variable horizontal resolution. To take advantage of these properties, a finite-element representation of the linearized, steady-state, barotropic potential vorticity equation is developed. The Stommel problem is used as an initial test for the model. A fourth-order eddy viscosity term is then added, and the resulting problem is solved in both simply and multiply connected domains under both slip and no-slip boundary conditions. The beta-plane assumption is then relaxed, and the model is reformulated in spherical coordinates. A realistic geography and topography version of this model is also used to examine the barotropic circulation in the No...

Abstract: Sufficient conditions for solution Lipschitz continuity, piecewise differentiability, and directional differentiability are presented for parametric nonlinear programs and variational inequalities using the idea of continuous selections. The gaps between the sufficient conditions obtained here and the weakest possible conditions for the corresponding conclusions are discussed and measured by known regularity conditions.

Abstract: When manipulating parts, it is important to determine the orientation of the part with respect to the gripper. This orientation may not be known precisely or may be disturbed by the act of grasping. In some cases, it is possible to use mechanical compliance to orient parts during grasping. Goldberg (1993) showed that any part with polygonal boundary can be oriented and grasped in this manner using a parallel-jaw gripper. Many of the curves currently used in engineering design are algebraic but nonlinear. Although these curves can be approximated as polygons for the purpose of visualization, such approximations can lead to false conclusions about mechanical behavior. In this paper we consider the class of parts whose planar projection has a piecewise algebraic convex hull. Our primary result is a proof that a grasp plan exists for any such part. We give a planning algorithm that produces the shortest plan and runs in time O(n/sup 2/ log n+N), where n is the number of transitions in the grasp function and N is the length of the plan produced. We believe this to be the first paper to address the problem of manipulating algebraic parts when nothing is know about their initial orientation. >

Abstract: Numerical methods composed of upwind-difference operators on uniform meshes are shown analytically to be defective for solving singularly perturbed differential equations, in the sense that, as the mesh is refined, the error in the numerical approximation increases until the mesh parameter has decreased to the same order of magnitude as the singular perturbation parameter. It is also shown that the same is true for upwind-difference operators on piecewise uniform meshes having a transition point that is dependent solely on the singular perturbation parameter. It is then shown, with specific analytic examples, that both upwind- and central-difference operators on specially designed piecewise-uniform meshes give numerical methods which do not suffer from this defect. Conditions are also given on the structure of a piecewise uniform mesh that are necessary if the numerical method, composed of this mesh and an upwind-difference operator, is to be convergent uniformly with respect to the singular perturbation parameter

Abstract: Motivated by problems in computational geometry, this paper investigates the complexity of finding a fixed-point of the composition of two or three continuous functions that are defined piecewise. It shows that certain cases require nested binary search taking Θ(log2 n) time. Other cases can be solved in logarithmic time by using a prune-and-search technique that may make tentative discards and later revoke or certify them. This work finds application in optimal subroutines that compute approximations to convex polygons, dense packings, and Voronoi vertices for Euclidean and polygonal distance functions.

Abstract: The construction of a surface from arbitrarily scattered data is an important problem in many applications. When there are a large number of data points, the surface representations generated by interpolation methods may be inefficient in both storage and computational requirements. This paper describes an adaptive method for smooth surface approximation from scattered 3D points. The approximating surface is represented by a piecewise cubic triangular Bezier surface possessing C 1 continuity. The method begins with a rough surface interpolating only boundary points and, in the successive steps, refines it by adding the maximum error point at a time among the remaining internal points until the desired approximation accuracy is reached. Our method is simple in concept and efficient in computational time, yet realizes efficient data reduction. Some experimental results are given to show that surface representations constructed by our method are compact and faithful to the original data points.

Abstract: A conservative and monotonic remapping algorithm is developed, which could be used as a component of a semi-Lagrangian transport scheme for numerical models of the atmosphere. The algorithm is a version of the monotonic piecewise parabolic interpolations combined with a “cascade” approach, where only one-dimensional operators are used for formulation of the multidimensional scheme.