Showing papers on "Maxima and minima published in 1996"

First Step Towards Planning of Syntheses in Solid-State Chemistry: Determination of Promising Structure Candidates by Global Optimization

Abstract: A method is presented here that allows, in principle, the prediction of the existence and structure of (meta)stable solid compounds. It is based on a set of adjustable modules that are applied to the study of the energy function of the chemical system of interest. The main elements are a set of routines for global optimization and local minimization, as well as algorithms for the investigation of the phase space structure near local minima of the potential energy, and the analysis and characterization of the structure candidates. The current implementation focuses on ionic compounds, for which empirical potentials are used for the evaluation of the energy function in the first stage, and a Hartree–Fock algorithm for refinements. The global optimization is performed with a stochastic simulated annealing algorithm, and the local minimization employs stochastic quenches and gradient methods. The neighborhoods of the local minima are studied with the threshold algorithm. The results of this approach are illustrated with a number of examples: compounds of binary noble gases, and binary and ternary ionic compounds. These include several substances that have not been synthesized yet, but should stand a fair chance of being kinetically stable, for example further alkali metal nitrides besides Li3N, as well as Ca3SiBr2 or SrTi2O5.

Radiation pattern evaluation from near-field intensities on planes

Abstract: This paper describes, analyzes, and implements an approach to far-field determination from phaseless measurements over two planar surfaces. A proper formulation of the problem is considered as a quadratic inverse one whose data is the square amplitude of the near field. A solution is introduced as the global minimum of an appropriate functional. Next, to perform such a minimization procedure, a finite dimensional representation of the field radiated by sources whose plane-wave spectrum becomes negligible outside a fixed angular domain is used. A detailed investigation of the properties of the mapping connecting the unknowns to the data makes it possible to analyze how to escape from the local minima possibly met in the course of the minimization procedure. To this end, the crucial role of the availability of phaseless data over two different sarfa,ces and of appropriate weights in the functional definition is emphasized, and a reliable iterative procedure converging on the solution, regardless of the starting point, is thus obtained. This property is confirmed by experimental results concerning near-zone data from a shaped reflector at 9 Ghz. It can be readily appreciated that when only the field intensity is detected the complexity and the cost of the equipment required for near-field techniques in antenna testing and diagnostics can be reduced to a very large extent.

Gradient watersheds in morphological scale-space

Abstract: After introducing several results relating to the modification of the homotopy of gradient functions based on extrema in the base image and building on earlier results in morphological scale-space, we introduce a scale-space monotonicity theorem for regions of an image defined by watersheds of a gradient function modified to retain only the local minima or maxima of its smoothed parent image. We then illustrate the theorem with an example of the scale-space extraction of texture features from the nuclei of cervical cells.

Global Stability Analysis and Calculation of Liquid−Liquid Equilibrium in Multicomponent Mixtures†

Abstract: A global algorithm for implementing the Gibbs tangent plane stability test in multicomponent, multiphase liquid mixtures is described. The algorithm is self-starting and significantly improves the reliability and robustness of multiphase equilibrium calculations. The main improvement is a result of a new approach for locating the stationary points of the tangent plane distance function. This relies on the geometric properties of the tangent plane distance function but is independent of the specific fluid model. For ternary mixtures, it is shown that stationary points in the tangent plane distance function satisfy a global topological constraint Nmax + Nmin − Nsad = 1 where Nmax is the number of maxima, Nmin is the number of minima, and Nsad is the number of saddles. This provides a global consistency check on the numerical results that is independent of the specific model or parameters used to describe the mixture. The method has been implemented and tested on a variety of problems with three and four com...

Studying the energy hypersurface of continuous systems - the threshold algorithm

Abstract: A new method is presented for the study of the structure of the energy hypersurface of continuous systems. This so-called threshold algorithm is an adaptation of the `lid method' introduced by Sibani and co-workers in 1993 (Sibani P et al 1993 Europhys. Lett. 22 479 - 85) for the investigation of discrete energy landscapes. The algorithm produces an estimate of the local densities of states near deep-lying local minima of the potential energy of the system together with the barrier heights around these minima. This allows the computation of, for example, specific heats, the estimation of the kinetic stability of configurations that represent metastable minima of the potential energy, and the description of the relaxation behaviour of the system. As an example, a two-dimensional neon crystal is studied, where for example the calculated specific heat is found to agree with the experimental values of the three-dimensional case scaled to two dimensions.

The tunneling method for global optimization in multidimensional scaling

Abstract: This paper focuses on the problem of local minima of the STRESS function. It turns out that unidimensional scaling is particularly prone to local minima, whereas full dimensional scaling with Euclidean distances has a local minimum that is global. For intermediate dimensionality with Euclidean distances it depends on the dissimilarities how severe the local minimum problem is. For city-block distances in any dimensionality many different local minima are found. A simulation experiment is presented that indicates under what conditions local minima can be expected. We introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances. The tunneling method alternates a local search step, in which a local minimum is sought, with a tunneling step in which a different configuration is sought with the same STRESS as the previous local minimum. In this manner successively better local minima are obtained, and experimentation so far shows that the last one is often a global minimum.

Long-distance potentials: an approach to the multiple-minima problem in ligand-receptor interaction

Abstract: The multiple-minima problem is a classical problem in molecular structure prediction. For ligand-receptor systems, a possible direction to alleviate this major obstacle is to simplify the objective function (intermolecular energy) and smooth its profile. We introduce long-distance atom-atom potentials for ligand-receptor interactions. The longer ranges result in averaging of the energy potential at a given point. Our simplified force field is based on a trivial empirical representation of interatomic interactions as a step function. We demonstrate that the intermolecular energy calculation by a systematic search with such a simplified long-distance force field delivers the global minimum (crystallographically determined position of the ligand) by radically suppressing local minima (or false-positive fits). The effectiveness of the approach is demonstrated on different molecular complexes of known structure.

Simulated annealing wavelet estimation via fourth-order cumulant matching

Abstract: The fourth-order cumulant matching method has been developed recently for estimating a mixed-phase wavelet from a convolutional process. Matching between the trace cumulant and the wavelet moment is done in a minimum mean-squared error sense under the assumption of a non-Gaussian, stationary, and statistically independent reflectivity series. This leads to a highly nonlinear optimization problem, usually solved by techniques that require a certain degree of linearization, and that invariably converge to the minimum closest to the initial model. Alternatively, we propose a hybrid strategy that makes use of a simulated annealing algorithm to provide reliability of the numerical solutions by reducing the risk of being trapped in local minima. Beyond the numerical aspect, the reliability of the derived wavelets depends strongly on the amount of data available. However, by using a multidimensional taper to smooth the trace cumulant, we show that the method can be used even in a trace-by-trace implementation, which is very important from the point of view of stationarity and consistency. We demonstrate the viability of the method under several reflectivity models. Finally, we illustrate the hybrid strategy using marine and field real data examples. The consistency of the results is very encouraging because the improved cumulant matching strategy we describe can be effectively used with a limited amount of data.

On the local minima in phase reconstruction algorithms

Abstract: Signal reconstruction from phaseless data occurs in electromagnetics both in the diagnostics of large reflector antennas and in the near-field testing of microwave and millimeter wave antennas. The main difficulty in solving such problems arises from its inherent nonlinearity. Because of the latter, in fact, the solution procedure can converge to a function which has nothing to deal with the actual solution. In this paper we review and summarize the properties of a recently proposed approach to phase retrieval problems which allows to understand and possibly avoid the false solution problem. To this end, ill-posedness of the problem is briefly recalled, and a convenient generalized solution is introduced as the global minimum of a proper functional whose local minima are indeed the false solutions of the problem. Then the geometrical meaning of such a functional is considered, and properties of the solution are discussed according to the formulation of the problem as a quadratic inverse one. In particular, the relevance of using nonredundant representations for the unknown of the problem and as many independent data as possible are discussed in full detail from both a theoretical and a numerical point of view. Finally, it is shown how the proposed solution procedure, descending from the adopted formulation based on a weaker nonlinearity with respect to the other ones, has indeed a better behavior as far as local minima problem is concerned.

The virtual springs method: path planning and collision avoidance for redundant manipulators

Abstract: Potential-field based methods for path planning have the ad vantage of speed, but also a reputation for getting stuck in local minima when used with complex systems such as re dundant manipulators. We have developed a novel method of setting up a potential-field system that replaces rigid links by stiff (virtual) springs. The method is fast, directly suitable for parallel processing, and experimentally much less prone to the danger of local minima. Although quite general, it is par ticularly suited to use with manipulators whose links are thin compared with their length.

A new potential field-based algorithm for path planning

Abstract: In this paper, the path-planning problem is considered. We introduce a new potential function for path planning that has the remarkable feature that it is free from any local minima in the free space irrespective of the number of obstacles in the configuration space. The only global minimum is the goal configuration whose region of attraction extends over the whole free space. We also propose a new method for path optimization using an expanding sphere that can be used with any potential or penalty function. Simulations using a point mobile robot and smooth obstacles are presented to demonstrate the qualities of the new potential function. Finally, practical considerations are also discussed for nonpoint robots

The dynamics of minima and contours

Abstract: The dynamics has been introduced for ranking the minima of a topographic surface according to their contrast Constructing the watershed associated to the set of markers with a dynamic higher than a given threshold will produce a tesselation of the space As the threshold becomes higher, neighboring regions merge: the contours which vanish may be labeled by the dynamics for which the merging occurs

Conjugate points and shocks in nonlinear optimal control

Abstract: In this paper the authors use the method of characteristics to extend the Jacobi conjugate points theory to the Bolza problem arising in nonlinear optimal control. This yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation. The same approach allows to investigate as well smoothness of the value function.

Studying the energy hypersurface of multi‐minima systems — the threshold and the lid algorithm

Abstract: A method is presented that allows the detailed study of regions of multi-minima energy landscapes close to some deep-lying minimum. Two versions are described the lid algorithm and the threshold algorithm for discrete and continuous landscapes, respectively. Applications of these algorithms to the travelling salesman problem (TSP), a two-dimensional layer of Ne atoms, and the system Mg 2 F 4 are sketched.

A New Merit Function and A Successive Quadratic Programming Algorithm for Variational Inequality Problems

Abstract: Recently, various merit functions for variational inequality problems have been proposed and their properties have been studied. Unfortunately, these functions may not be easy to evaluate unless the constraints of the problem have a relatively simple structure. In this paper, a new merit function for variational inequality problems with general convex constraints is proposed. At each point, the proposed function is defined as an optimal value of a quadratic programming problem whose constraints consist of a linear approximation of the given nonlinear constraints. It is shown that the set of constrained minima of the proposed merit function coincides with the set of solutions to the original variational inequality problem. It is also shown that this function is directionally differentiable in all directions and, under suitable assumptions, any stationary point of the function over the constraint set actually solves the original variational inequality problem. Finally, a descent method for solving the variational inequality problem is proposed and its convergence is proved. The method is closely related to a successive quadratic programming method for solving nonlinear programming problems.

Length- and cost-dependent local minima of unconstrained blind channel equalizers

Abstract: Baud-rate linear blind equalizers may converge to undesirable stable equilibria due to different mechanisms One such mechanism is the use of linear FIR filters as equalizers It is shown that this type of local minima exist for all unconstrained blind equalizers whose cost functions satisfy two general conditions The local minima generated by this mechanism are thus called length-dependent local minima Another mechanism is generated by the cost function adopted by the blind algorithm itself This type of local minima are called cost-dependent local minima It is shown that several well-designed algorithms do not have cost-dependent local minimum, whereas other algorithms, such as the decision-directed equalizer and the stop-and-go algorithm (SGA), do Unlike many existing convergence analysis, the convergence of the Godard (1980) algorithms (GAs) and standard cumulant algorithms (SCAs) under Gaussian noise is also presented Computer simulations are used to verify the analytical results

An algebraic solution to the time difference of arrival equations

Abstract: A closed form, algebraic solution to the time difference of arrival equations is presented. Prior derivations assumed that the receiving antennas are coplanar. Our derivation allows the receiving antennas to be located anywhere in three dimensional space. Previous techniques address the over-determined problem by considering a weighted average of the set of minimally determined solutions. Our technique employs the generalized inverse as a means of forming an optimal solution based on the geometry of the receiving sites. Our solution is additionally compared with a gradient search method to demonstrate its ability to avoid local minima. Finally, analysis is presented to demonstrate the effect of noisy measurements on the solution.

Coherent optical neural networks that have optical-frequency-controlled behavior and generalization ability in the frequency domain

Abstract: Coherent optical neural networks that have optical-frequency-controlled behavior are proposed as sophisticated optical neural systems. The coherent optical neural-network system consists of an optical complex-valued neural network, a phase reference path, and coherent detectors for self-homodyne detection. The learning process is realized by adjusting the delay time and the transparency of neural connections in the optical neural network with the optical frequency as a learning parameter. Generalization ability in frequency space is also analyzed. Information geometry in the learning process is discussed for obtaining a parameter range in which a reasonable generalization is realized in frequency space. It is found that there are error-function minima periodically both in the delay-time domain and the input-signal-frequency domain. Because of this reason, the initial connection delay should be within a certain range for a meaningful generalization. Simulation experiments demonstrate that a stable learning and a reasonable generalization in the frequency domain are successfully realized in a parameter range obtained in the theory.

Modeling complex reservoir geometries with multiple-point statistics

Abstract: Large-scale reservoir architecture constitutes first-order reservoir heterogeneity and dietates to a large extent reservoir flow behavior. It also manifests geometric characteristics beyond the capability of traditional geostatistical models conditioned only on single-point and two-point statistics. Multiple-point statistics, as obtained by scanning a training image deemed representative of the actual reservoir, if reproduced properly provides stochastic models that better capture the essence of the heterogeneity. A “growth” algorithm, coupled with an optimization procedure, is proposed to reproduce target multiple-point histograms. The growth algorithm makes an analogy between geological accretion process and stochastic process and amounts to restricting the random path of sequential simulation at any given stage to a set of eligible nodes (immediately adjacent to a previously simulated node or sand grain). The proposed algorithm, combined with a multiple-grid approach, is shown to reproduce effectively the geometric essence of complex training images exhibiting long-range and curvilinear structures. Also, by avoiding a rigorous search for global minimum and accepting local minima, the proposed algorithm improves CPU time over traditional optimization procedures by several orders of magnitude. Average flow responses run on simulated realizations are shown to bracket correctly the reference responses of the training image.

A sensor-based obstacle avoidance for a redundant manipulator using a velocity potential function

Abstract: A new approach based on artificial potential function is proposed for the obstacle avoidance of redundant manipulators. Unlike the so-called "global" path planning method, which requires expensive computations for the path search before the manipulator starts to move, this new approach, called the "local" path planning, searches the path in real-time using the local distance information. Previous use of artificial potential functions has exhibited local minima in some complex environments. This paper proposes a potential function that has no local minima even for a cluttered environment. The proposed potential function has been implemented for the collision avoidance of a redundant robot in simulation. A simulation is demonstrated on an algorithm that prevents collisions with obstacles by calculating the repulsive potential exerted on links, based on the shortest distance to an object.