Powell Algorithm Research Articles

Optimal design of composite ChamberCore structures

An optimization procedure has been developed to uniquely and efficiently determine the “best” local geometry design of a new composite ChamberCore structure. This procedure is based on minimization of the total mass of a single composite ChamberCore subject to a set of design and stress constraints. The stress constraints are obtained in closed form based on the composite box-beam model for various composite lamination designs and loading conditions. The optimization problem statement is constructed and then solved using the VMCON optimization program, which is an iterative sequential quadratic programming (SQP) technique based on Powell's algorithm. The sensitivity of the solution of the optimal geometry to the values of parameters that characterize the structural durability and the failure mechanism is discussed.

Optimization for Aircraft Engines with Regression and Neural-Network Analysis Approximators

The NASA Engine Performance Program (NEPP) can configure and analyze almost any type of gas turbine engine that can be generated through the interconnection of a set of standard physical components. In addition, the code can optimize engine performance by changing adjustable variables under a set of constraints. However, for engine cycle problems at certain operating points, the NEPP code can encounter difficulties: nonconvergence in the currently implemented Powell's optimization algorithm and deficiencies in the Newton-Raphson solver during engine balancing. A project was undertaken to correct these deficiencies. Nonconvergence was avoided through a cascade optimization strategy, and deficiencies associated with engine balancing were eliminated through neural network and linear regression methods. An approximation-interspersed cascade strategy was used to optimize the engine's operation over its flight envelope. Replacement of Powell's algorithm by the cascade strategy improved the optimization segment of the NEPP code. The performance of the linear regression and neural network methods as alternative engine analyzers was found to be satisfactory. This report considers two examples-a supersonic mixed-flow turbofan engine and a subsonic waverotor-topped engine-to illustrate the results, and it discusses insights gained from the improved version of the NEPP code.

Two optimizing procedures for the solution of complex systems of equations: a powerful tool for modelling and simulation of metabolism

Introduction Standard calculations for the evaluation of indirect calorimetry (IC) are based on two-dimensional nonlinear systems of equations. For a more sophisticated evaluation metabolic models can be used, which are described by complex systems of equations. Since the solutions are multidimensional, a concrete result must be selected by means of constraints, using optimizing procedures. These multidimensional optimizations are critical concerning processing time and reproducibility of minimum detection. Methods In order to simulate the status of metabolism of ICU patients on the basis of IC data, a complex model of metabolism was developed. The model was described by a system of equations consisting of 13 equations and 25 variables. The final result is selected out of the nine-dimensional space of solutions by optimization, considering a total of 69 constraints. Two optimizing procedures (Simplex algorithm [1] SPX, Powell procedure [2] POW) were modified, and analysed referring to processing time (PT), variance and reproducibility (VK, VC) of results. Test data as well as long-term IC measurements in ICU patients were evaluated. Modifications including ‘meta procedures’, which multiply restart the ‘core’ of the optimization (SPX) or find an optimal starting point for the core by a deformation of optimal sets of IC data with already known solutions, were necessary. VC and VK were tested for four representative parameters of the metabolic simulation (enzymes pyruvate dehydrogenase PDH, citrate synthase CS, ketoglutarate decarboxylase KDC, malate dehydrogenase MDH) by means of seven sets of calculated IC data. Penalty values as a measure of the quality of the simulation and processing time (Pentium 120 MHz) were determined using real data of a ventilated patient. Results Both optimization procedures showed a high reproducibility and a low variance (all but one VK/VC values below 10%); there were no differences between Simplex and Powell algorithm (see Table 1). Evaluating real data, we found a lower frequency of outliers (nonevaluable data) for SPX (3.7% vs. 5.7%), while processing time was shorter for POW (1.95 ± 0.87 vs. 8.88 ± 3.06 min; both P < 0.05; n = 800). Table 1. Variance (VK) and reproducibility (VC) for the Simplex algorithm (SPX) and the Powell procedure (POW) for four different enzymes (PDH, CS, KDC, MDH) VC VK SPX POW SPX POW PDH 3.5% ± 1.7% 2.1% ± 1.7% 5.1% ± 1.7% 2.8% ± 2.3% CS 3.6% ± 1.6% 1.4% ± 1.4% 10.1% ± 2.1% 4.7% ± 5.8% KDC 0.1% ± 0.0% 0.0% ± 0.0% 1.5% ± 0.0% 0.0% ± 0.0% MDH 1.5% ± 0.7% 0.5% ± 0.4% 3.0% ± 1.3% 2.5% ± 1.5% Conclusion The evaluation of IC data using a model for the simulation of metabolism was successful even in clinical situations, where standard procedures usually fail. Multidimensional nonlinear systems of equations, as used for this simulation, are a considerable extension of available tools for the interpretation of IC data, if a powerful optimizing procedure is used. Both of the analysed procedures have shown to be suitable to reduce the nine-dimensional space of solutions of the system of equations to a concrete result, and can be applied to online metabolic simulations, if fast PC hardware is available.

Designing experiments for single output multilayer perceptrons

Where should a researcher conduct experiments to provide training data for a multilayer perceptron This question is investigated and a statistical method for selecting optimal experimental design points for multilayer perceptrons is introduced. Two-class discrimination problems are examined in which the multilayer perceptron is viewed as a univariate nonlinear regression model. A selection criterion is developed based on the volume of the joint confidence ellipsoid for the weights in a multilayer perceptron. The criterion is minimized to find optimal design points. Minimization is accomplished using Powell's algorithm when the feature space is continuous and a discrete exchange algorithm when the feature space is discrete. An example is used to demonstrate the superiority of optimally selected design points over randomly chosen points and points chosen in a grid pattern. In addition, two measures are developed to rank the design points in terms of their relative importance.

Optimization of via formation in photosensitive dielectric layers using neural networks and genetic algorithms

Via formation using photosensitive polymer technology can reduce process cost by reducing process complexity and is hence of great interest in electronics packaging substrate fabrication. However, to overcome technical difficulties and to facilitate low-cost manufacturing, process modeling, optimization and control are required. In this paper, a process optimization approach for via formation in dielectric layers composed of photosensitive benzocyclobutene (BCB) for high density interconnect (HDI) in MCM-L/D substrates is presented. A series of designed experiments are used to characterize the via formation workcell (which consists of the spin coat, soft bake, expose, develop, cure, and plasma de-scum unit process steps). Neural network process models are then constructed to characterize via yield and geometry, as well as film thickness, retention, and uniformity. These models are used for process optimization using genetic algorithms (GA's) and hybrid combinations of GA's with the Powell algorithm and with the simplex algorithm. The optimized process recipes are verified experimentally. Comparison of the three approaches reveals that the hybrid GA/simplex method yields superior recipes.

A quasi-newton method for minimum trace factor analysis

In the past several algorithms have been given to solve the minimum trace factor analysis (MTFA) and the constrained minimum trace factor analysis (CMTFA) problems. Some of these algorithms, depending on the initial value, may converge to points that are not the solution to the above problems, some converge linearly, and some are quadratically convergent but are somewhat difficult to implement. In this paper we propose modified Han–Powell algorithms to solve the MTFA and CMTFA problems. The modifications deal with the problem of multiple eigenvalues. The proposed algorithms are globally convergent and their speed is locally superlinear. We also give a modified Han–Powell algorithm to solve the weighted minimum trace factor analysis (WMTFA) problem. This method is also locally superlinear and is simpler to implement as compared to methods proposed earlier. Four examples are given to show the performance of the proposed algorithms. More generally, our experience with these algorithms shows that, starting at arbitrary points, they converge to the solution in a small number of iterations and reasonable time.

A discrete energy formulation for the definition of optimal tension structures

Based upon basic energy concepts, a simple method is proposed for the analysis (specifically the more difficult case of form-finding) of cablenet and membrane structures. A truncated strain expression is used to define the total potential energy. By using a comparative approach from finite element techniques it is shown that a pseudo load potential can be defined. The final energy form is minimized using the Powell algorithm. Example solutions are included.

Hydro-thermal performance of unsaturated bentonite-sand buffer material

Several series of one-dimensional heat and moisture flow tests were performed to examine the moisture and temperature distributions in the buffer material compacted to a dry density of 1.67 Mg m −3 and water content of 17.7%. In all tests, water was allowed to infiltrate into a horizontal soil column from one end under a constant hydrostatic head of 276 kPa. Also the specimens were heated from the other end by the heater to a constant temperature. It is experimentally demonstrated that the moisture moves from both ends toward the mid part of the soil column due to both thermal gradient from one end and hydraulic gradient from the other end. It was observed that, in spite of no overall volume change, local volume change occurs in the system. The measured temperatures along the length of the specimen indicate that temperature distributions stabilize within a short period of time. The time required for the temperature to stabilize decreases as the heater skin temperature increases. The diffusivity parameters are calculated using the measured moisture and temperature profiles combined with the finite difference method. Powell's optimization algorithm was used to determine the material parameters. Good agreements between experimentally measured and calibrated volumetric water content shows that the diffusion parameters can be expressed in a linear function of the volumetric water content and temperature.

Using neural network process models to perform PECVD silicon dioxide recipe synthesis via genetic algorithms

Silicon oxide (SiO/sub 2/) films have extensive applications in integrated circuit fabrication technology, including passivation layers for integrated circuits, diffusion or photolithographic masks, and interlayer dielectrics for metal-insulator structures such as MOS transistors or multichip modules. The properties of SiO/sub 2/ films deposited by plasma enhanced chemical vapor deposition (PECVD) are determined by the nature and composition of the plasma, which is in turn controlled by the deposition variables involved in the PECVD process. The complex nature of particle dynamics within a plasma makes it very difficult to quantify the exact relationship between deposition conditions and critical output parameters reflecting film quality. In this study, the synthesis and optimization of process recipes using genetic algorithms is introduced. In order to characterize the PECVD of SiO/sub 2/ films deposited under varying conditions, a central composite designed experiment has been performed. Data from this experiment was then used to develop neural network based process models. A recipe synthesis procedure was then performed using the optimized neural network models to generate the necessary deposition conditions to obtain several novel film qualities, including zero stress, 100% uniformity, low permittivity, and minimal impurity concentration. This synthesis procedure utilized genetic algorithms, Powell's algorithm, the simplex method, and hybrid combinations thereof. Recipes predicted by these techniques were verified by experiment, and the performance of each synthesis method are compared. It was found that the genetic algorithm-based recipes generally produced films of superior quality. Deposition was carried out in a Plasma Therm 700 series PECVD system.

Automated image registration for FDOPA PET studies

In this study, various image registration methods are investigated for their suitability for registration of L-6-[18F]-fluoro-DOPA (FDOPA) PET images. Five different optimization criteria including sum of absolute difference (SAD), mean square difference (MSD), cross-correlation coefficient (CC), standard deviation of pixel ratio (SDPR), and stochastic sign change (SSC) were implemented and Powell's algorithm was used to optimize the criteria. The optimization criteria were calculated either unidirectionally (i.e. only evaluating the criteria for comparing the resliced image 1 with the original image 2) or bidirectionally (i.e. averaging the criteria for comparing the resliced image 1 with the original image 2 and those for the sliced image 2 with the original image 1). Monkey FDOPA images taken at various known orientations were used to evaluate the accuracy of different methods. A set of human FDOPA dynamic images was used to investigate the ability of the methods for correcting subject movement. It was found that a large improvement in performance resulted when bidirectional rather than unidirectional criteria were used. Overall, the SAD, MSD and SDPR methods were found to be comparable in performance and were suitable for registering FDOPA images. The MSD method gave more adequate results for frame-to-frame image registration for correcting subject movement during a dynamic FDOPA study. The utility of the registration method is further demonstrated by registering FDOPA images in monkeys before and after amphetamine injection to reveal more clearly the changes in spatial distribution of FDOPA due to the drug intervention.

A comparative study of several chemometric methods applied to the treatment of two-way kinetic-spectral data for mixture resolution

A comparative study was conducted to investigate the performance of several chemometric methods applied to the treatment of two-way kinetic-spectral data with the aim of resolving mixtures. The methods involved are non-linear least squares regression performed using the Powell algorithm, the linear and extended Kalman filter, and partial least squares regression. Both simulated and experimental data were processed. The coupling reaction of diazotized sulfanilamide with arylamines to give azodyes was monitored spectrophotometrically. Binary mixtures of the substrates with different values of the rate constant ratio and with varied degrees of spectral overlap were resolved. The effects of several influence factors have been studied using numeric simulation. The advantages and the limitations of each method have been evaluated.

Resource allocation in state-dependent emergency evacuation networks

Most of the previous studies on the Emergency Evacuation Problem (EEP) assume that the length and widths of the circulation spaces are fixed. This assumption is only true if one is evaluating facilities that are already built. However, when designing the network for the first time, the size of the circulation space is not known to the designer, in fact it is one of several design parameters. After the routes have been established, it seems that the next logical question is to find out whether or not the system circulation spaces are capable of accommodating the traffic for both normal circulation and in an emergency. The problem of designing emergency evacuation networks is very complex and it is only recently that queueing networks are now being used to model this problem. Recent advances include state-dependent queueing network models that incorporate the mean value analysis algorithm to capture the non-linearities in the problem. We extend these models by incorporating the mean value analysis algorithm within Powell's derivative free unconstrained optimization algorithm. The effect of varying circulation widths on throughput will be discussed and a methodology for solving the resource allocation problem is proposed and demonstrated on several examples. The computational experience of the new methodology illustrates its usefulness in network design problems.

Synthesis of shaped beam antenna patterns using implicitly constrained current elements

The amplitude and phase values of the current elements of a linear array which produce a shaped-beam antenna pattern are found by a search over constrained space rather than through placement of pattern zeroes. The current elements must be constrained to realize physically producible patterns. Implicitly constraining the current elements enables Powell's direction set algorithm (DSA) to be used rather than a more elaborate explicitly constrained algorithm. In contrast with conventional methods, the zeroes of the pattern function in the sidelobe region are not constrained to lie on the Schelkunoff unit circle. To assess the utility of the proposed technique, a comparison is made with previously published results. Compared to conventional approaches, the resulting current element-to-element variation is significantly less, which minimizes mutual coupling difficulties. Both traveling-wave and center-fed designs are possible with the technique outlined in this paper.

Determination of mass transfer coefficients during freeze drying using modeling and parameter estimation techniques

An approach using mathematical modeling and parameter estimation techniques was established to determine the best-fit mass transfer coefficients of the dried-layer resistance during primary drying of pharmaceuticals, in order to minimize experimental efforts. For modeling of the primary drying process, the equations associated with heat and mass transfer mechanisms established by Pikal were used as the model equations, whereas the parameter estimation was accomplished by Powell's nonlinear least-squares minimization algorithm. The advantages of this approach include that it can be applied to analyze various types of measurable quantities, such as the cumulative mass of sublimation M t, the temperature profiles at the bottom center of the frozen layer T b, and the pressure profiles of the vials. It can also be used for these quantities measured using the vials with different heat transfer coefficients. To test the performance of the proposed approach, hypothetical values of M t and T b with perturbed errors, were simulated using the modeling algorithm and a random number generator. These values were in turn used as the input data for parameter estimation. The results show that the best-fit mass transfer coefficients are successfully obtained using either the hypothetical M t or T b profiles, with appropriate initial guesses of the parameters. All computations in modeling and parameter estimation were developed in FORTRAN and compiled using Microsoft FORTRAN compiler, of which the source codes and documentation, with detailed mathematical equations and computation steps, are available upon request.

Arrangement of discs in 2d binary assemblies

We study arrangements of the two species of discs in binary assemblies at an intermediate scale. Small discs rearrange along large ones in clusters whose mass and compactness are analyzed with the tools of percolation. The assemblies are generated analogically on an air table or numerically from RSA or Powell algorithms. At a given packing fraction, an infinite cluster of small discs exists above a critical composition; a phenomenological expression for this threshold is proposed. Like in usual percolation problems, the number of inner links in a cluster is a linear function of its mass, with a slope depending both on the packing fraction, the composition of the mixture and the building procedure. An approximate expression is derived for it.

An integrated design strategy for flow-injection analysis based on the coupling of mathematical modelling and optimization algorithms

A methodology to design a flow- injection system with optimized performance characteristics for different experimental requirements is presented which requires minimal experimental effort. Departing from basic experimental knowledge of the hydrodynamic parameters and rates of the reactions involved, the coupling of a mathematical model describing the flow system with the Powell's optimization algorithm allows the automated generation of a complete set of flow- injection configurations, each of them being optimal with respect to its objective function. From this set of configurations, the performance of the flow- injection system is easily visualized, allowing the user to select the system that better suits the requirements of his particular application and exploiting the potentiality of multiple flow- injection optimal configurations.

Surface Approximation of a Cloud of 3D Points

We present an implementation of deformable models to approximate a 3-D surface given by a cloud of 3D points. It is an extension of our previous work on "B-snakes" (S. Menet, P. Saint-Marc, and G. Medioni, in Proceedings of Image Understanding Workshop, Pittsburgh, 1990, pp. 720-726; and C. W. Liao and G. Medioni, in Proceedings of International Conference on Pattern Recognition, Hague, Netherlands, 1992, pp. 745-748), which approximates curves and surfaces using B-splines. The user (or the system itself) provides an initial simple surface, such as closed cylinder, which is subject to internal forces (describing implicit continuity properties such as smoothness) and external forces which attract it toward the data points. The problem is cast in terms of energy minimization. We solve this nonconvex optimization problem by using the web-known Powell algorithm which guarantees convergence and does not require gradient information. The variables are the positions of the control points. The number of control points processed by Powell at one time is controlled. This methodology leads to a reasonable complexity, robustness, and good numerical stability. We keep the time and space complexities in check through a coarse-to-fine approach and a partitioning scheme. We handle closed surfaces by decomposing an object into two caps and an open cylinder, smoothly connected. The process is controlled by two parameters only, which are constant for all our experiments. We show results on real range images to illustrate che applicability of our approach. The advantages of this approach are that it provides a compact representation of the approximated data and lends itself to applications such as nonrigid motion tracking and object recognition. Currently, our algorithm gives only a C 0 continuous analytical description of the data, but because the output of our algorithm is in rectangular mesh format, a C 1 or C 2 surface can be constructed easily by existing algorithms (F. J. M. Schmitt, B. A. Barsky, and W.-H. Du, in ACM SIGGRAPH 86, pp. 179-1988).

A general technique for interstudy registration of multifunction and multimodality images

A technique that can register anatomic/structural brain images (e.g., MRI) with various functional images (e.g., PET-FDG and PET-FDOPA) of the same subject has been developed. The procedure of this technique includes the following steps: (1) segmentation of MRI brain images into gray matter (GM), white matter (WM), cerebral spinal fluid (CSF), and, muscle (MS) components, (2) assignment of appropriate radio-tracer concentrations to various components depending on the kind of functional image that is being registered, (3) generation of simulated functional images to have a spatial resolution that is comparable to that of the measured ones, (4) alignment of the measured functional images to the simulated ones that are based on MRI images. A self-organization clustering method is used to segment the MRI images. The image alignment is based on the criterion of least squares of the pixel-by-pixel differences between the two sets of images that are being matched and on the Powell's algorithm for minimization. The technique was applied successfully for registering the MRI, PET-FDG, and PET-FDOPA images. This technique offers a general solution to the registration of structural images to functional images and to the registration of different functional images of markedly different distributions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A kinetic model for the pectin hydrolysis using an endo-acting pectinase from Rhizopus

Enzyme kinetics of pectin hydrolysis reaction catalysed by an endo-polygalacturonase from Rhizopus have been analysed experimentally in a stirred batch reactor at pH=4.5 and T=34°C using a sodium salt of polygalacturonic acid as substrate. The reaction products, consisting of oligomers of galacturonic acid, were classified in four classes of oligomeric fractions employing three hollow fibre ultrafiltration units characterised by different molecular weight cut-off ( MW co). On the basis of the time-concentration data, a kinetic model of the reacting system has been proposed in terms of a series-side reaction pattern. Identification of kinetic parameters was carried out by a non-linear regression optimization procedure based on the Powell's algorithm of conjugated directions. Finally, the kinetic model was used to predict the best reactor configuration able to control the product distribution for a given enzyme/substrate concentration ratio.

Structural optimization using unconstrained nonlinear goal programming algorithm

This paper presents a method for solving structural optimization problems using nonlinear goal programming techniques. The developed method removes the difficulty of having to define an objective function and constraints. It also has the capacity of handling rank ordered design objectives or goals. The formulation of the structural optimization problem into a goal programming form is discussed. The resulting optimization problem is solved using Powell's unconstrained conjugate direction search algorithm. To demonstrate the effectiveness of the method, as a design tool, the solutions of some numerical test cases are included.