Search NASA⌕ Search

SEARCH · Search NASA

Results for “convergence and optimality analysis”

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.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1997

Convergence Estimates for Multidisciplinary Analysis and Optimization

A quantitative analysis of coupling between systems of equations is introduced. This analysis is then applied to problems in multidisciplinary analysis, sensitivity, and optimization. For the sensitivity and optimization problems both multidisciplinary and single discipline feasibility schemes are considered. In all these cases a "convergence factor" is estimated in terms of the Jacobians and Hessians of the system, thus it can also be approximated by existing disciplinary analysis and optimization codes. The convergence factor is identified with the measure for the "coupling" between the disciplines in the system. Applications to algorithm development are discussed. Demonstration of the convergence estimates and numerical results are given for a system composed of two non-linear algebraic equations, and for a system composed of two PDEs modeling aeroelasticity.

Arian, Eyal↗
NASA NTRS2005

Surrogate-based Analysis and Optimization

A major challenge to the successful full-scale development of modem aerospace systems is to address competing objectives such as improved performance, reduced costs, and enhanced safety. Accurate, high-fidelity models are typically time consuming and computationally expensive. Furthermore, informed decisions should be made with an understanding of the impact (global sensitivity) of the design variables on the different objectives. In this context, the so-called surrogate-based approach for analysis and optimization can play a very valuable role. The surrogates are constructed using data drawn from high-fidelity models, and provide fast approximations of the objectives and constraints at new design points, thereby making sensitivity and optimization studies feasible. This paper provides a comprehensive discussion of the fundamental issues that arise in surrogate-based analysis and optimization (SBAO), highlighting concepts, methods, techniques, as well as practical implications. The issues addressed include the selection of the loss function and regularization criteria for constructing the surrogates, design of experiments, surrogate selection and construction, sensitivity analysis, convergence, and optimization. The multi-objective optimal design of a liquid rocket injector is presented to highlight the state of the art and to help guide future efforts.

Queipo, Nestor V.↗
DOE OSTI2025

Analysis of the SiMPL Method for Density-Based Topology Optimization

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides pointwise bound-preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [D. Kim et al., Struct. Multidiscip. Optim., 68 (2025), 74]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. Here, we refer to the new algorithm as the SiMPL method (pronounced “simple”), which stands for Sigmoidal Mirror descent with a Projected Latent variable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Local convergence analysis of an inexact trust-region method for nonsmooth optimization

In Baraldi, we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. Furthermore, this algorithm has demonstrated excellent performance and scalability with problem size. In this paper, we enrich the convergence analysis for this algorithm, proving strong convergence of the iterates with guaranteed rates. In particular, we demonstrate that the trust-region algorithm recovers superlinear, even quadratic, convergence rates when using a second-order Taylor approximation of the smooth objective function term.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2012

Creation of an Upper Stage Trajectory Capability Boundary to Enable Booster System Trade Space Exploration

The problem of trajectory optimization is important in all space missions. The solution of this problem enables one to specify the optimum thrust steering program which should be followed to achieve a specified mission objective, simultaneously satisfying the constraints.1 It is well known that whether or not the ascent trajectory is optimal can have a significant impact on propellant usage for a given payload, or on payload weight for the same gross vehicle weight.2 Consequently, ascent guidance commands are usually optimized in some fashion. Multi-stage vehicles add complexity to this analysis process as changes in vehicle properties in one stage propagate to the other stages through gear ratios and changes in the optimal trajectory. These effects can cause an increase in analysis time as more variables are added and convergence of the optimizer to system closure requires more analysis iterations. In this paper, an approach to simplifying this multi-stage problem through the creation of an upper stage capability boundary is presented. This work was completed as part of a larger study focused on trade space exploration for the advanced booster system that will eventually form a part of NASA s new Space Launch System.3 The approach developed leverages Design of Experiments and Surrogate Modeling4 techniques to create a predictive model of the SLS upper stage performance. The design of the SLS core stages is considered fixed for the purposes of this study, which results in trajectory parameters such as staging conditions being the only variables relevant to the upper stage. Through the creation of a surrogate model, which takes staging conditions as inputs and predicts the payload mass delivered by the SLS upper stage to a reference orbit as the response, it is possible to identify a "surface" of staging conditions which all satisfy the SLS requirement of placing 130 metric tons into low-Earth orbit (LEO).3 This identified surface represents the 130 metric ton capability boundary for the upper stage, such that if the combined first stage and boosters can achieve any one staging point on that surface, then the design is identified as feasible. With the surrogate model created, design and analysis of advanced booster concepts is streamlined, as optimization of the upper stage trajectory is no longer required in every design loop.

Walsh, Ptrick↗
DOE OSTI2023

An Online Joint Optimization–Estimation Architecture for Distribution Networks

Here in this article, we propose an optimal joint optimization-estimation architecture for distribution networks, which jointly solves the optimal power flow (OPF) problem and static state estimation (SE) problem through an online gradient-based feedback algorithm. The main objective is to enable a fast and timely interaction between the OPF decisions and state estimators with limited sensor measurements. First, convergence and optimality of the proposed algorithm are analytically established. Then, the proposed gradient-based algorithm is modified by introducing statistical information of the inherent estimation and linearization errors for an improved and robust performance of the online OPF decisions. Overall, the proposed method eliminates the traditional separation of operation and monitoring, where optimization and estimation usually operate at distinct layers and different time scales. Hence, it enables a computationally affordable, efficient, and robust online operational framework for distribution networks under time-varying settings.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS2022

A NASA-Air Force Precipitation Analysis for Near-Real-Time Operations

This article describes a new precipitation analysis algorithm developed by NASA for time-sensitive operations at the United States Air Force. Implemented as part of the Land Information System—a land modeling and data assimilation software framework—this NASA–Air Force Precipitation Analysis (NAFPA) combines numerical weather prediction model outputs with rain gauge measurements and satellite estimates to produce global, gridded 3-h accumulated precipitation fields at approximately 10-km resolution. Input observations are subjected to quality control checks before being used by the Bratseth analysis algorithm that converges to optimal interpolation. NAFPA assimilates up to 3.5 million observations without artificial data thinning or selection. To evaluate this new approach, a multiyear reanalysis is generated and intercompared with eight alternative precipitation products across the contiguous United States, Africa, and the monsoon region of eastern Asia. NAFPA yields superior accuracy and correlation over low-latency (up to 14 h) alternatives (numerical weather prediction and satellite retrievals), and often outperforms high-latency (up to 3.5 months) products, although the details for the latter vary by region and product. The development of NAFPA offers a high-quality, near-real-time product for use in meteorological, land surface, and hydrological research and applications.

Africa↗
NASA NTRSDate not supplied

Optimal Control using Composite Bernstein Approximants

In this work, we present composite Bernstein polynomials as a direct collocation method for approximating optimal control problems. An analysis of the convergence properties of composite Bernstein polynomials is provided, and beneficial properties of composite Bernstein polynomials for the solution of optimal control problems are discussed. The efficacy of the proposed approximation method is demonstrated through a bang-bang example. Lastly, we apply this method to a motion planning problem, offering a practical solution that emphasizes the ability of this method to solve complex optimal control problems.

Gage MacLin↗
NASA NTRS1989

A zonal method for modeling powered-lift aircraft flow fields

A zonal method for modeling powered-lift aircraft flow fields is based on the coupling of a three-dimensional Navier-Stokes code to a potential flow code. By minimizing the extent of the viscous Navier-Stokes zones the zonal method can be a cost effective flow analysis tool. The successful coupling of the zonal solutions provides the viscous/inviscid interations that are necessary to achieve convergent and unique overall solutions. The feasibility of coupling the two vastly different codes is demonstrated. The interzone boundaries were overlapped to facilitate the passing of boundary condition information between the codes. Routines were developed to extract the normal velocity boundary conditions for the potential flow zone from the viscous zone solution. Similarly, the velocity vector direction along with the total conditions were obtained from the potential flow solution to provide boundary conditions for the Navier-Stokes solution. Studies were conducted to determine the influence of the overlap of the interzone boundaries and the convergence of the zonal solutions on the convergence of the overall solution. The zonal method was applied to a jet impingement problem to model the suckdown effect that results from the entrainment of the inviscid zone flow by the viscous zone jet. The resultant potential flow solution created a lower pressure on the base of the vehicle which produces the suckdown load. The feasibility of the zonal method was demonstrated. By enhancing the Navier-Stokes code for powered-lift flow fields and optimizing the convergence of the coupled analysis a practical flow analysis tool will result.

Roberts, D. W.↗
DOE OSTI2024

Variational Quantum Algorithms for Semidefinite Programming

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., N$\gg$M, where N is the dimension of the input matrices and M is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Convergence Analysis for an Online Data-Driven Feedback Control Algorithm

This paper presents convergence analysis of a novel data-driven feedback control algorithm designed for generating online controls based on partial noisy observational data. The algorithm comprises a particle filter-enabled state estimation component, estimating the controlled system’s state via indirect observations, alongside an efficient stochastic maximum principle-type optimal control solver. By integrating weak convergence techniques for the particle filter with convergence analysis for the stochastic maximum principle control solver, we derive a weak convergence result for the optimization procedure in search of optimal data-driven feedback control. Numerical experiments are performed to validate the theoretical findings.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Nonlinear optimal recovery in Hilbert spaces

Here, this paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear optimal recovery problem, establishing its well-posedness and proving its convergence to the true solution as the number of measurements increases. However, the resulting formulation might not have a finite-dimensional solution in general. We thus present a sufficient condition for the finite dimensionality of the solution, applicable to problems with well-defined point evaluation measurements. To address the broader setting, we introduce a relaxed nonlinear optimal recovery and provide a detailed convergence analysis. An illustrative example is given to demonstrate that our formulations and theoretical findings offer a comprehensive framework for solving nonlinear problems in infinite-dimensional spaces with limited data.

convergence↗
NASA NTRS1997

Gradient-Based Aerodynamic Shape Optimization Using ADI Method for Large-Scale Problems

A gradient-based shape optimization methodology, that is intended for practical three-dimensional aerodynamic applications, has been developed. It is based on the quasi-analytical sensitivities. The flow analysis is rendered by a fully implicit, finite volume formulation of the Euler equations.The aerodynamic sensitivity equation is solved using the alternating-direction-implicit (ADI) algorithm for memory efficiency. A flexible wing geometry model, that is based on surface parameterization and platform schedules, is utilized. The present methodology and its components have been tested via several comparisons. Initially, the flow analysis for for a wing is compared with those obtained using an unfactored, preconditioned conjugate gradient approach (PCG), and an extensively validated CFD code. Then, the sensitivities computed with the present method have been compared with those obtained using the finite-difference and the PCG approaches. Effects of grid refinement and convergence tolerance on the analysis and shape optimization have been explored. Finally the new procedure has been demonstrated in the design of a cranked arrow wing at Mach 2.4. Despite the expected increase in the computational time, the results indicate that shape optimization, which require large numbers of grid points can be resolved with a gradient-based approach.

Pandya, Mohagna J.↗
DOE OSTI2024

An Empirical Quantile Estimation Approach for Chance-Constrained Nonlinear Optimization Problems

We investigate an empirical quantile estimation approach to solve chance-constrained nonlinear optimization problems. Our approach is based on the reformulation of the chance constraint as an equivalent quantile constraint to provide stronger signals on the gradient. In this approach, the value of the quantile function is estimated empirically from samples drawn from the random parameters, and the gradient of the quantile function is estimated via a finite-difference approximation on top of the quantile-function-value estimation. We establish a convergence theory of this approach within the framework of an augmented Lagrangian method for solving general nonlinear constrained optimization problems. The foundation of the convergence analysis is a concentration property of the empirical quantile process, and the analysis is divided based on whether or not the quantile function is differentiable. In contrast to the sampling-and-smoothing approach used in the literature, the method developed in this paper does not involve any smoothing function and hence the quantile-function gradient approximation is easier to implement and there are less accuracy-control parameters to tune. Furthermore, we demonstrate the effectiveness of this approach and compare it with a smoothing method for the quantile-gradient estimation. Numerical investigation shows that the two approaches are competitive for certain problem instances.

Applied Probability↗
NASA NTRS1996

An Optimal Order Nonnested Mixed Multigrid Method for Generalized Stokes Problems

A multigrid algorithm is developed and analyzed for generalized Stokes problems discretized by various nonnested mixed finite elements within a unified framework. It is abstractly proved by an element-independent analysis that the multigrid algorithm converges with an optimal order if there exists a 'good' prolongation operator. A technique to construct a 'good' prolongation operator for nonnested multilevel finite element spaces is proposed. Its basic idea is to introduce a sequence of auxiliary nested multilevel finite element spaces and define a prolongation operator as a composite operator of two single grid level operators. This makes not only the construction of a prolongation operator much easier (the final explicit forms of such prolongation operators are fairly simple), but the verification of the approximate properties for prolongation operators is also simplified. Finally, as an application, the framework and technique is applied to seven typical nonnested mixed finite elements.

Deng, Qingping↗
NASA NTRS1971

An analysis and comparison of several trajectory optimization methods

The sensitivities of the convergence characteristics of the methods to initially assumed parameters and trial solution, convergence times, computer logic, and storage requirements are discussed. Numerical comparison of the convergence characteristics is made by considering a minimum time, low thrust, Earth-Mars transfer trajectory. A modified quasi-linearization method reduces convergence time by approximately 70% when compared with the generalized Newton-Raphson method and allows the terminal boundary to be specified by a general function of the problem variables. A uniquely specified and easily determined, time dependent weighting matrix for the gradient techniques accelerates the shaping of the optimal control program and improves the convergence characteristics during the terminal iterations. Convergence envelopes, indicating how sensitive the convergence characteristics are to initially assumed parameters, are plotted for the perturbation and quasi-linearization methods. Several iteration schemes are proposed which increase the size of the convergence envelopes and decrease the sensitivity of the method to initially assumed parameters.

Lewallen, J. M.↗
NASA NTRS1995

Parallel Computational Environment for Substructure Optimization

Design optimization of large structural systems can be attempted through a substructure strategy when convergence difficulties are encountered. When this strategy is used, the large structure is divided into several smaller substructures and a subproblem is defined for each substructure. The solution of the large optimization problem can be obtained iteratively through repeated solutions of the modest subproblems. Substructure strategies, in sequential as well as in parallel computational modes on a Cray YMP multiprocessor computer, have been incorporated in the optimization test bed CometBoards. CometBoards is an acronym for Comparative Evaluation Test Bed of Optimization and Analysis Routines for Design of Structures. Three issues, intensive computation, convergence of the iterative process, and analytically superior optimum, were addressed in the implementation of substructure optimization into CometBoards. Coupling between subproblems as well as local and global constraint grouping are essential for convergence of the iterative process. The substructure strategy can produce an analytically superior optimum different from what can be obtained by regular optimization. For the problems solved, substructure optimization in a parallel computational mode made effective use of all assigned processors.

Gendy, Atef S.↗
DOE OSTI2025

Electrical Load Forecasting Over Multihop Smart Metering Networks With Federated Learning

Electric load forecasting is essential for power management and stability in smart grids. This is mainly achieved via advanced metering infrastructure, where smart meters (SMs) record household energy data. Traditional machine learning (ML) methods are often employed for load forecasting, but require data sharing, which raises data privacy concerns. Federated learning (FL) can address this issue by running distributed ML models at local SMs without data exchange. However, current FL-based approaches struggle to achieve efficient load forecasting due to imbalanced data distribution across heterogeneous SMs. Here, this article presents a novel personalized FL (PFL) method for high-quality load forecasting in metering networks. A meta-learning-based strategy is developed to address data heterogeneity at local SMs in the collaborative training of local load forecasting models. Moreover, to minimize the load forecasting delays in our PFL model, we study a new latency optimization problem based on optimal resource allocation at SMs. A theoretical convergence analysis is also conducted to provide insights into FL design for federated load forecasting. Extensive simulations from real-world datasets show that our method outperforms existing approaches regarding better load forecasting and reduced operational latency costs.

Rahman, Ratun [Univ. of Alabama, Huntsville, AL (U↗