Layne T. Watson

Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms

There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK provides three qualitatively different algorithms for tracking the homotopy zero curve: ordinary differential equation-based, normal flow, and augmented Jacobian matrix. Separate routines are also provided for dense and sparse Jacobian matrices. A high-level driver is included for the special case of polynomial systems.

The Topographic Primal Sketch

A complete mathematical treatment is given for describing the topographic primal sketch of the underlying gray tone intensity surface of a digital image. Each picture element is independently classified and assigned a unique descriptive label, invariant under monotonically increasing gray tone transformations from the set (peak, pit, ridge, ravine, saddle, flat, and hillside), with hillside having subcategories (inflection point, slope, convex hill, concave hill, and saddle hill). The topographic classification is based on the first and second directional derivatives of the estimated image- intensity surface. A local, facet model, two-dimensional, cubic polynomial fit is done to estimate the image-intensity surface. Zero-crossings of the first directional derivative are identified as locations of interest in the image.

A facet model for image data

Abstract Image processing algorithms implicitly or explicitly assume an idealized form for the image data on which they operate. The degree to which the observed data meets the assumed idealized form is typically not examined or accounted for. This causes processing errors often attributed to noise. In this paper we discuss a facet model for image data which has the potential for fitting the form of the real idealized image, and for describing how the observed image differs from the idealized form. It is also an appropriate form for a variety of image processing algorithms. We give a relaxation procedure, and prove its convergence, for determining an estimate of the ideal image from observed image data.

Improved Genetic Algorithm for the Design of Stiffened Composite Panels

The design of composite structures against buckling presents two major challenges to the designer. First, the problem of laminate stacking sequence design is discrete in nature, involving a small set of fiber orientations, which complicates the solution process. Therefore, the design of the stacking sequence is a combinatorial optimization problem which is suitable for genetic algorithms. Second, many local optima with comparable performance may be found. Most optimization algorithms find only a single optimum, while often a designer would want to obtain all the local optima with performance close to the global optimum. Genetic algorithms can easily find many near optimal solutions. However, they usually require very large computational costs. Previous work by the authors on the use of genetic algorithms for designing stiffened composite panels revealed both the above strength and weakness of the genetic algorithm. The present paper suggests several changes to the basic genetic algorithm developed previously, and demonstrates reduced computational cost and increased reliability of the algorithm due to these changes. Additionally, for a stiffened composite panel used in this study, designs lighter by about 4 percent compared to previous results were obtained.

COMPOSITE LAMINATE DESIGN OPTIMIZATION BY GENETIC ALGORITHM WITH GENERALIZED ELITIST SELECTION

Abstract Genetic algorithms with elitist selection based on cloning a best single individual (SI) from one generation to the next are popular, but generalized elitist selection (GES) procedures have been proposed and tried in the past. The present paper explores several generalized elitist procedures for the design of composite laminates. It is shown that GES procedures are superior to an SI procedure for two types of problems. The first type involves many global optima, and the GES procedure can find several global optima more efficiently than the SI procedure. This may give a designer more design freedom. The second type of problem involves an isolated optimum surrounded by many designs with performance that is very close to optimal. It is shown that GES procedures can find the optimum and near optimal designs much more easily and reliably than the SI procedure.

Artificial parameter homotopy methods for the DC operating point problem

Efficient and robust computation of one or more of the operating points of a nonlinear circuit is a necessary first step in a circuit simulator. The application of globally convergent probability-one homotopy methods to various systems of nonlinear equations that arise in circuit simulation is discussed. The coercivity conditions required for such methods are established using concepts from circuit theory. The theoretical claims of global convergence for such methods are substantiated by experiments with a collection of examples that have proved difficult for commercial simulation packages that do not use homotopy methods. Moreover, by careful design of the homotopy equations, the performance of the homotopy methods can be made quite reasonable. An extension to the steady-state problem in the time domain is also discussed. >

Genetic Algorithms with Local Improvement for Composite LaminateDesign

This paper describes the application of a genetic algorithm to the stacking sequence optimization of a laminated composite plate for buckling load maximization. Two approaches for reducing the number of analyses required by the genetic algorithm are described. First, a binary tree is used to store designs, affording an efficient way to retrieve them and thereby avoid repeated analyses of designs that appeared in previous generations. Second, a local improvement scheme based on approximations in terms of lamination parameters is introduced. Two lamination parameters are sufficient to define the flexural stiffness and hence the buckling load of a balanced, symmetrically laminated plate. Results were obtained for rectangular graphite-epoxy plates under biaxial in-plane loading. The proposed improvements are shown to reduce significantly the number of analyses required for the genetic optimization.

Photosynthetic Acclimation Is Reflected in Specific Patterns of Gene Expression in Drought-Stressed Loblolly Pine

Because the product of a single gene can influence many aspects of plant growth and development, it is necessary to understand how gene products act in concert and upon each other to effect adaptive changes to stressful conditions. We conducted experiments to improve our understanding of the responses of loblolly pine (Pinus taeda) to drought stress. Water was withheld from rooted plantlets of to a measured water potential of -1 MPa for mild stress and -1.5 MPa for severe stress. Net photosynthesis was measured for each level of stress. RNA was isolated from needles and used in hybridizations against a microarray consisting of 2,173 cDNA clones from five pine expressed sequence tag libraries. Gene expression was estimated using a two-stage mixed linear model. Subsequently, data mining via inductive logic programming identified rules (relationships) among gene expression, treatments, and functional categories. Changes in RNA transcript profiles of loblolly pine due to drought stress were correlated with physiological data reflecting photosynthetic acclimation to mild stress or photosynthetic failure during severe stress. Analysis of transcript profiles indicated that there are distinct patterns of expression related to the two levels of stress. Genes encoding heat shock proteins, late embryogenic-abundant proteins, enzymes from the aromatic acid and flavonoid biosynthetic pathways, and from carbon metabolism showed distinctive responses associated with acclimation. Five genes shown to have different transcript levels in response to either mild or severe stress were chosen for further analysis using real-time polymerase chain reaction. The real-time polymerase chain reaction results were in good agreement with those obtained on microarrays.

Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms

HOMPACK90 is a Fortran 90 version of the Fortran 77 package HOMPACK (Algorithm 652), a collection of codes for finding zeros or fixed points of nonlinear systems using globally convergent probability-one homotopy algorithms. Three qualitatively different algorithms— ordinary differential equation based, normal flow, quasi-Newton augmented Jacobian matrix—are provided for tracking homotopy zero curves, as well as separate routines for dense and sparse Jacobian matrices. A high level driver for the special case of polynomial systems is also provided. Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.

Robust Window Operators

It is a common practice in computer vision and image processing to convolve rectangular constant coefficient windows with digital images to perform local smoothing and derivative estimation for edge detection and other purposes. If all data points in each image window belong to the same statistical population, this practice is reasonable and fast. But, as is well known, constant coefficient window operators produce incorrect results if more than one statistical population is present within a window, for example, if a gray-level or gradient discontinuity is present. This paper shows one way to apply the theory of robust statistics to the data smoothing and derivative estimation problem. A robust window operator is demonstrated that preserves gray-level and gradient discontinuities in digital images as it smooths and estimates derivatives.