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Results for “quadratic optimization”

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.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1984

Development and testing of stable, invariant, isoparametric curvilinear 2- and 3-D hybrid-stress elements

Linear and quadratic Serendipity hybrid-stress elements are examined in respect of stability, coordinate invariance, and optimality. A formulation based upon symmetry group theory successfully addresses these issues in undistorted geometries and is fully detailed for plane elements. The resulting least-order stable invariant stress polynomials can be applied as astute approximations in distorted cases through a variety of tensor components and variational principles. A distortion sensitivity study for two- and three-dimensional elements provides favorable numerical comparisons with the assumed displacement method.

Punch, E. F.↗
NASA NTRS1978

Design of a compensation for an ARMA model of a discrete time system

The design of an optimal dynamic compensator for a multivariable discrete time system is studied. Also the design of compensators to achieve minimum variance control strategies for single input single output systems is analyzed. In the first problem the initial conditions of the plant are random variables with known first and second order moments, and the cost is the expected value of the standard cost, quadratic in the states and controls. The compensator is based on the minimum order Luenberger observer and it is found optimally by minimizing a performance index. Necessary and sufficient conditions for optimality of the compensator are derived. The second problem is solved in three different ways; two of them working directly in the frequency domain and one working in the time domain. The first and second order moments of the initial conditions are irrelevant to the solution. Necessary and sufficient conditions are derived for the compensator to minimize the variance of the output.

Mainemer, C. I.↗
NASA NTRS1999

Optimized System Identification

In system identification, one usually cares most about finding a model whose outputs are as close as possible to the true system outputs when the same input is applied to both. However, most system identification algorithms do not minimize this output error. Often they minimize model equation error instead, as in typical least-squares fits using a finite-difference model, and it is seen here that this distinction is significant. Here, we develop a set of system identification algorithms that minimize output error for multi-input/multi-output and multi-input/single-output systems. This is done with sequential quadratic programming iterations on the nonlinear least-squares problems, with an eigendecomposition to handle indefinite second partials. This optimization minimizes a nonlinear function of many variables, and hence can converge to local minima. To handle this problem, we start the iterations from the OKID (Observer/Kalman Identification) algorithm result. Not only has OKID proved very effective in practice, it minimizes an output error of an observer which has the property that as the data set gets large, it converges to minimizing the criterion of interest here. Hence, it is a particularly good starting point for the nonlinear iterations here. Examples show that the methods developed here eliminate the bias that is often observed using any system identification methods of either over-estimating or under-estimating the damping of vibration modes in lightly damped structures.

Juang, Jer-Nan↗
DOE OSTI2020

An augmented Lagrangian filter method

Here, we introduce a filter mechanism to enforce convergence for augmented Lagrangian methods for nonlinear programming. In contrast to traditional augmented Lagrangian methods, our approach does not require the use of forcing sequences that drive the first-order error to zero. Instead, we employ a filter to drive the optimality measures to zero. Our algorithm is flexible in the sense that it allows for equality-constrained quadratic programming steps to accelerate local convergence. We also include a feasibility restoration phase that allows fast detection of infeasible problems. We provide a convergence proof that shows that our algorithm converges to first-order stationary points. We provide preliminary numerical results that demonstrate the effectiveness of our proposed method.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1989

Algorithm For Optimal Control Of Large Structures

Cost of computation appears competitive with other methods. Problem to compute optimal control of forced response of structure with n degrees of freedom identified in terms of smaller number, r, of vibrational modes. Article begins with Hamilton-Jacobi formulation of mechanics and use of quadratic cost functional. Complexity reduced by alternative approach in which quadratic cost functional expressed in terms of control variables only. Leads to iterative solution of second-order time-integral matrix Volterra equation of second kind containing optimal control vector. Cost of algorithm, measured in terms of number of computations required, is of order of, or less than, cost of prior algoritms applied to similar problems.

Salama, Moktar A.↗
DOE OSTI2014

SU-E-T-573: The Robustness of a Combined Margin Recipe for Uncertainties During Radiotherapy

Purpose: To investigate the variability of a safety margin recipe that combines CTV and PTV margins quadratically, with several tumor, treatment, and user related factors. Methods: Margin recipes were calculated by monte-carlo simulations in 5 steps. 1. A spherical tumor with or without isotropic microscopic was irradiated with a 5 field dose plan2. PTV: Geometric uncertainties were introduced using systematic (Sgeo) and random (sgeo) standard deviations. CTV: Microscopic disease distribution was modelled by semi-gaussian (Smicro) with varying number of islets (Ni)3. For a specific uncertainty set (Sgeo, sgeo, Smicro(Ni)), margins were varied until pre-defined decrease in TCP or dose coverage was fulfilled. 4. First, margin recipes were calculated for each of the three uncertainties separately. CTV and PTV recipes were then combined quadratically to yield a final recipe M(Sgeo, sgeo, Smicro(Ni)).5. The final M was verified by simultaneous simulations of the uncertainties.Now, M has been calculated for various changing parameters like margin criteria, penumbra steepness, islet radio-sensitivity, dose conformity, and number of fractions. We subsequently investigated A: whether the combined recipe still holds in all these situations, and B: what the margin variation was in all these cases. Results: We found that the accuracy of the combined margin recipes remains on average within 1mm for all situations, confirming the correctness of the quadratic addition. Depending on the specific parameter, margin factors could change such that margins change over 50%. Especially margin recipes based on TCP-criteria are more sensitive to more parameters than those based on purely geometric Dmin-criteria. Interestingly, measures taken to minimize treatment field sizes (by e.g. optimizing dose conformity) are counteracted by the requirement of larger margins to get the same tumor coverage. Conclusion: Margin recipes combining geometric and microscopic uncertainties quadratically are accurate under varying circumstances. However margins can change up to 50% for different situations.

60 APPLIED LIFE SCIENCES↗
NASA NTRS1991

Optimal uniform-damping ratio controller for sequential design of multivariable systems

An optimal uniform-damping ratio controller is developed for the sequential design of a multivariable control system so that the designed closed-loop poles of the respective multivariable system and reduced-order observer are exactly placed on the negative real axis and/or the boundaries of desired sectors with constant-damping ratios. The functions in the quadratic performance index to be minimized are chosen as a combination of the weighted outputs, reduced states and inputs. Also, the optimal uniform-damping ratio controller is a combination of optimal output-feedback and optimal reduced-order state-feedback controllers. A numerical example is given to demonstrate the design procedure.

Shieh, Leang G.↗
DOE OSTI2026

Classical-Quantum Algorithm for Solving Stochastic Programs

Stochastic programming provides a rigorous mathematical framework for making decisions under uncertainty in a risk-aware manner. Two-stage stochastic programming is, perhaps, the simplest form of this framework. Here the first-stage variables represent decisions that must be made "here and now" in the face of uncertainty, while the second-stage variables are decisions made after uncertain events. However, the broad adoption of stochastic programming has been hindered by computational challenges caused by the two-stage stochastic programming formulation which requires solving an ensemble of optimization problems. Using quantum amplitude estimation (QAE), quantum computers have shown the theoretic ability to compute expectations with Monte-Carlo methods with quadratically fewer samples than classical methods. In this work, we present a quantum algorithm for computing the expectation term using QAE for given first-stage decisions. Further, we detail methods of computing gradient information from the quantum calculation enabling the application of classical gradient-based optimization techniques. The result is a classical-quantum hybrid method of solving two-stage stochastic programs. These techniques are demonstrated with computational experiments based an engineering optimization problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Solution of monotone complementarity and general convex programming problems using a modified potential reduction interior point method

We present a homogeneous algorithm equipped with a modified potential function for the monotone complementarity problem. We show that this potential function is reduced by at least a constant amount if a scaled Lipschitz condition (SLC) is satisfied. A practical algorithm based on this potential function is implemented in a software package named iOptimize. The implementation in iOptimize maintains global linear and polynomial time convergence properties, while achieving practical performance. It either successfully solves the problem, or concludes that the SLC is not satisfied. When compared with the mature software package MOSEK (barrier solver version 6.0.0.106), iOptimize solves convex quadratic programming problems, convex quadratically constrained quadratic programming problems, and general convex programming problems in fewer iterations. Moreover, several problems for which MOSEK fails are solved to optimality. In addition, we also find that iOptimize detects infeasibility more reliably than the general nonlinear solvers Ipopt (version 3.9.2) and Knitro (version 8.0).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A variational framework for residual-based adaptivity in neural PDE solvers and operator learning

Residual-based adaptive strategies are widely used in scientific machine learning yet remain largely heuristic. We introduce a variational framework that formalizes these methods through convex transformations of the residual, where different transformations correspond to distinct objective functionals. For instance, exponential weights target uniform error minimization, while linear weights recover quadratic error minimization. This perspective reveals adaptive weighting as a means of selecting sampling distributions that optimize a primal objective, directly linking discretization choices to error metrics. This principled approach yields three key benefits: it enables systematic design of adaptive schemes, reduces discretization error by lowering estimator variance, and enhances learning dynamics by improving gradient signal-to-noise ratio. Extending the framework to operator learning, we demonstrate substantial performance gains across diverse optimizers and architectures. Our results provide a theoretical perspective for residual-based adaptivity and establish a foundation for principled discretization and training.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1989

A new adaptive control approach for aerospace vehicles with parameter uncertainties

A new stochastic adaptive control structure is developed for the problem of combined parameter estimation and control of aerospace vehicles with changing parameters. Parameter uncertainties are modeled as first-order Gauss-Markov processes, and are introduced to the system dynamics through a small parameter. It is assumed that an accurate inertial measurement unit gives perfect measurements of the state variables. Since the stochastic system is assumed to be Gauss-Markov, the density function of the parameters given these measurements is conditionally Gaussian. Based on this conditionally Gaussian density, the problem of minimizing a quadratic cost over an infinite time horizon can be set up within the framework of stochastic optimal control theory. The optimal feedback control law is derived from a straightforward expansion of the Hamilton-Jacobi-Bellman equation, based on the LQG solution. The resulting nonlinear controller is applied to the pitch axis control of a space platform with uncertain moments of inertia and is shown to produce marked improvement over a fixed controller.

Hahn, Yungsun↗
NASA NTRS1991

Minimum-fuel rescue trajectories for the Extravehicular Excursion Unit

The problem of determining minimum-fuel trajectories for rescuing astronauts or equipment which become separated from a Space Station is addressed. Using the Clohessy-Wiltshire equations of relative motion and assuming impulsive Delta-Vs, the minimum-fuel rescue problem is shown to be a parameter optimization problem. Minimum-fuel rescue trajectories are found for seventeen test cases using a recursive quadratic programming algorithm. The results are analyzed and general rules for astronaut rescue and equipment retrieval are developed.

Fowler, W. T.↗
NASA NTRS1991

Orientation and shape control of a weight optimum free-free beam in a circular orbit

In this study, the vibration and orientation control of large space structures using the linear quadratic regulator technique is investigated. Emphasis is placed on the control of both a class of optimally designed structures and uniform structures meeting the mission requirements using a long free-free beam in orbit as an example. The open loop and closed loop dynamics are compared and the transient responses are obtained to determine the effectiveness of the control system design.

Bainum, Peter M.↗
DOE OSTI2016

SU-F-T-01: Optimization of the Accelerated Partial Breast Brachytherapy Fractionation with Consideration of Physical Doses to Tumor and Organ at Risk

Purpose: The accelerated partial breast irradiation (APBI) with brachytherapy prescribes 34Gy to be delivered in 10 fractions over 5 consecutive working days without considering the physical dose to the target and organs at risk (OARs) for an individual patient. The purpose of this study is to optimize the fractionation scheme by evaluating the radiation effect on tumor and OARs with a modified linear-quadratic (LQ) model based on dose-volume histograms (DVHs). Methods: Five breast patients treated with multilumen balloon brachytherapy were selected. The minimum skin and rib spacing were ranged from 2.5mm to 14.3mm and from 1.0mm to 25.0mm, respectively. The LQ model parameters were set as: (1) breast: α=0.08, β=0.028, doubling time Tpot=14.4 days, and starting time Tk=21days; (2) skin: acute reaction α=0.101, β=0.009; late reaction α=0.064, β=0.029; (3) rib: α=0.3, β=0.12. Boundary dose Dt was 6 Gy for both target and OARs. The relation between radiation effects on the tumor (ET) and OARs (EOAR) were plotted for fraction number from 1 to 20. Results: The value of radiation effect from routine 3.4Gyx10 fractions was used as reference, ETref and EOARref. If set ET=ETref, the fractionation that results in minimum EOAR values correspond to the optimal fractionation. For these patients, the optimal numbers are 10 fractions for skin acute reaction, 18 fractions for skin and rib late reaction while the doses per fraction are 3.4Gy and 2.05–2.10Gy, respectively. If set EOAR=EOARref, the fractionation that results in a maximum ET value corresponds to the optimal fractionation. The optimal fractionation is 3.4Gyx10 fractions for skin acute reaction, and 2.10–2.25Gyx18 fractions for skin late reaction and rib. Conclusion: For APBI brachytherapy, the routine 3.4Gyx10 fractions is optimal fractionation for skin acute reaction, while 2.05–2.25Gyx18 fractions is optimal fractionation for late reaction of skin and rib.

60 APPLIED LIFE SCIENCES↗
NASA NTRS1979

Towards fault-tolerant optimal control

The paper considers the design of fault-tolerant controllers that may endow systems with dynamic reliability. Results for jump linear quadratic Gaussian control problems are extended to include random jump costs, trajectory discontinuities, and a simple case of non-Markovian mode transitions.

Chizeck, H. J.↗
DOE OSTI2021

Optimal Cosmic Microwave Background Lensing Reconstruction and Parameter Estimation with SPTpol Data

Here, we perform the first simultaneous Bayesian parameter inference and optimal reconstruction of the gravitational lensing of the cosmic microwave background (CMB), using 100 deg 2 of polarization observations from the SPTpol receiver on the South Pole Telescope. These data reach noise levels as low as 5.8 μK arcmin in polarization, which are low enough that the typically used quadratic estimator (QE) technique for analyzing CMB lensing is significantly suboptimal. Conversely, the Bayesian procedure extracts all lensing information from the data and is optimal at any noise level. We infer the amplitude of the gravitational lensing potential to be ${A}_{\phi }=0.949\,\pm \,0.122$ using the Bayesian pipeline, consistent with our QE pipeline result, but with 17% smaller error bars. The Bayesian analysis also provides a simple way to account for systematic uncertainties, performing a similar job as frequentist "bias hardening" or linear bias correction, and reducing the systematic uncertainty on A Φ due to polarization calibration from almost half of the statistical error to effectively zero. Finally, we jointly constrain A Φ along with A L , the amplitude of lensing-like effects on the CMB power spectra, demonstrating that the Bayesian method can be used to easily infer parameters both from an optimal lensing reconstruction and from the delensed CMB, while exactly accounting for the correlation between the two. These results demonstrate the feasibility of the Bayesian approach on real data, and pave the way for future analysis of deep CMB polarization measurements with SPT-3G, Simons Observatory, and CMB-S4, where improvements relative to the QE can reach 1.5 times tighter constraints on A Φ and seven times lower effective lensing reconstruction noise.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1989

Guidance and Control strategies for aerospace vehicles

A neighboring optimal guidance scheme was devised for a nonlinear dynamic system with stochastic inputs and perfect measurements as applicable to fuel optimal control of an aeroassisted orbital transfer vehicle. For the deterministic nonlinear dynamic system describing the atmospheric maneuver, a nominal trajectory was determined. Then, a neighboring, optimal guidance scheme was obtained for open loop and closed loop control configurations. Taking modelling uncertainties into account, a linear, stochastic, neighboring optimal guidance scheme was devised. Finally, the optimal trajectory was approximated as the sum of the deterministic nominal trajectory and the stochastic neighboring optimal solution. Numerical results are presented for a typical vehicle. A fuel-optimal control problem in aeroassisted noncoplanar orbital transfer is also addressed. The equations of motion for the atmospheric maneuver are nonlinear and the optimal (nominal) trajectory and control are obtained. In order to follow the nominal trajectory under actual conditions, a neighboring optimum guidance scheme is designed using linear quadratic regulator theory for onboard real-time implementation. One of the state variables is used as the independent variable in reference to the time. The weighting matrices in the performance index are chosen by a combination of a heuristic method and an optimal modal approach. The necessary feedback control law is obtained in order to minimize the deviations from the nominal conditions.

Hibey, J. L.↗
NASA NTRS1991

Neighboring optimal guidance for aeroassisted noncoplanar orbital transfer

The fuel-optimal control problem in aeroassisted noncoplanar orbital transfer is addressed. The equations of motion for the atmospheric maneuver are nonlinear and the optimal (nominal) trajectory and control are obtained. In order to follow the nominal trajectory under actual conditions, a neighboring optimum guidance scheme is designed using linear quadratic regulator theory for onboard real-time implementation. One of the state variables is used as the independent variable in preference to the time. The weighting matrices in the performance index are chosen by a combination of a heuristic method and an optimal modal approach. The necessary feedback control law is obtained in order to minimize the deviations from the nominal conditions. The results are presented to a typical aeroassisted noncoplanar orbital transfer problem.

Naidu, D. S.↗