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NASA NTRS2018

An Active Subspace Method for Accelerating Convergence in Delaunay-Based Optimization via Dimension Reduction

Delaunay-based derivative-free optimization, ∆DOGS, is an efﬁcient and provably-convergent global optimization method for the problems which has computationally expensive objection function and the analytical expression for the objective function is not available. ∆-DOGS is a novel optimization scheme in the family of response surface methods (RSMs); however, it suffers from the curse of dimensionality since the computational cost increases dramatically as the number of design parameters increases. As a result, the number of design parameters in ∆-DOGS algorithm is relatively low (n.10). To avoid such problems, this paper proposes a combination of derivative-free optimization, seeking the global minimizer of an expensive and nonconvex objective function f(x) and active subspace method, detecting the directions of the most variability using evaluations of the gradient. The contribution of other directions to the objective function is bounded by a sufﬁciently small constant. This new algorithm iteratively applied Delaunay-based derivative-free optimization to seek the minimizer on the d-dimensional active subspace that has most function variation. Inverse mapping is needed to project data from active subspace to full-model for evaluating function values. This task is overcome by solving an inequality constrained problem that curves the response surface of the objective function. The test results show that this strategy is effective on a handful of optimization problems.

Bewley, Thomas R.↗

NASA NTRS1991

Method and apparatus for configuration control of redundant robots

A method and apparatus to control a robot or manipulator configuration over the entire motion based on augmentation of the manipulator forward kinematics is disclosed. A set of kinematic functions is defined in Cartesian or joint space to reflect the desirable configuration that will be achieved in addition to the specified end-effector motion. The user-defined kinematic functions and the end-effector Cartesian coordinates are combined to form a set of task-related configuration variables as generalized coordinates for the manipulator. A task-based adaptive scheme is then utilized to directly control the configuration variables so as to achieve tracking of some desired reference trajectories throughout the robot motion. This accomplishes the basic task of desired end-effector motion, while utilizing the redundancy to achieve any additional task through the desired time variation of the kinematic functions. The present invention can also be used for optimization of any kinematic objective function, or for satisfaction of a set of kinematic inequality constraints, as in an obstacle avoidance problem. In contrast to pseudoinverse-based methods, the configuration control scheme ensures cyclic motion of the manipulator, which is an essential requirement for repetitive operations. The control law is simple and computationally very fast, and does not require either the complex manipulator dynamic model or the complicated inverse kinematic transformation. The configuration control scheme can alternatively be implemented in joint space.

Seraji, Homayoun↗

DOE OSTI2024

Urban Land Surface Temperature Downscaling in Chicago: Addressing Ethnic Inequality and Gentrification

In this study, we developed a XGBoost-based algorithm to downscale 2 km-resolution land surface temperature (LST) data from the GOES satellite to a finer 70 m resolution, using ancillary variables including NDVI, NDBI, and DEM. This method demonstrated a superior performance over the conventional TsHARP technique, achieving a reduced RMSE of 1.90 °C, compared to 2.51 °C with TsHARP. Our approach utilizes the geostationary GOES satellite data alongside high-resolution ECOSTRESS data, enabling hourly LST downscaling to 70 m—a significant advancement over previous methodologies that typically measure LST only once daily. Applying these high-resolution LST data, we examined the hottest days in Chicago and their correlation with ethnic inequality. Our analysis indicated that Hispanic/Latino communities endure the highest LSTs, with a maximum LST that is 1.5 °C higher in blocks predominantly inhabited by Hispanic/Latino residents compared to those predominantly occupied by White residents. This study highlights the intersection of urban development, ethnic inequality, and environmental inequities, emphasizing the need for targeted urban planning to mitigate these disparities. The enhanced spatial and temporal resolution of our LST data provides deeper insights into diurnal temperature variations, crucial for understanding and addressing the urban heat distribution and its impact on vulnerable communities.

Lee, Jangho (ORCID:0000000289421092)↗

DOE OSTI2024

Clustering and Cliques in Preferential Attachment Random Graphs with Edge Insertion

In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step t, either a new vertex is added with probability f(t), or an edge is added between two existing vertices with probability 1 – f(t). We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that f(t) is a regularly varying function at infinity with index of regular variation –$\gamma$, where $\gamma$ $\in$ [0, 1). Finally, we also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1991

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. As an alternative to the positive real lemma and to the s-domain inequalities, these conditions provide a recursive algorithm for testing positive realness which result in a set of simple algebraic conditions. By relating the positive real property to the associated variational problem, a unified derivation of necessary and sufficient conditions for optimality of both singular and nonsingular problems is derived.

Wang, Q.↗

NASA NTRS1990

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. As an alternative to the positive real lemma and to the s-domain inequalities, these conditions provide a recursive algorithm for testing positive realness which results in a set of simple algebraic conditions. By relating the positive real property to the associated variational problem, a unified derivation of necessary and sufficient conditions for optimality of both singular and nonsingular problems is derived.

Wang, Q.↗

NASA NTRS1993

State-constrained booster trajectory solutions via finite elements and shooting

This paper presents an extension of a FEM formulation based on variational principles. A general formulation for handling internal boundary conditions and discontinuities in the state equations is presented, and the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions which only touch the state constraint and solutions which have a boundary arc of finite length are considered. Suitable shape and test functions are chosen for a FEM discretization. All element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form. The final form of the algebraic equations is then derived. A simple state-constrained problem is solved. Then, for a practical application of the use of the FEM formulation, a launch vehicle subject to a dynamic pressure constraint (a first-order state inequality constraint) is solved. The results presented for the launch-vehicle trajectory have some interesting features, including a touch-point solution.

Bless, Robert R.↗

DOE OSTI2021

Optimization-based algorithms for nonlinear mechanics and frictional contact

An optimization-based strategy for solving nonlinear mechanics problems is proposed. In contrast to typical nonlinear equation solver algorithms that aim to find zeros in the residual force function, we minimize an energy (or energy-like) function to encourage solutions which are locally stable equilibria. These smooth and potentially non-convex objective functions are minimized using a preconditioned conjugate-gradient trust-region algorithm. Contact is formulated as an inequality constrained minimization problem, and is solved with an augmented Lagrangian algorithm. Friction is included in the approach via a regularized quasi-potential energy, and other dissipative behavior is included through the use of variational constitutive updates. Finally, to accelerate convergence rates for the Lagrange multipliers, we propose a novel multiplier update algorithm utilizing the Fischer-Burmeister function, and demonstrate super-linear solver convergence for some applications.

42 ENGINEERING↗

NASA NTRS1979

The near-wake structure of the Atmosphere Explorer C /AE-C/ satellite - A parametric investigation

Measurements of ion current, electron temperature, and values of space potential obtained from the cylindrical electrostatic probe on board the Atmosphere Explorer C (AE-C) satellite were used to examine, in a parametric manner, the angular distribution of charge around the satellite. Interest is focused on nighttime equatorial data in the altitude range 275-620 km, which yields a wide range for the parameter R sub D (the radius of the satellite divided by the ambient value of the Debye lengths), including R sub D greater than 100, which is of practical significance to large space platforms. The variations of normalized ion current in the wake zone of the AE-C satellite appear to display an exponential dependence on R sub D for 'constant' values of other relevant parameters. The angular variations of electron temperature and space potential in the close vicinity of the satellite's surface were examined and compared with results from the Explorer 31 satellite. The variation of the ratio of measured to computed space potential with electron temperature was examined using data from both the AE-C and Explorer 31 satellites. It was found that the ratio is greater than unity. Possible causes for the above inequality are discussed.

Samir, U.↗

NASA NTRS1988

On the convergence of difference approximations to scalar conservation laws

A unified treatment is given for time-explicit, two-level, second-order-resolution (SOR), total-variation-diminishing (TVD) approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced to obtain results in terms of the latter. The existence of a cell entropy inequality is discussed, and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first-order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Osher, Stanley↗

DOE OSTI2022

Inequality in the availability of residential air conditioning across 115 US metropolitan areas

Continued climate change is increasing the frequency, severity, and duration of populations’ high temperature exposures. Indoor cooling is a key adaptation, especially in urban areas, where heat extremes are intensified—the urban heat island effect (UHI)—making residential air conditioning (AC) availability critical to protecting human health. In the United States, the differences in residential AC prevalence from one metropolitan area to another is well understood, but its intra-urban variation is poorly characterized, obscuring neighborhood-scale variability in populations’ heat vulnerability and adaptive capacity. We address this gap by constructing empirically derived probabilities of residential AC for 45,995 census tracts across 115 metropolitan areas. Within cities, AC is unequally distributed, with census tracts in the urban “core” exhibiting systematically lower prevalence than their suburban counterparts. Moreover, this disparity correlates strongly with multiple indicators of social vulnerability and summer daytime surface UHI intensity, highlighting the challenges that vulnerable urban populations face in adapting to climate-change driven heat stress amplification.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

McCormick envelopes in mixed-integer PDE-constrained optimization

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

Approximations and Expansions↗

NASA NTRS1984

Global variation of the para hydrogen fraction in Jupiter's atmosphere and implications for dynamics on the outer planets

A detailed analysis of the Voyager infrared spectrometer measurements on Jupiter's atmosphere is presented, and possible implications of para hydrogen disequilibrium for the energetics and dynamics of that atmosphere are examined. The method of data analysis is described, and results for the large scale latitude variation of the para hydrogen fraction are presented. The Jovian results show pronounced latitude variation, and are compared with other parameters including wind fields, thermal structure, and various indicators of atmospheric clouds. The problem of equilibration rate is reexamined, and it is concluded that on Jupiter the equilibration time is longer than the radiative time constant at the level of emission to space, but that this inequality reverses at greater depths. A model for the interaction of fluid motions with the ortho-para conversion process is presented, and a consistent mixing length theory for the reacting ortho-para mixture is developed. Several implications of the Jovian data for atmospheric energetics and stability on the outer planets are presented.

Conrath, B. J.↗

NASA NTRS1971

The influence of the great inequality on the secular disturbing function of the planetary system.

This paper derives the contribution by the great inequality to the secular disturbing function of the principal planets. Andoyer's expansion of the planetary disturbing function and von Zeipel's method of eliminating the periodic terms is employed; thereby, the corrected secular disturbing function for the planetary system is derived. The conclusion is drawn that the canonicity of the equations for the secular variation of the heliocentric elements can be preserved if there be retained, in the secular disturbing function, terms only of the second and fourth order relative to the eccentricity and inclinations. The Krylov-Bogoliubov method is suggested for eliminating periodic terms, if it is desired to include the secular perturbations of the fifth and higher order in the heliocentric elements. The additional part of the secular disturbing function derived in this paper can be included in existing theories of the secular effects of principal planets.

Musen, P.↗

NASA NTRS1972

N-burn optimal analytic trajectories.

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

Jezewski, D. J.↗

NASA NTRS1978

Expansion and broadening of coronal loop transients - A theoretical explanation

Consequences are examined of the assumption that an observed coronal loop transient is a twisted rope of magnetic-field lines expanding and broadening in the background coronal plasma and magnetic field. It is shown that the expansion can be accounted for by the azimuthal component of the field; the observed broadening of the loop as it moves outward can be accounted for by the longitudinal component of the field. In order to have a net outward force and at the same time avoid a classical pinch (sausage) instability, the two components of the field must satisfy a certain inequality. It is predicted that, as the loop rises, the width (h) of its top portion should vary proportionally with distance (R) from the sun's center. This is in good agreement with measurements that show h is proportional to the 0.8 power of R. The prediction that the radius of curvature of the top portion of the loop should be proportional to R differs from the measured variation. The difference could be accounted for by a drag due to the background coronal field that flattens the loop's top.

Mouschovias, T. CH.↗

DOE OSTI2020

Quantum maximin surfaces

We formulate a quantum generalization of maximin surfaces and show that a quantum maximin surface is identical to the minimal quantum extremal surface, introduced in the EW prescription. We discuss various subtleties and complications associated to a maximinimization of the bulk von Neumann entropy due to corners and unboundedness and present arguments that nonetheless a maximinimization of the UV-finite generalized entropy should be well-defined. We give the first general proof that the EW prescription satisfies entanglement wedge nesting and the strong subadditivity inequality. In addition, we apply the quantum maximin technology to prove that recently proposed generalizations of the EW prescription to nonholographic subsystems (including the so-called “quantum extremal islands”) also satisfy entanglement wedge nesting and strong subadditivity. Our results hold in the regime where backreaction of bulk quantum fields can be treated perturbatively in G N h, but we emphasize that they are valid even when gradients of the bulk entropy are of the same order as variations in the area, a regime recently investigated in new models of black hole evaporation in AdS/CFT.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1989

Flutter suppression using eigenspace freedoms to meet requirements

A constrained optimization methodology has been developed which allows specific use of eigensystem freedoms to meet design requirements. A subset of the available eigenvector freedoms was employed. The eigenvector freedoms associated with a particular closed-loop eigenvalue are coefficients of basis vectors which span the subspace in which that closed-loop vector must lie. Design requirements are included as a vector of inequality constraints. The procedure was successfully applied to develop an unscheduled controller which stabilizes symmetric flutter of an aeroelastic vehicle to a dynamic pressure 44 percent above the open-loop flutter point. The design process proceeded from full-state feedback to the inclusion of a full-order observer to the selection of an eighth-order controller which preserved the full-state sensitivity characteristics. Only a subset of the design freedoms was utilized (i.e., assuming full-state feedback only four out of 26 eigenvectors were used, and no variations were made in the closed-loop eigenvalues). Utilization of additional eigensystem freedoms could further improve the controller.

Adams, William M., Jr.↗