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All sourcesNASA NTRSDOE OSTI
NASA NTRS1980

Modal methods in optimal control synthesis

Efficient algorithms for solving linear smoother-follower problems with quadratic criteria are presented. For time-invariant systems, the algorithm consists of one backward integration of a linear vector equation and one forward integration of another linear vector equation. Furthermore, the backward and forward Riccati matrices can be expressed in terms of the eigenvalues and eigenvectors of the Euler-Lagrange equations. Hence, the gains of the forward and backward Kalman-Bucy filters and of the optimal state-feedback regulator can be determined without integration of matrix Riccati equations. A computer program has been developed, based on this method of determining the gains, to synthesize the optimal time-invariant compensator in the presence of random disturbance inputs and random measurement errors. The program also computes the rms state and control variables of the optimal closed-loop system.

Bryson, A. E., Jr.↗
NASA NTRS1989

Minimizing distortion and internal forces in truss structures by simulated annealing

Inaccuracies in the length of members and the diameters of joints of large truss reflector backup structures may produce unacceptable levels of surface distortion and member forces. However, if the member lengths and joint diameters can be measured accurately it is possible to configure the members and joints so that root-mean-square (rms) surface error and/or rms member forces is minimized. Following Greene and Haftka (1989) it is assumed that the force vector f is linearly proportional to the member length errors e(sub M) of dimension NMEMB (the number of members) and joint errors e(sub J) of dimension NJOINT (the number of joints), and that the best-fit displacement vector d is a linear function of f. Let NNODES denote the number of positions on the surface of the truss where error influences are measured. The solution of the problem is discussed. To classify, this problem was compared to a similar combinatorial optimization problem. In particular, when only the member length errors are considered, minimizing d(sup 2)(sub rms) is equivalent to the quadratic assignment problem. The quadratic assignment problem is a well known NP-complete problem in operations research literature. Hence minimizing d(sup 2)(sub rms) is is also an NP-complete problem. The focus of the research is the development of a simulated annealing algorithm to reduce d(sup 2)(sub rms). The plausibility of this technique is its recent success on a variety of NP-complete combinatorial optimization problems including the quadratic assignment problem. A physical analogy for simulated annealing is the way liquids freeze and crystallize. All computational experiments were done on a MicroVAX. The two interchange heuristic is very fast but produces widely varying results. The two and three interchange heuristic provides less variability in the final objective function values but runs much more slowly. Simulated annealing produced the best objective function values for every starting configuration and was faster than the two and three interchange heuristic.

Kincaid, Rex K.↗
DOE OSTI2015

A feasibility study: Selection of a personalized radiotherapy fractionation schedule using spatiotemporal optimization

Purpose: To investigate the impact of using spatiotemporal optimization, i.e., intensity-modulated spatial optimization followed by fractionation schedule optimization, to select the patient-specific fractionation schedule that maximizes the tumor biologically equivalent dose (BED) under dose constraints for multiple organs-at-risk (OARs). Methods: Spatiotemporal optimization was applied to a variety of lung tumors in a phantom geometry using a range of tumor sizes and locations. The optimal fractionation schedule for a patient using the linear-quadratic cell survival model depends on the tumor and OAR sensitivity to fraction size (α/β), the effective tumor doubling time (T{sub d}), and the size and location of tumor target relative to one or more OARs (dose distribution). The authors used a spatiotemporal optimization method to identify the optimal number of fractions N that maximizes the 3D tumor BED distribution for 16 lung phantom cases. The selection of the optimal fractionation schedule used equivalent (30-fraction) OAR constraints for the heart (D{sub mean} ≤ 45 Gy), lungs (D{sub mean} ≤ 20 Gy), cord (D{sub max} ≤ 45 Gy), esophagus (D{sub max} ≤ 63 Gy), and unspecified tissues (D{sub 05} ≤ 60 Gy). To assess plan quality, the authors compared the minimum, mean, maximum, and D{sub 95} of tumor BED, as well as the equivalent uniform dose (EUD) for optimized plans to conventional intensity-modulated radiation therapy plans prescribing 60 Gy in 30 fractions. A sensitivity analysis was performed to assess the effects of T{sub d} (3–100 days), tumor lag-time (T{sub k} = 0–10 days), and the size of tumors on optimal fractionation schedule. Results: Using an α/β ratio of 10 Gy, the average values of tumor max, min, mean BED, and D{sub 95} were up to 19%, 21%, 20%, and 19% larger than those from conventional prescription, depending on T{sub d} and T{sub k} used. Tumor EUD was up to 17% larger than the conventional prescription. For fast proliferating tumors with T{sub d} less than 10 days, there was no significant increase in tumor BED but the treatment course could be shortened without a loss in tumor BED. The improvement in the tumor mean BED was more pronounced with smaller tumors (p-value = 0.08). Conclusions: Spatiotemporal optimization of patient plans has the potential to significantly improve local tumor control (larger BED/EUD) of patients with a favorable geometry, such as smaller tumors with larger distances between the tumor target and nearby OAR. In patients with a less favorable geometry and for fast growing tumors, plans optimized using spatiotemporal optimization and conventional (spatial-only) optimization are equivalent (negligible differences in tumor BED/EUD). However, spatiotemporal optimization yields shorter treatment courses than conventional spatial-only optimization. Personalized, spatiotemporal optimization of treatment schedules can increase patient convenience and help with the efficient allocation of clinical resources. Spatiotemporal optimization can also help identify a subset of patients that might benefit from nonconventional (large dose per fraction) treatments that are ineligible for the current practice of stereotactic body radiation therapy.

60 APPLIED LIFE SCIENCES↗
NASA NTRS1990

A superlinear interior points algorithm for engineering design optimization

We present a quasi-Newton interior points algorithm for nonlinear constrained optimization. It is based on a general approach consisting of the iterative solution in the primal and dual spaces of the equalities in Karush-Kuhn-Tucker optimality conditions. This is done in such a way to have primal and dual feasibility at each iteration, which ensures satisfaction of those optimality conditions at the limit points. This approach is very strong and efficient, since at each iteration it only requires the solution of two linear systems with the same matrix, instead of quadratic programming subproblems. It is also particularly appropriate for engineering design optimization inasmuch at each iteration a feasible design is obtained. The present algorithm uses a quasi-Newton approximation of the second derivative of the Lagrangian function in order to have superlinear asymptotic convergence. We discuss theoretical aspects of the algorithm and its computer implementation.

Herskovits, J.↗
NASA NTRS1993

Space transportation vehicle design evaluation using saturated designs

An important objective in the preliminary design and evaluation of space transportation vehicles is to find the best values of design variables that optimize the performance characteristic (e.g. dry weight). For a given configuration, the vehicle performance can be determined by the use of complex sizing and performance evaluation computer programs. These complex computer programs utilize iterative algorithms and they are generally too expensive and/or difficult to use directly in multidisciplinary design optimization. An alternative is to use response surface methodology (RSM) and obtain quadratic polynomial approximations to the functional relationships between performance characteristics and design variables. In RSM, these approximation models are then used to determine optimum design parameter values and for rapid sensitivity studies. Constructing a second-order model requires that 'n' design parameters be studied at least at 3 levels (values) so that the coefficients in the model can be estimated. There, 3(n) factorial experiments (point designs or observations) may be necessary. For small values of 'n' such as two or three, this design works well. However, when a large number of design parameters are under study, the number of design points required for a full-factorial design may become excessive. Fortunately, these quadratic polynomial approximations can be obtained by selecting an efficient design matrix using central composite designs (CCD) from design of experiments theory. Each unique point design from the CCD matrix is then conducted using computerized analysis tools (e.g. POST, CONSIZ, etc.). In the next step, least squares regression analysis is used to calculate the quadratic polynomial coefficients from the data. However, in some multidisciplinary applications involving a large number of design variables and several disciplines, the computerized performance synthesis programs may get too time consuming and expensive to run even with the use of efficient central composite designs. In such cases, it may be preferable to keep the number of design points to an absolute minimum and trade some model accuracy with cost. For this purpose, another class of experimental designs, called saturated D-optimal designs may be utilized for generating a matrix of vehicle designs. A design is called saturated when the number of design points is exactly equal to the number of terms in the model to be fitted. As a result, saturated designs require the absolute minimum number of design points ((n+1)(n+2)/s) to estimate the quadratic polynomial model coefficients. Saturated designs can be generated using the D-optimality criterion. A good saturated design should give rise to least squares estimates with minimum generalized variance.

Unal, Resit↗
NASA NTRS1990

Optimal helicopter trajectory planning for terrain following flight

Helicopters operating in high threat areas have to fly close to the earth surface to minimize the risk of being detected by the adversaries. Techniques are presented for low altitude helicopter trajectory planning. These methods are based on optimal control theory and appear to be implementable onboard in realtime. Second order necessary conditions are obtained to provide a criterion for finding the optimal trajectory when more than one extremal passes through a given point. A second trajectory planning method incorporating a quadratic performance index is also discussed. Trajectory planning problem is formulated as a differential game. The objective is to synthesize optimal trajectories in the presence of an actively maneuvering adversary. Numerical methods for obtaining solutions to these problems are outlined. As an alternative to numerical method, feedback linearizing transformations are combined with the linear quadratic game results to synthesize explicit nonlinear feedback strategies for helicopter pursuit-evasion. Some of the trajectories generated from this research are evaluated on a six-degree-of-freedom helicopter simulation incorporating an advanced autopilot. The optimal trajectory planning methods presented are also useful for autonomous land vehicle guidance.

Menon, P. K. A.↗
NASA NTRS1989

Optimal force distribution for payload positioning using a planar dual-arm robot

This paper presents a parameter optimization technique for deciding the force distribution on a payload being transported along a predetermined trajectory using two planar manipulator arms. The methodology begins by transforming the singular dynamics of two-arm transport to an ordinary set of differential equations and then proceeds to obtain a relation between the torques exerted by each arm. This relation is then used in a quadratic torque cost which is subsequently minimized to yield an optimal torque distribution. Significant savings in energy were found to occur when the arms were allowed to interact by transmission of forces through the payload. Even more significant are the savings found over one-arm transport of payloads where the arm torques are fixed by the prescribed trajectory.

Carignan, C. R.↗
NASA NTRS1994

Automation of reverse engineering process in aircraft modeling and related optimization problems

During the year of 1994, the engineering problems in aircraft modeling were studied. The initial concern was to obtain a surface model with desirable geometric characteristics. Much of the effort during the first half of the year was to find an efficient way of solving a computationally difficult optimization model. Since the smoothing technique in the proposal 'Surface Modeling and Optimization Studies of Aerodynamic Configurations' requires solutions of a sequence of large-scale quadratic programming problems, it is important to design algorithms that can solve each quadratic program in a few interactions. This research led to three papers by Dr. W. Li, which were submitted to SIAM Journal on Optimization and Mathematical Programming. Two of these papers have been accepted for publication. Even though significant progress has been made during this phase of research and computation times was reduced from 30 min. to 2 min. for a sample problem, it was not good enough for on-line processing of digitized data points. After discussion with Dr. Robert E. Smith Jr., it was decided not to enforce shape constraints in order in order to simplify the model. As a consequence, P. Dierckx's nonparametric spline fitting approach was adopted, where one has only one control parameter for the fitting process - the error tolerance. At the same time the surface modeling software developed by Imageware was tested. Research indicated a substantially improved fitting of digitalized data points can be achieved if a proper parameterization of the spline surface is chosen. A winning strategy is to incorporate Dierckx's surface fitting with a natural parameterization for aircraft parts. The report consists of 4 chapters. Chapter 1 provides an overview of reverse engineering related to aircraft modeling and some preliminary findings of the effort in the second half of the year. Chapters 2-4 are the research results by Dr. W. Li on penalty functions and conjugate gradient methods for quadratic programming problems.

Li, W.↗
DOE OSTI2024

Efficient Parameterization of Density Functional Tight-Binding for 5 f -Elements: A Th–O Case Study

Density functional tight binding (DFTB) models for f-element species are challenging to parametrize owing to the large number of adjustable parameters. The explicit optimization of the terms entering the semiempirical DFTB Hamiltonian related to f orbitals is crucial to generating a reliable parametrization for f-block elements, because they play import roles in bonding interactions. However, since the number of parameters grows quadratically with the number of orbitals, the computational cost for parameter optimization is much more expensive for the f-elements than for the main group elements. In this work we present a set of efficient approaches for mitigating the hurdle imposed by the large size of the parameter space. A novel group-by-orbital correction functions for two-center bond integrals was developed. With this approach the number of parameters is reduced, and it grows linearly with the number of elements, maintaining the accuracy and the number of parameters, in the case of f elements, by more than 40%. The parameter optimization step was accelerated by means of the mini-batch BFGS method. This method allows parameter optimizations with much larger training sets than other single batch methods. A stochastic optimizer was employed that helped overcome shallow local minima in the objective function. The proposed algorithm was used to parametrize the DFTB Hamiltonian for the Th–O system, which was subsequently applied to the study of ThO 2 nanoparticles. The training set consisted of 6322 unique structures, which is barely feasible with conventional optimization methods. The optimized parameter set, LANL-ThO, displays good agreement with DFT-calculated properties such as energies, forces, and structures for both clusters and bulk ThO 2 . Benefiting from the fewer number of parameters and lower computational costs for objective function evaluations, this new approach shows its potential applications in DFTB parametrization for elements with high angular momentum, which present a challenge to conventional methods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

A trust region algorithm for PDE-constrained optimization with bound constraints using reduced order modeling

This paper presents a novel trust region algorithm that relies on proper orthogonal de-composition techniques to construct accurate reduced order models during optimization.The algorithm samples high fidelity snapshots to compute the POD functions that are used to generate reduced order models. The reduced order models are used to replace the computationally intensive high fidelity finite element evaluations during optimization.The proposed algorithm employs a trust region framework to detect loss in predictive ac-curacy in the reduced order model and automatically update the POD functions during optimization. The trust region framework allows the algorithm to use sound mathematical metrics to effectively improve the accuracy and robustness of the reduced order models during optimization. The algorithm also employs a projected gradient algorithm to model bound constraints and compute optimal and feasible controls.This paper also presents an accurate Hessian formulation for topology optimization problems. The proposed trust region framework relies on a quadratic model to update the control. This quadratic model needs reliable second order derivative information to predict the behavior of the objective function within a suitable trust region. If a nonlinear Hessian formulation is used, the computational effort increases due to additional finite element evaluations. The proposed linear Hessian formulation reduces the computational effort and enables the calculation of the second order derivative information without additional finite element model evaluations. Examples in topology optimization are presented to demonstrate the applicability of the proposed algorithm and linear Hessian formulation for large-scale PDE-constrained optimization.

bound constraints↗
NASA NTRS1983

Approximation of the optimal compensator for a large space structure

This paper considers the approximation of the optimal compensator for a Large Space Structure. The compensator is based upon a solution to the Linear Stochastic Quadratic Regulator problem. Colocation of sensors and actuators is assumed. A small gain analytical solution for the optimal compensator is obtained for a single input/single output system, i.e., certain terms in the compensator can be neglected for sufficiently small gain. The compensator is calculated in terms of the kernel to a Volterra integral operator using a Neumann series. The calculation of the compensator is based upon the C sub 0 semigroup for the infinite dimensional system. A finite dimensional approximation of the compensator is, therefore, obtained through analysis of the infinite dimensional compensator which is a compact operator.

Mackay, M. K.↗
NASA NTRS2002

Linear-Quadratic-Gaussian Regulator Developed for a Magnetic Bearing

Linear-Quadratic-Gaussian (LQG) control is a modern state-space technique for designing optimal dynamic regulators. It enables us to trade off regulation performance and control effort, and to take into account process and measurement noise. The Structural Mechanics and Dynamics Branch at the NASA Glenn Research Center has developed an LQG control for a fault-tolerant magnetic bearing suspension rig to optimize system performance and to reduce the sensor and processing noise. The LQG regulator consists of an optimal state-feedback gain and a Kalman state estimator. The first design step is to seek a state-feedback law that minimizes the cost function of regulation performance, which is measured by a quadratic performance criterion with user-specified weighting matrices, and to define the tradeoff between regulation performance and control effort. The next design step is to derive a state estimator using a Kalman filter because the optimal state feedback cannot be implemented without full state measurement. Since the Kalman filter is an optimal estimator when dealing with Gaussian white noise, it minimizes the asymptotic covariance of the estimation error.

Choi, Benjamin B.↗
NASA NTRS1978

Specific-optimal control with a dual minimal-order observer-based compensator

The steady-state optimal control of a linear time-invariant stochastic system by means of a minimal-order dual-observer-based compensator is considered in this paper. The structure of the compensator is fixed while the associated gains are to be chosen so as to minimize a quadratic penalty on the plant state. Necessary and sufficient conditions for optimality are given, and an explicit solution is displayed. Salient features pertaining to the optimal system are: a decoupling property, a projection property, and an innovation property. Finally, it is shown that this design corresponds to a singular LQG problem, which is precisely the dual of another singular LQG problem: namely Newmann's problem. A complete picture is then given showing clearly the correspondence between the two designs.

Blanvillain, P. J.↗
DOE OSTI2025

A unified funnel restoration SQP algorithm

We consider nonlinearly constrained optimization problems and discuss a generic double-loop framework consisting of basic algorithmic ingredients that unifies a broad range of nonlinear optimization solvers. This framework has been implemented in the open-source solver Uno, a Swiss Army knife-like C++ optimization framework that unifies many nonlinearly constrained nonconvex optimization solvers. We illustrate the framework with a sequential quadratic programming (SQP) algorithm that maintains an acceptable upper bound on the constraint violation, called a funnel, that is monotonically decreased to control the feasibility of the iterates. Infeasible quadratic subproblems are handled by a feasibility restoration strategy. Globalization is controlled by a line search or a trust-region method. We prove global convergence of the trust-region funnel SQP method, building on known results from filter methods. We implement the algorithm in Uno, and we provide extensive test results for the trust-region line-search funnel SQP on small CUTEst instances.

Kiessling, David [Katholieke Univ. Leuven, Heverle↗
DOE OSTI2018

Gaussian Mean Field Lattice Gas

We study rigorously a lattice gas version of the Sherrington–Kirckpatrick spin glass model. In discrete optimization literature this problem is known as unconstrained binary quadratic programming and it belongs to the class NP-hard. We prove that the fluctuations of the ground state energy tend to vanish in the thermodynamic limit, and we give a lower bound of such ground state energy. Then we present a heuristic algorithm, based on a probabilistic cellular automaton, which seems to be able to find configurations with energy very close to the minimum, even for quite large instances.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Application-specific machine-learned interatomic potentials: exploring the trade-off between DFT convergence, MLIP expressivity, and computational cost

Machine-learned interatomic potentials (MLIPs) are revolutionizing computational materials science and chemistry by offering an efficient alternative to ab initio molecular dynamics (MD) simulations. However, fitting high-quality MLIPs remains a challenging, time-consuming, and computationally intensive task where numerous trade-offs have to be considered, e.g., How much and what kind of atomic configurations should be included in the training set? Which level of ab initio convergence should be used to generate the training set? Which loss function should be used for fitting the MLIP? Which machine learning architecture should be used to train the MLIP? The answers to these questions significantly impact both the computational cost of MLIP training and the accuracy and computational cost of subsequent MLIP MD simulations. In this study, we use a configurationally diverse beryllium dataset and quadratic spectral neighbor analysis potential. We demonstrate that joint optimization of energy versus force weights, training set selection strategies, and convergence settings of the ab initio reference simulations, as well as model complexity can lead to a significant reduction in the overall computational cost associated with training and evaluating MLIPs. This opens the door to computationally efficient generation of high-quality MLIPs for a range of applications which demand different accuracy versus training and evaluation cost trade-offs.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Wake forces in a background of quadratically coupled mediators

Two particles can exert forces on each other when embedded in a sea of weakly coupled particles. These “wake forces” occur whenever the source and target particles have quadratic interactions with the mediating particles; they are proportional to the ambient energy density and typically have a range of order the characteristic de Broglie wavelength of the background. The effect can be understood as source particles causing a disturbance in the background waves—a wake—which subsequently interacts with the target particles. Wake forces can be mediated by bosons or fermions, can have spin dependence, may be attractive or repulsive, and have a generally anisotropic spatial profile and range that depends on the phase-space distribution of the ambient particles. In this work, I investigate the application of wake forces to dark matter searches, recast existing limits on short-range forces into leading constraints on dark matter with quadratic couplings, and sketch out potential experimental modifications to optimize sensitivity. Wake forces occur in the Standard Model: the presence of the cosmic neutrino background induces a millimeter-range force about 22 orders of magnitude weaker than gravity. Wake forces may also be relevant in condensed-matter and atomic physics. Published by the American Physical Society 2024

Astronomy & Astrophysics↗