Search NASA⌕ Search

SEARCH · Search NASA

Results for “Quadratic convergence”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

204 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

A homotopy algorithm for digital optimal projection control GASD-HADOC

The linear-quadratic-gaussian (LQG) compensator was developed to facilitate the design of control laws for multi-input, multi-output (MIMO) systems. The compensator is computed by solving two algebraic equations for which standard closed-loop solutions exist. Unfortunately, the minimal dimension of an LQG compensator is almost always equal to the dimension of the plant and can thus often violate practical implementation constraints on controller order. This deficiency is especially highlighted when considering control-design for high-order systems such as flexible space structures. This deficiency motivated the development of techniques that enable the design of optimal controllers whose dimension is less than that of the design plant. A homotopy approach based on the optimal projection equations that characterize the necessary conditions for optimal reduced-order control. Homotopy algorithms have global convergence properties and hence do not require that the initializing reduced-order controller be close to the optimal reduced-order controller to guarantee convergence. However, the homotopy algorithm previously developed for solving the optimal projection equations has sublinear convergence properties and the convergence slows at higher authority levels and may fail. A new homotopy algorithm for synthesizing optimal reduced-order controllers for discrete-time systems is described. Unlike the previous homotopy approach, the new algorithm is a gradient-based, parameter optimization formulation and was implemented in MATLAB. The results reported may offer the foundation for a reliable approach to optimal, reduced-order controller design.

Collins, Emmanuel G., Jr.↗
NASA NTRS2000

Primate translational vestibuloocular reflexes. II. Version and vergence responses to fore-aft motion

To maintain binocular fixation on near targets during fore-aft translational disturbances, largely disjunctive eye movements are elicited the amplitude and direction of which should be tuned to the horizontal and vertical eccentricities of the target. The eye movements generated during this task have been investigated here as trained rhesus monkeys fixated isovergence targets at different horizontal and vertical eccentricities during 10 Hz fore-aft oscillations. The elicited eye movements complied with the geometric requirements for binocular fixation, although not ideally. First, the corresponding vergence angle for which the movement of each eye would be compensatory was consistently less than that dictated by the actual fixation parameters. Second, the eye position with zero sensitivity to translation was not straight ahead, as geometrically required, but rather exhibited a systematic dependence on viewing distance and vergence angle. Third, responses were asymmetric, with gains being larger for abducting and downward compared with adducting and upward gaze directions, respectively. As frequency was varied between 4 and 12 Hz, responses exhibited high-pass filter properties with significant differences between abduction and adduction responses. As a result of these differences, vergence sensitivity increased as a function of frequency with a steeper slope than that of version. Despite largely undercompensatory version responses, vergence sensitivity was closer to ideal. Moreover, the observed dependence of vergence sensitivity on vergence angle, which was varied between 2.5 and 10 MA, was largely linear rather than quadratic (as geometrically predicted). We conclude that the spatial tuning of eye velocity sensitivity as a function of gaze and viewing distance follows the general geometric dependencies required for the maintenance of foveal visual acuity. However, systematic deviations from ideal behavior exist that might reflect asymmetric processing of abduction/adduction responses perhaps because of different functional dependencies of version and vergence eye movement components during translation.

NASA Program Biomedical Research and Countermeasur↗
NASA NTRS2004

Real-Time Adaptive Color Segmentation by Neural Networks

Artificial neural networks that would utilize the cascade error projection (CEP) algorithm have been proposed as means of autonomous, real-time, adaptive color segmentation of images that change with time. In the original intended application, such a neural network would be used to analyze digitized color video images of terrain on a remote planet as viewed from an uninhabited spacecraft approaching the planet. During descent toward the surface of the planet, information on the segmentation of the images into differently colored areas would be updated adaptively in real time to capture changes in contrast, brightness, and resolution, all in an effort to identify a safe and scientifically productive landing site and provide control feedback to steer the spacecraft toward that site. Potential terrestrial applications include monitoring images of crops to detect insect invasions and monitoring of buildings and other facilities to detect intruders. The CEP algorithm is reliable and is well suited to implementation in very-large-scale integrated (VLSI) circuitry. It was chosen over other neural-network learning algorithms because it is better suited to realtime learning: It provides a self-evolving neural-network structure, requires fewer iterations to converge and is more tolerant to low resolution (that is, fewer bits) in the quantization of neural-network synaptic weights. Consequently, a CEP neural network learns relatively quickly, and the circuitry needed to implement it is relatively simple. Like other neural networks, a CEP neural network includes an input layer, hidden units, and output units (see figure). As in other neural networks, a CEP network is presented with a succession of input training patterns, giving rise to a set of outputs that are compared with the desired outputs. Also as in other neural networks, the synaptic weights are updated iteratively in an effort to bring the outputs closer to target values. A distinctive feature of the CEP neural network and algorithm is that each update of synaptic weights takes place in conjunction with the addition of another hidden unit, which then remains in place as still other hidden units are added on subsequent iterations. For a given training pattern, the synaptic weight between (1) the inputs and the previously added hidden units and (2) the newly added hidden unit is updated by an amount proportional to the partial derivative of a quadratic error function with respect to the synaptic weight. The synaptic weight between the newly added hidden unit and each output unit is given by a more complex function that involves the errors between the outputs and their target values, the transfer functions (hyperbolic tangents) of the neural units, and the derivatives of the transfer functions.

Duong, Tuan A.↗
DOE OSTI2020

A quadratic programming flux correction method for high-order DG discretizations of S transport

In this work, we present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin (DG) discretizations of the S N transport equation, and we demonstrate the compatibility of this approach with the Variable Eddington Factor (VEF) method. The new fixup approach is sweep-compatible: during a transport sweep (block Gauss-Seidel iteration in which the scattering source is lagged), a local quadratic programming (QP) problem is solved in each spatial element to ensure that the solution satisfies certain physical constraints, including local particle balance. In this paper, we describe two choices of physical constraints, resulting in two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). In QPZ, the finite element coefficients of the solution are constrained to be nonnegative. In QPMP, they are constrained to adhere to an approximate discrete maximum principle. There are two primary takeaways in this paper. First, when the positive Bernstein basis is used for DG discretization, the QPMP method eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The latter feature – the dampening of unphysical oscillations – is an improvement upon standard, simpler fixup approaches such as the approach described in (denoted as the “zero and rescale” (ZR) method in this paper). This improvement comes at a moderate computational cost, but it is not prohibitive. Our results show that, even in an unrealistic worst-case scenario where 83% of the spatial elements require a fixup, the computational cost of performing a transport sweep with fixup is only ~31% greater than performing one without fixup. The second takeaway is that the VEF method can be used to accelerate the convergence of transport sweeps even when a fixup is applied. When optically thick regions are present, transport sweeps converge slowly, regardless of whether a fixup is applied, and acceleration is needed. However, attempting to apply standard diffusion synthetic acceleration (DSA) to fixed-up transport sweeps results in divergence for optically thick problems. Our results show that the same is not true for VEF. When VEF is combined with fixed-up transport sweeps, the result is a scheme that produces a nonnegative solution, converges independently of the mean free path, and, in the case of the QPMP fixup, adheres to an approximate discrete maximum principle.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

A new look at the simultaneous analysis and design of structures

The minimum weight optimization of structural systems, subject to strength and displacement constraints as well as size side constraints, was investigated by the Simultaneous ANalysis and Design (SAND) approach. As an optimizer, the code NPSOL was used which is based on a sequential quadratic programming (SQP) algorithm. The structures were modeled by the finite element method. The finite element related input to NPSOL was automatically generated from the input decks of such standard FEM/optimization codes as NASTRAN or ASTROS, with the stiffness matrices, at present, extracted from the FEM code ANALYZE. In order to avoid ill-conditioned matrices that can be encountered when the global stiffness equations are used as additional nonlinear equality constraints in the SAND approach (with the displacements as additional variables), the matrix displacement method was applied. In this approach, the element stiffness equations are used as constraints instead of the global stiffness equations, in conjunction with the nodal force equilibrium equations. This approach adds the element forces as variables to the system. Since, for complex structures and the associated large and very sparce matrices, the execution times of the optimization code became excessive due to the large number of required constraint gradient evaluations, the Kreisselmeier-Steinhauser function approach was used to decrease the computational effort by reducing the nonlinear equality constraint system to essentially a single combined constraint equation. As the linear equality and inequality constraints require much less computational effort to evaluate, they were kept in their previous form to limit the complexity of the KS function evaluation. To date, the standard three-bar, ten-bar, and 72-bar trusses have been tested. For the standard SAND approach, correct results were obtained for all three trusses although convergence became slower for the 72-bar truss. When the matrix displacement method was used, correct results were still obtained, but the execution times became excessive due to the large number of constraint gradient evaluations required. Using the KS function, the computational effort dropped, but the optimization seemed to become less robust. The investigation of this phenomenon is continuing. As an alternate approach, the code MINOS for the optimization of sparse matrices can be applied to the problem in lieu of the Kreisselmeier-Steinhauser function. This investigation is underway.

Striz, Alfred G.↗
DOE OSTI2026

Implementing a unified solver for nonlinearly constrained optimization

SQP and interior-point methods (also referred to as Lagrange-Newton methods) typically share key algorithmic components, such as strategies for computing descent directions and mechanisms that promote global convergence. Building on this insight, we introduce a unifying framework with eight building blocks that abstracts the workflows of Lagrange-Newton methods. We then present Uno, a modular C++ solver that implements our unifying framework and allows the automatic combination of a wide range of strategies with no programming effort from the user. Uno is meant to (1) organize mathematical optimization strategies into a coherent hierarchy; (2) offer a wide range of efficient and robust methods that can be compared for a given instance; (3) enable researchers to experiment with novel optimization strategies; and (4) reduce the cost of development and maintenance of multiple optimization solvers. Uno’s software design allows user to compose new customized solvers for emerging optimization areas such as robust optimization or optimization problems with complementarity constraints, while building on reliable nonlinear optimization techniques. We demonstrate that Uno is highly competitive against state-of-the-art solvers filterSQP, IPOPT, SNOPT, MINOS, LANCELOT, LOQO, and CONOPT on a subset of 429 small problems from the CUTE collection. Uno is available as open-source software under the MIT license at https://github.com/cvanaret/Uno and via its C, Julia, Python, Fortran, and AMPL interfaces.

97 MATHEMATICS AND COMPUTING↗