Showing papers on "Piecewise linear function published in 1992"

Surface reconstruction from unorganized points

Abstract: This thesis describes a general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points. Instances of surface reconstruction arise in numerous scientific and engineering applications, including reverse-engineering--the automatic generation of CAD models from physical objects. Previous surface reconstruction methods have typically required additional knowledge, such as structure in the data, known surface genus, or orientation information. In contrast, the method outlined in this thesis requires only the 3D coordinates of the data points. From the data, the method is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners. The reconstruction method has three major phases: (1) initial surface estimation, (2) mesh optimization, and (3) piecewise smooth surface optimization. A key ingredient in phase 3, and another principal contribution of this thesis, is the introduction of a new class of piecewise smooth representations based on subdivision. The effectiveness of the three-phase reconstruction method is demonstrated on a number of examples using both simulated and real data. Phases 2 and 3 of the surface reconstruction method can also be used to approximate existing surface models. By casting surface approximation as a global optimization problem with an energy function that directly measures deviation of the approximation from the original surface, models are obtained that exhibit excellent accuracy to conciseness trade-offs. Examples of piecewise linear and piecewise smooth approximations are generated for various surfaces, including meshes, NURBS surfaces, CSG models, and implicit surfaces.

On the Convergence of Reflective Newton Methods for Large-scale Nonlinear Minimization Subject to Bounds

Abstract: We consider a new algorithm, a reflective Newton method, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generates strictly feasible iterates by following piecewise linear paths ("reflection" paths) to generate improved iterates. The reflective Newton approach does not require identification as an "activity set." In this report we establish that the reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of this general reflective path approach suitable for large-scale and sparse problems.

Topological lattice models in four-dimensions

Abstract: We define a lattice statistical model on a triangulated manifold in four dimensions associated to a group G. When G=SU(2), the statistical weight is constructed from the 15j-symbol as well as the 6j-symbol for recombination of angular momenta, and the model may be regarded as the four-dimensional version of the Ponzano-Regge model. We show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold. For an orientable manifold, the model is related to the so-called BF model. The q-analog of the model is also constructed, and it is argued that its partition function is invariant under the Alexander moves. It is discussed how to realize the 't Hooft operator in these models associated to a closed surface in four dimensions as well as the Wilson operator associated to a closed loop. Correlation functions of these operators in the q-deformed version of the model would define a new type of invariants of knots and links in four dimensions.

Multilevel preconditioning

Abstract: This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets.

Analog reconstruction circuit and bar code reading apparatus employing same

Abstract: A bar code detection circuit accepts as input the discretized analog output of a CCD array, and performs piecewise linear reconstruction to produces a continuous polylinear output signal. In the region of a bar/space transition, the output signal is a close approximation of the reflectance function of a bar code symbol convolved with the system transfer function of the bar code reader. Linear interpolation is performed in order to determine the offset of a given threshold value from an edge of the CCD analog output.

A linear discretization of the volume conductor boundary integral equation using analytically integrated elements (electrophysiology application)

Abstract: A method is presented to compute the potential distribution on the surface of a homogeneous isolated conductor of arbitrary shape. The method is based on an approximation of a boundary integral equation as a set of linear algebraic equations. The potential is described as a piecewise linear or quadratic function. The matrix elements of the discretized equation are expressed as analytical formulas. >

Groups of piecewise linear homeomorphisms

Abstract: In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups

Optimal Semi-Active Suspension with Preview Based on a Quarter Car Model

Abstract: This paper deals with the synthesis of an optimal yet practical finite preview controller for a semi-active dissipative suspension system based on a two - degree - of - freedom (2-DOF) vehicle model. The proposed controller utilises knowledge about approaching road disturbances obtained from preview sensors to minimise the effect of these disturbances. A truly optimal control law, which minimises a quadratic performance index under passivity constraints, is derived using a variational approach. The optimal closed loop system becomes piecewise linear varying between two passive systems and a fully active one. It is shown that the steady state system response to a periodic input is also periodic and its amplitude is proportional to the amplitude of the input. Therefore, frequency domain characteristics in a classical sense can be generated. The problem formulation and the analytical solution are given in a general form and hence they apply to any bilinear system with system disturbances that are a priori unknown but some preview information is possible. The results of this analysis are applied to a quarter car model with semi-active suspension whose frequency domain and time domain performances are evaluated and compared to those of fully active and passive models. The effect of preview time on the system performance is also examined.

An efficient algorithm for finding all solutions of piecewise-linear resistive circuits

Abstract: An efficient algorithm for finding all solutions of piecewise-linear resistive circuits is presented. First, a technique that substantially reduces the number of function evaluations needed in the piecewise-linear modeling process is proposed. Then a simple and efficient sign test is proposed that remarkably reduces the number of linear simultaneous equations to be solved for finding all solutions. An effective technique that makes the sign test even more efficient is introduced. All of the techniques exploit the separability of nonlinear mappings. Some numerical examples are given, and it is shown that all solutions are computed rapidly. The algorithm is simple and efficient, and can be easily programmed. >

Finite-state models in the alignment of macromolecules.

Abstract: Minimum message length encoding is a technique of inductive inference with theoretical and practical advantages. It allows the posterior odds-ratio of two theories or hypotheses to be calculated. Here it is applied to problems of aligning or relating two strings, in particular two biological macromolecules. We compare the r-theory, that the strings are related, with the null-theory, that they are not related. If they are related, the probabilities of the various alignments can be calculated. This is done for one-, three-, and five-state models of relation or mutation. These correspond to linear and piecewise linear cost functions on runs of insertions and deletions. We describe how to estimate parameters of a model. The validity of a model is itself an hypothesis and can be objectively tested. This is done on real DNA strings and on artificial data. The tests on artificial data indicate limits on what can be inferred in various situations. The tests on real DNA support either the three- or five-state models over the one-state model. Finally, a fast, approximate minimum message length string comparison algorithm is described.

The complete canonical piecewise-linear representation. I. The geometry of the domain space

Abstract: Continuous piecewise-linear functions from R/sup n/ to R/sup m/ are analyzed in terms of the dimensions of their domain space and of degenerate kth-order intersections of region boundaries. The theory developed demonstrates how these two quantities are connected. Moreover, the exact number of independent parameters is demonstrated for boundary configurations containing degenerate intersections of arbitrary orders. >

A comparison of piecewise linear model descriptions

Abstract: Current methods for storing piecewise-linear mappings are discussed and compared. To do so, the model descriptions are all transformed into a general form. Among the aspects compared are the ease of modeling, the class of functions that can be modeled using a certain model description, and the suitability of using the models in simulators. No best model description is found, but it is made apparent that some models are better suited for certain applications than others

Nonlinear dynamics and symbolic dynamics of neural networks

Abstract: A piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks. A symbolic representation of the dynamics in this equation is given as a directed graph on an N-dimensional hypercube. This provides a formal link with discrete neural networks such as the original Hopfield models. Analytic criteria are given to establish steady states and limit cycle oscillations independent of network dimension. Model networks that display multiple stable limit cycles and chaotic dynamics are discussed. The results show that such equations are a useful and efficient method of investigating the behavior of neural networks.

A simplex algorithm for piecewise-linear programming III: computational analysis and application

Abstract: The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustarate these arguments; the piecewise-linear simplex algorithm is observed to run 2–6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linear programming algorithm is provided by a survey of many varied applications.

A boundary-crossing result for Brownian motion

Abstract: In this paper we obtain a new crossing result for Brownian motion. The boundary studied is a piecewise linear function consisting of two lines. The expression obtained for the boundary crossing probability is of a simple directly computable form.

Variational parameter estimation for a two-dimensional numerical tidal model

Abstract: SUMMARY It is shown that the parameters in a two-dimensional (depth-averaged) numerical tidal model can be estimated accurately by assimilation of data from tide gauges. The tidal model considered is a semilinearized one in which kinematical non-linearities are neglected but non-linear bottom friction is included. The parameters to be estimated (bottom friction coefficient and water depth) are assumed to be positiondependent and are approximated by piecewise linear interpolations between certain nodal values. The numerical scheme consists of a two-level leapfrog method. The adjoint scheme is constructed on the assumption that a certain norm of the difference between computed and observed elevations at the tide gauges should be minimized. It is shown that a satisfactory numerical minimization can be completed using either the Broyden-Fletcher-GoldfarbShanno (BFGS) quasi-Newton algorithm or Nash's truncated Newton algorithm. On the basis of a number of test problems, it is shown that very effective estimation of the nodal values of the parameters can be achieved provided the number of data stations is sufficiently large in relation to the number of nodes.

Ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2-torus

Abstract: We study generic piecewise linear hyperbolic automorphisms of the 2-torus. We explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations.

Projection functions for particle‐grid methods

Abstract: Random walk particle methods (RWPM) can be used in operator splitting schemes to simulate reactive solute transport in porous media. Projection functions are used to transfer particle location and mass information to concentrations at selected spatial points. Because of the stochastic nature of RWPM, concentration estimates made from particle distributions include a “noisy” error component. In some cases of reactive or density-dependent flows, this type of error may be propagated forward in time. It can be reduced by using larger numbers of particles or by using different projection functions. The effects of using different projection functions or numbers of particles in different flow regimes or dimensions are explored using concentration solutions for a set of one-, two-, and three-dimensional nonreactive test problems. Resulting solutions are compared with analytic results and classical random walk error estimates. A piecewise linear projection function provides a reasonable improvement in accuracy over the more convenient box methods at a modest increase in cost. The support of the projection functions should be O(Δx) to avoid excessive smearing. Multidimensional projection functions may be advantageously formed by products of different one-dimensional projection functions.

Piecewise Linear Models for Switch-Level Simulation

Abstract: Rsim is an efficient logic plus timing simulator that employs the switched resistor transistor model and RC tree analysis to simulate efficiently MOS digital circuits at the transistor level. We investigate the incorporation of piecewise linear transistor models and generalized moments matching into this simulation framework. General piecewise linear models allow more accurate MOS models to be used to simulate circuits that are hard for Rsim. Additionally, they enable the simulator to handle circuits containing bipolar transistors such as ECL and BiCMOS. Nonetheless, the switched resistor model has proved to be efficient and accurate for a large class of MOS digital circuits. Therefore, it is retained as just one particular model available for use in this framework. The use of piecewise linear models requires the generalization of RC tree analysis. Unlike switched resistors, more general models may incorporate gain and floating capacitance. Additionally, we extend the analysis to handle non-tree topologies and feedback. Despite the increased generality, for many common MOS and ECL circuits the complexity remains linear. Thus, this timing analysis can be used to simulate, efficiently, those portions of the circuit that are well described by traditional switch level models, while simultaneously simulating, more accurately, those portions that are not. We present preliminary results from a prototype simulator, Mom. We demonstrate its use on a number of MOS, ECL, and BiCMOS circuits.

Characteristic, local grid refinement techniques for reservoir flow problems

Abstract: Using a sequential time-marching approach, we describe and implement effective solution strategies for the equations governing immiscible displacement in a porous medium. By combining adaptive grid refinement with an operator-splitting technique based on the modified method of characteristics, the saturation equation is treated in a consistent and accurate way. For the pressure equation we construct accurate piecewise linear velocity components from a piecewise linear pressure approximation.

A global and quadratically convergent method for linear l ∞ problems

Abstract: A new globally and quadratically convergent algorithm is proposed for the linear $l_\infty $ problem. This method works on the piecewise linear $l_\infty $, problem directly by generating descent directions—via a sequence of weighted least squares problems—and using a piecewise linear line search to ensure a decrease in the $l_\infty $ function at every step. It is proven that ultimately full Newton-like steps are taken where the Newton step is based on the complementary slackness condition holding at the solution. Numerical results suggest a very promising method; the number of iterations required to achieve high accuracy is relatively insensitive to problem size.

A piecewise-linear function approximation using current mode circuits

Abstract: The technique is based on the utilization of current mirrors as basic building blocks. The resulting circuits are compact, modular, and programmable and can operate at very high frequencies. Experimental results are presented that verify the proposed theoretical technique. >

Gauss-seidel method for least-distance problems

Abstract: In this paper, we reformulate the least-distance problems with bounded inequality constraints as an unconstrained convex minimization problem, which is equivalent to a system of piecewise linear equationsA(a+A T y) c d =b. The proposed Gauss-Seidel method for solving the problems is easy to implement and behaves very well when the number of rows ofA is much less than the number of columns ofA. Moreover, we prove that the Gauss-Seidel method has a linear convergence rate.

An effective method for computing regression quantiles

Abstract: Regression quantiles were introduced in Koenker & Bassett [7] as quantities of interest in developing robust estimation procedures. They can be computed by linear programming combined with post optimality techniques. Here an effective alternative is presented based on the reduced gradient algorithm for l 1 fitting as described in [8] combined with a piecewise linear homotopy. There is a close connection between the two approaches (the new method essentially describes what is going on in the linear programming), but it is argued that the new approach is preferable. Its robustness as a computational procedure is illustrated by several examples which give rise to a variety of different behaviours

Orienting generalized polygonal parts

Abstract: K.Y. Goldberg (1990) described an algorithm for orienting planar polygonal parts using a modified parallel-jaw gripper. The authors extend this algorithm to handle parts with curved edges. They define a generalized polygon as a planar figure made up of piecewise linear and circular edges, and a generalized polygonal part as one of constant cross section, the convex hull of which is a generalized polygon. For this class of parts a complete algorithm for determining an optimal parts-orienting strategy is presented. >

Superconvergence of recovered gradients of discrete time/piecewise linear Galerkin approximations for linear and nonlinear parabolic problems

Abstract: Superconvergent error estimates in l2(H1) and l∞(H1) norms are derived for recovered gradients of finite difference in time/piecewise linear Galerkin approximations in space for linear and quasinonlinear parabolic problems in two space dimensions The analysis extends previous results for elliptic problems to the parabolic context, and covers problems in regions with nonsmooth boundaries under certain assumptions on the regularity of the solutions © 1994 John Wiley & Sons, Inc