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Results for “Bernstein polynomials”

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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27 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Three-dimensional aerodynamic shape optimization using discrete sensitivity analysis

An aerodynamic shape optimization procedure based on discrete sensitivity analysis is extended to treat three-dimensional geometries. The function of sensitivity analysis is to directly couple computational fluid dynamics (CFD) with numerical optimization techniques, which facilitates the construction of efficient direct-design methods. The development of a practical three-dimensional design procedures entails many challenges, such as: (1) the demand for significant efficiency improvements over current design methods; (2) a general and flexible three-dimensional surface representation; and (3) the efficient solution of very large systems of linear algebraic equations. It is demonstrated that each of these challenges is overcome by: (1) employing fully implicit (Newton) methods for the CFD analyses; (2) adopting a Bezier-Bernstein polynomial parameterization of two- and three-dimensional surfaces; and (3) using preconditioned conjugate gradient-like linear system solvers. Whereas each of these extensions independently yields an improvement in computational efficiency, the combined effect of implementing all the extensions simultaneously results in a significant factor of 50 decrease in computational time and a factor of eight reduction in memory over the most efficient design strategies in current use. The new aerodynamic shape optimization procedure is demonstrated in the design of both two- and three-dimensional inviscid aerodynamic problems including a two-dimensional supersonic internal/external nozzle, two-dimensional transonic airfoils (resulting in supercritical shapes), three-dimensional transport wings, and three-dimensional supersonic delta wings. Each design application results in realistic and useful optimized shapes.

Burgreen, Gregory W.↗
NASA NTRS1990

New Bernstein type inequalities for polynomials on ellipses

New and sharp estimates are derived for the growth in the complex plane of polynomials known to have a curved majorant on a given ellipse. These so-called Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. Also presented are some new results for approximation problems of this type.

Freund, Roland↗
NASA NTRS1989

On Bernstein type inequalities and a weighted Chebyshev approximation problem on ellipses

A classical inequality due to Bernstein which estimates the norm of polynomials on any given ellipse in terms of their norm on any smaller ellipse with the same foci is examined. For the uniform and a certain weighted uniform norm, and for the case that the two ellipses are not too close, sharp estimates of this type were derived and the corresponding extremal polynomials were determined. These Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. Some new results were also presented for a weighted approximation problem of this type.

Freund, Roland↗
NASA NTRS2012

Bounding the Failure Probability Range of Polynomial Systems Subject to P-box Uncertainties

This paper proposes a reliability analysis framework for systems subject to multiple design requirements that depend polynomially on the uncertainty. Uncertainty is prescribed by probability boxes, also known as p-boxes, whose distribution functions have free or fixed functional forms. An approach based on the Bernstein expansion of polynomials and optimization is proposed. In particular, we search for the elements of a multi-dimensional p-box that minimize (i.e., the best-case) and maximize (i.e., the worst-case) the probability of inner and outer bounding sets of the failure domain. This technique yields intervals that bound the range of failure probabilities. The offset between this bounding interval and the actual failure probability range can be made arbitrarily tight with additional computational effort.

Crespo, Luis G.↗
NASA NTRS2015

Kodiak: An Implementation Framework for Branch and Bound Algorithms

Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements.

Smith, Andrew P.↗
NASA NTRSDate not supplied

Spatial and Temporal Deconfliction of Trajectories in the Presence of Uncertainties

Demonstration of conflict-free movement for multi-agent teams in challenging scenarios is crucial in developing trust and trustworthiness in an autonomous transport system. Tolerance verification queries are explored as a mechanism to enforce spatial and temporal deconfliction for a cooperating team of Unmanned Aerial Systems (UAS) with prescribed heterogeneous path-following performance. Obstacles in the environment are modelled as set of polyhedra, whereas each vehicle’s trajectory is represented as a sequence of polynomial curves with C2 continuity, expressed in a Bernstein basis. Each vehicle is modelled as a point mass and a safety distance, informed by the geometry of the UAS and the worst-case path-following error. This defines a tube around the trajectories where each UAS is most likely to fly through. In addition, obstacles in the environment have an associated safety buffer around them to account for the uncertainty in their location and geometric description. The tolerance verification queries explored in this paper combine the safety distance information from each UAS and environmental hazard to compute trajectories that are contained within the safe configuration space. Tolerance verification is also compared with other proximity queries to determine the suitability of each method along the different steps of the trajectory generation algorithm. This paper analyzes the fitness and performance of three proximity queries – collision, tolerance verification, and distance computations – between polyhedral and polynomial curves to ensure deconfliction between obstacles and vehicles, but also between polynomial curves to guarantee safe separation among cooperating UAS.

trajectory generation↗
NASA NTRS1992

On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices

The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.

Fischer, Bernd↗
NASA NTRS2011

Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions

This paper proposes an uncertainty analysis framework based on the characterization of the uncertain parameter space. This characterization enables the identification of worst-case uncertainty combinations and the approximation of the failure and safe domains with a high level of accuracy. Because these approximations are comprised of subsets of readily computable probability, they enable the calculation of arbitrarily tight upper and lower bounds to the failure probability. A Bernstein expansion approach is used to size hyper-rectangular subsets while a sum of squares programming approach is used to size quasi-ellipsoidal subsets. These methods are applicable to requirement functions whose functional dependency on the uncertainty is a known polynomial. Some of the most prominent features of the methodology are the substantial desensitization of the calculations from the uncertainty model assumed (i.e., the probability distribution describing the uncertainty) as well as the accommodation for changes in such a model with a practically insignificant amount of computational effort.

Crespo, Luis G.↗