Steepest Descent Technique Research Articles

Optimization of isothermal-isobaric chemical vapor infiltration

This study addresses the issue of determining optimal operating conditions for isothermal isobaric chemical vapor infiltration for obtaining products of maximum densification. The optimal conditions are determined by a conventional steepest descent technique which guarantee at least a local minimum for a cost function related to the residual porosity. This work shows that the optimal operating temperature is not necessarily a flat profile.

Perturbed steepest-descent technique in multiextremal problems

The steepest-descent technique dealing with the perturbed values of the objective function and its gradients and with nonexact line searches is considered. Attention is given to the case where the perturbations do not decrease on the algorithm trajectories; the aim is to investigate how perturbations at every iteration of the algorithm perturb its original attractor set. Based on the Liapunov direct method for attraction analysis of discrete-time processes, a sharp estimation of the attractor set generated by a perturbed steepest-descent technique with respect to the perturbation magnitudes is obtained. Some global optimization properties of finite-difference analogues of the gradient method are discovered. These properties are not inherent in methods which use exact gradients.

On-line state estimation and adaptive optimization using state equations for continuous production of bioethanol

On-line optimization of bioprocesses is a difficult task due to the absence of accurate mathematical models and complex non-linear dynamic behaviour. In the present investigation, state equations were used to develop an algorithm to estimate unmeasurable state variables and time-varying parameters which were then used for adaptive, dynamic optimization of a continuous bioreactor. The manipulated variable was adjusted based on the steepest-descent technique to continuously drive the process to the optimal operating conditions in a reasonable time period. The estimation algorithm is shown to predict the unmeasured state variables to within ± 5% during dynamic fluctuations caused by both sudden changes in dilution rate and gradual changes in nutrient feed supply. The control algorithm is shown to quickly and smoothly adjust the manipulated variable to keep the bioreactor operating at its highest productivity as compared to steady state conditions even during inverse-response dynamic conditions.

Some properties of microcanonical rate constants

Abstract An isolated body is characterized by conserved properties such as its energy and angular momentum. The rates of processes occurring in that body can thus be investigated as a function of these properties. On the other hand the rates may be better known, both conceptually and experimentally, as a function of temperature. This paper surveys relations between rate descriptions in the two domains, as deduced using steepest descent techniques. The rudiments of these techniques are given in an appendix.

Simultaneous Optimization of Sensor/Actuator Placement and Control Law in Flexible Beams.

This paper concerns design methods of optimal placement for locating sensors and actuators on flexible beams. Recently, modern control theory, which is based on the LQ control or H∞ control theory, has been applied to control the vibration of flexible strutures. However, most of reports are concerned with model reduction and spillover problems, but few reports have discussed whether located sensor/actuator of the flexible structures are suitably placed for the designed controller. Therefore, it is necessary to consider optimization through the entire design procedure including sensor/actuator placement. In this paper an iterative algorithm is proposed to search for the optimal sensor/actuator placement for stabilization of flexible beams. This method is based on the steepest descent techniques using sensitivity analysis and output feedback control. It is shown that the system controlled by the proposed method improves closed-loop performance in frequency domain,

A novel method for optimizing quantum Monte Carlo wave functions

We propose an algorithm for optimizing quantum Monte Carlo wave functions. An improved steepest-descent technique is used, with step size automatically adjustable to obtain a procedure that converges superlinearly. We also propose a novel trial function, which has both correct electron–electron cusp conditions and correct electron–nucleus cusp conditions. To test the optimization procedure and the optimized trial function, the ground states for CH4 and H2O molecules were investigated using variational Monte Carlo (VMC) and fixed-node quantum Monte Carlo (FNQMC) calculations. For CH4 and H2O, the VMC recovered 73.3% and 57.9% of the correlation energy, respectively, and the FNQMC recovered 99.3% and 92.8%, respectively. The optimization procedure is three to five times faster than conventional usual steepest-descent procedures. The trial functions optimized are more accurate than prior trial functions of similar complexity.

General steady creep analysis for various constitutive laws. Part 1: theory and formulation

Generalized formulations of the finite element method with penalty multiplier for steady creep problems are proposed as being suitable for various constitutive relations. A simple expression is derived to obtain mean stress by identifying the physical quantity of the penalty multiplier. To solve the nonlinear problem, the Newton-Raphson or modified Newton-Raphson process with steepest descent technique is adopted. The detailed expressions of secant and tangential constitutive matrixes are deduced for Norton, Prandtl, Dorn and Garofalo creep laws. Some advice on numerical implementation is given.

Shape optimization of skeleton structures using mixed-discrete variables

The shape and size optimization problem of a structural body is solved by a mixed-discrete algorithm, where a continuous variable is handled as a pseudo-continuous one. It is found that handling a continuous variable in the pseudo-continuous sense can reduce analytical difficulties. The mixed-discrete algorithm uses two different techniques of which one is the gradient based steepest descent technique and the other is the gradient free Rosenbrock orthogonalization procedure. Both techniques are modified to suit discrete or pseudo-continuous variables. Hybridization of two different types of optimization techniques enables the algorithm to solve different kinds of optimization problems.

Improving the convergence of iterative filtered backprojection algorithms

Several authors have proposed variations of the iterative filtered backprojection (IFBP) reconstruction algorithms claiming fast initial convergence rates. We have found that these algorithms are trying to minimize an unusual squared-error criterion in a suboptimal way. As a result, existing IFBP algorithms are inefficient in the minimization of the criterion, and may become unstable at higher iteration numbers. We show that existing IFBP algorithms can be modified to use the steepest descent technique by simply optimizing the step size at each iteration. Further gains in convergence rates can be achieved with conjugate gradient IFBP algorithms derived from the same criterion. The steepest descent and conjugate gradient IFBP algorithms are guaranteed to converge, unlike some IFBP algorithms, and will do so in fewer iterations than existing IFBP algorithms.

Optimization of an industrial microalgae fermentation

Optimization of cellular productivity of an industrial microalgae fermentation was investigated. The fermentation was carried out at Coors Biotech Products Company, Fort Collins, Colorado. A mathematical model was developed based on the data collected from pilot plant test runs at different operating conditions. Pontryagin's maximum principle was used for determining the optimal feed policy. A feedback control algorithm was also studied for maximizing the cellular productivity. During continuous operation, the optimum dilution rate was determined by an adaptive optimization scheme based on the steepest descent technique and a recursive least squares estimation of model parameters. A direct search algorithm was also applied to determine the optimum feed rate. Comparison of the theoretical results of the different optimization schemes revealed that the direct search algorithm was preferable because of its simplicity. The experimental results of real time application of the feedback algorithm agreed fairly well with those of the theoretical analyses. (c) 1994 John Wiley & Sons, Inc.

Learning Algorithms for Neural Networks Based on Quasi-Newton Methods With Self-Scaling

Previous studies have suggested that, for moderate sized neural networks, the use of classical Quasi-Newton methods yields the best convergence properties among all the state-of-the-art [1]. This paper describes a set of even better learning algorithms based on a class of Quasi-Newton optimization techniques called Self-Scaling Variable Metric (SSVM) methods. One of the characteristics of SSVM methods is that they provide a set of search directions which are invariant under the scaling of the objective function. With an XOR benchmark and an encoder benchmark, simulations using the SSVM algorithms for the learning of general feedforward neural networks were carried out to study their performance. Compared to classical Quasi-Newton methods, it is shown that the SSVM method reduces the number of iterations required for convergence by 40 percent to 60 percent that of the classical Quasi-Newton methods which, in general, converge two to three orders of magnitude faster than the steepest descent techniques.

New adaptive algorithms based on the method of steepest descent

Abstract A new simple formulation for governing the choice of the convergence parameter µ in adaptive algorithms using the method of steepest descent techniques is given. Two algorithms are derived and examined here. The first, the steepest descent algorithm, results in a time-varying µ which is the same for each filter but is updated at each iteration to yield optimum performance. The second, the fast convergence algorithm, has a time-varying convergence parameter which is chosen suitably for each adaptive filter coefficient and is updated at each iteration. These algorithms have been tested successfully to yield faster convergence and more accurate adaptation compared with the existing least-mean-square (LMS) algorithm that uses a constant value of µ. Substantial improvements in transient behaviour in comparison to stochastic-gradient or LMS algorithms are efficiently achieved by the presented algorithms. This true solution is recursively calculated at a relatively modest increase in computational requi...

A real-time learning algorithm for a multilayered neural network based on the extended Kalman filter

A novel real-time learning algorithm for a multilayered neural network is derived from the extended Kalman filter (EKF). Since this EKF-based learning algorithm approximately gives the minimum variance estimate of the linkweights, the convergence performance is improved in comparison with the backwards error propagation algorithm using the steepest descent techniques. Furthermore, tuning parameters which crucially govern the convergence properties are not included, which makes its application easier. Simulation results for the XOR and parity problems are provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Regge calculus in anisotropic quantum cosmology

The authors investigate the no-boundary path integral prescription in a Regge calculus model in three spacetime dimensions. The simplicial manifold is topologically the 'doughnut' D*S1, where D is the two-dimensional closed disc. The discretization and the edge lengths are chosen symmetrically so that the boundary 2-metric is completely described by two independent edge lengths, and the interior 3-metric is completely described by these plus two further independent edge lengths. They pass to a limit in which the discretization on the boundary becomes infinitely dense. The classical solutions in the resulting semi-continuum model are in qualitative agreement with the solutions to the continuum Einstein equations with the same boundary data. Convergent contours for the no-boundary integral in the semi-continuum model are sought in analogy with the conformal rotation prescription of Gibbons et al (1978) and using steepest-descent techniques to analyse the contours for the conformal factor. A convergent prescription is found in the case of a vanishing cosmological constant, and the resulting wavefunction is shown to have the expected semiclassical behaviour. Possible reasons for the authors not being able to find a similar prescription with a non-vanishing cosmological constant are discussed.

ON-LINE OPTIMIZATION OF DISTILLATION COLUMNS IN SERIES

Abstract This paper presents an on-line optimization algorithm for distillation columns which consists of a steepest descent technique based on a simple model of product recovery with on-line estimation of the critical model parameter. The algorithm is developed for a single column and then for a two-column train. A dynamic programming approach is used to reduce the optimization problem to a single parameter search for each column in the train. The resulting algorithm is simple and computer resource requirements are small. The algorithm has been successfully used in two industrial applications, one consisting of two columns in series and the other of three columns in series.

Steepest-descent approximation theory for guided modes of weakly guiding optical waveguides and fibers

The steepest-descent approximation theory (SDAT) is developed for the calculation of dispersion relations and field distribution functions in optical waveguides and fibers. Based on the functional-extremum principle, this new method improves the form of the trial functions for the modal field distributions directly, instead of optimizing parameters as in conventional variational approaches. It does so by making use of the concept of the functional derivative of Rayleigh’s quotient and the steepest-descent technique. The SDAT also provides iterative procedures for generating better approximations for both the propagation constants and the modal fields. Several low-order modes of an optical fiber with a Gaussian refractive-index profile are studied, demonstrating the usefulness of this new method. Improved analytical expressions for the field distributions are obtained with improved values for the propagation constants.

On the EMC dipole feed-line parasitic radiation

A rigorous analysis of the electromagnetically coupled microstrip dipole based on potential integral equations, Green's functions, and the moment method is presented. The computations of the antenna's radiated field using the steepest descent technique is then detailed, and the theoretical results are compared with experimental measurements in the X-band. By considering the feed as an antenna part, the excitation line parasitic radiation is shown clearly. Two possibilities to reduce this parasitic phenomenon are proposed. Also, it is shown that in any microstrip structure (one or two layers) excited by a microstripline, the feed parasitic radiation is nonnegligible.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Self-Consistent Methods in Hückel and Extended Hückel Theories

A self-consistent procedure for the Huckel theory, first indicated by Harris [J. Chem. Phys. 48, 4027 (1968)] has been fully developed. To avoid time consuming repeated orthogonal-nonorthogonal transformations and diagonalization of large matrices, a steepest descent technique has been suggested.

Optimal Thorium-Fueled CANDU Nuclear Reactor Fuel Management

The optimization of in-core fuel management for a thorium-fueled Canada deuterium uranium (CANDU) nuclear reactor was investigated by minimizing the total refueling rate at equilibrium with respect to criticality and power-peaking constraints. The computer code ASTERIX was written to solve the optimization problem, using a steepest descent technique with a moderate number of diffusion calculations required. Because of the presence of /sup 233/Pa in the fuel, the diffusion calculations are nonlinear and are solved numerically by the specially written program CALYPSO. Simulation was performed on simple models of a CANDU 600-MW reactor, with the core divided into two or four refueling zones. Results indicated that the optimization method investigated did work out well and that potential savings of up to 14% in the feed rate are possible for the self-sufficient equilibrium thorium cycle fuel, with an optimum refueling rate of 1.372 X 10/sup -4/ MgHE (heavy elements)/MWd. Sensitivity of the optimal discharge burnups to the value of the power-peaking constraint was significant.

Self‐Consistent methods in Hückel and extended Hückel theories

AbstractA self‐consistent procedure for the Hückel theory, first indicated by Harris [J. Chem. Phys. 48, 4027 (1968)] has been fully developed. To avoid time consuming repeated orthogonal‐nonorthogonal transformations and diagonalization of large matrices, a steepest descent technique has been suggested.