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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 NTRS1995

Properties of a center/surround retinex. Part 2: Surround design

The last version of Edwin Land's retinex model for human vision's lightness and color constancy has been implemented. Previous research has established the mathematical foundations of Land's retinex but has not examined specific design issues and their effects on the properties of the retinex operation. We have sought to define a practical implementation of the retinex without particular concern for its validity as a model for human lightness and color perception. Here we describe issues involved in designing the surround function. We find that there is a trade-off between rendition and dynamic range compression that is governed by the surround space constant. Various functional forms for the retinex surround are evaluated and a Gaussian form is found to perform better than the inverse square suggested by Land. Preliminary testing led to the design of a Gaussian surround with a space constant of 80 pixels as a reasonable compromise between dynamic range compression and rendition.

Jobson, Daniel J.↗
NASA NTRS1999

Structural Reliability Analysis and Optimization: Use of Approximations

This report is intended for the demonstration of function approximation concepts and their applicability in reliability analysis and design. Particularly, approximations in the calculation of the safety index, failure probability and structural optimization (modification of design variables) are developed. With this scope in mind, extensive details on probability theory are avoided. Definitions relevant to the stated objectives have been taken from standard text books. The idea of function approximations is to minimize the repetitive use of computationally intensive calculations by replacing them with simpler closed-form equations, which could be nonlinear. Typically, the approximations provide good accuracy around the points where they are constructed, and they need to be periodically updated to extend their utility. There are approximations in calculating the failure probability of a limit state function. The first one, which is most commonly discussed, is how the limit state is approximated at the design point. Most of the time this could be a first-order Taylor series expansion, also known as the First Order Reliability Method (FORM), or a second-order Taylor series expansion (paraboloid), also known as the Second Order Reliability Method (SORM). From the computational procedure point of view, this step comes after the design point identification; however, the order of approximation for the probability of failure calculation is discussed first, and it is denoted by either FORM or SORM. The other approximation of interest is how the design point, or the most probable failure point (MPP), is identified. For iteratively finding this point, again the limit state is approximated. The accuracy and efficiency of the approximations make the search process quite practical for analysis intensive approaches such as the finite element methods; therefore, the crux of this research is to develop excellent approximations for MPP identification and also different approximations including the higher-order reliability methods (HORM) for representing the failure surface. This report is divided into several parts to emphasize different segments of the structural reliability analysis and design. Broadly, it consists of mathematical foundations, methods and applications. Chapter I discusses the fundamental definitions of the probability theory, which are mostly available in standard text books. Probability density function descriptions relevant to this work are addressed. In Chapter 2, the concept and utility of function approximation are discussed for a general application in engineering analysis. Various forms of function representations and the latest developments in nonlinear adaptive approximations are presented with comparison studies. Research work accomplished in reliability analysis is presented in Chapter 3. First, the definition of safety index and most probable point of failure are introduced. Efficient ways of computing the safety index with a fewer number of iterations is emphasized. In chapter 4, the probability of failure prediction is presented using first-order, second-order and higher-order methods. System reliability methods are discussed in chapter 5. Chapter 6 presents optimization techniques for the modification and redistribution of structural sizes for improving the structural reliability. The report also contains several appendices on probability parameters.

Grandhi, Ramana V.↗
NASA NTRS2001

Implementation of Altimetry Data in the GIPSY POD Software Package

Altimetry data has been used extensively to acquire data about characteristics of the Earth, the Moon, and Mars. More recently, the idea of using altimetry for orbit determination has also been explored. This report discusses modifications to JPL's GIPSY/OASIS II software to include altimetry data as an observation type for precise orbit determination. The mathematical foundation of using altimetry for the purpose of orbit determination is presented, along with results.

Stauch, Jason R.↗
NASA NTRS1995

Development and Application of the Collaborative Optimization Architecture in a Multidisciplinary Design Environment

Collaborative optimization is a design architecture applicable in any multidisciplinary analysis environment but specifically intended for large-scale distributed analysis applications. In this approach, a complex problem is hierarchically de- composed along disciplinary boundaries into a number of subproblems which are brought into multidisciplinary agreement by a system-level coordination process. When applied to problems in a multidisciplinary design environment, this scheme has several advantages over traditional solution strategies. These advantageous features include reducing the amount of information transferred between disciplines, the removal of large iteration-loops, allowing the use of different subspace optimizers among the various analysis groups, an analysis framework which is easily parallelized and can operate on heterogenous equipment, and a structural framework that is well-suited for conventional disciplinary organizations. In this article, the collaborative architecture is developed and its mathematical foundation is presented. An example application is also presented which highlights the potential of this method for use in large-scale design applications.

Braun, R. D.↗
NASA NTRS2005

Coevolutionary Free Lunches

Recent work on the mathematical foundations of optimization has begun to uncover its rich structure. In particular, the "No Free Lunch" (NFL) theorems state that any two algorithms are equivalent when their performance is averaged across all possible problems. This highlights the need for exploiting problem-specific knowledge to achieve better than random performance. In this paper we present a general framework covering more search scenarios. In addition to the optimization scenarios addressed in the NFL results, this framework covers multi-armed bandit problems and evolution of multiple co-evolving players. As a particular instance of the latter, it covers "self-play" problems. In these problems the set of players work together to produce a champion, who then engages one or more antagonists in a subsequent multi-player game. In contrast to the traditional optimization case where the NFL results hold, we show that in self-play there are free lunches: in coevolution some algorithms have better performance than other algorithms, averaged across all possible problems. We consider the implications of these results to biology where there is no champion.

Wolpert, David H.↗
NASA NTRS2011

Grazing Incidence Wavefront Sensing and Verification of X-Ray Optics Performance

Evaluation of interferometrically measured mirror metrology data and characterization of a telescope wavefront can be powerful tools in understanding of image characteristics of an x-ray optical system. In the development of soft x-ray telescope for the International X-Ray Observatory (IXO), we have developed new approaches to support the telescope development process. Interferometrically measuring the optical components over all relevant spatial frequencies can be used to evaluate and predict the performance of an x-ray telescope. Typically, the mirrors are measured using a mount that minimizes the mount and gravity induced errors. In the assembly and mounting process the shape of the mirror segments can dramatically change. We have developed wavefront sensing techniques suitable for the x-ray optical components to aid us in the characterization and evaluation of these changes. Hartmann sensing of a telescope and its components is a simple method that can be used to evaluate low order mirror surface errors and alignment errors. Phase retrieval techniques can also be used to assess and estimate the low order axial errors of the primary and secondary mirror segments. In this paper we describe the mathematical foundation of our Hartmann and phase retrieval sensing techniques. We show how these techniques can be used in the evaluation and performance prediction process of x-ray telescopes.

Saha, Timo T.↗
NASA NTRS2006

Analyzing Aeroelastic Stability of a Tilt-Rotor Aircraft

Proprotor Aeroelastic Stability Analysis, now at version 4.5 (PASTA 4.5), is a FORTRAN computer program for analyzing the aeroelastic stability of a tiltrotor aircraft in the airplane mode of flight. The program employs a 10-degree- of-freedom (DOF), discrete-coordinate, linear mathematical model of a rotor with three or more blades and its drive system coupled to a 10-DOF modal model of an airframe. The user can select which DOFs are included in the analysis. Quasi-steady strip-theory aerodynamics is employed for the aerodynamic loads on the blades, a quasi-steady representation is employed for the aerodynamic loads acting on the vibrational modes of the airframe, and a stability-derivative approach is used for the aerodynamics associated with the rigid-body DOFs of the airframe. Blade parameters that vary with the blade collective pitch can be obtained by interpolation from a user-defined table. Stability is determined by examining the eigenvalues that are obtained by solving the coupled equations of motions as a matrix eigenvalue problem. Notwithstanding the relative simplicity of its mathematical foundation, PASTA 4.5 and its predecessors have played key roles in a number of engineering investigations over the years.

Kvaternil, Raymond G.↗
NASA NTRS2017

Depth from Stereo and Polarization of Opaque Non-Lambertian Surfaces

This paper presents a new approach to depth from stereo in scenes with non-Lambertian surfaces. While these surfaces usually prove to be hard to incorporate in photo-consistency energy minimization, we retrieve additional information about their surface normals from applying a polarization-based approach that computes the degree and angle of polarization at each pixel. Therefore, we set up a stereo camera rig that captures four different rotations of a linear polarizer simultaneously for each camera. While most existing polarization-based approaches to shape reconstruction require a setup with known lighting, our approach uses arbitrary unpolarized illumination. We describe a regularization-based stereo approach that constrains specular surface normals in the scene to lie in the plane of their observed angle of polarization. We derive the mathematical foundations of this approach and present results on virtually generated stereo and real-world stereo imagery.

Berger, Kai↗
NASA NTRS2022

A Survey of Mathematical Structures for Lunar Networks

To sustain the current and increasing accessibility of space, a scalable communications infrastructure (i.e. the Solar System Internet, SSI) is necessary. The goal of this paper is to begin the discovery of the fundamental underlying mathematical structure of space networks to help the research community harness these structures for algorithm development and optimization. To ensure the applicability of the research, the approaches are considered through the lens of simulated scenarios inspired by the Artemis Back-to-the-Moon mission set for 2024. We note that any approach to an SSI must fit under the umbrella of Delay Tolerant Networking (DTN), due to celestial mobility, high link latencies, high variance in link latencies, disconnections, lack of end-to-end paths, and so on. These difficulties are exacerbated by the fact that the underlying structure of a space network is a time-evolving network and may experience multiple discontinuities in its topology. In this paper we propose several novel approaches to a mathematical foundation for Delay Tolerant Networking Theory that fall outside the traditional scope of temporal network theory. These techniques include methods from Topological Data Analysis, Dynamic Graph Analysis, Applied Algebraic Geometry, Probability Theory, and Game Theory. Some of these methods include tools adapted to the study of dynamic metric spaces, such as zigzag persistent homology and their higher parameter analogs. We find that several of these methods target desired engineering outcomes such as discovery and automatic sub-netting. While each approach is theoretical, they are also algorithmic in nature and offer immediate practical applications. The paper concludes with comparisons of the various methods along with suggestions for future work.

Delay tolerant networking↗
DOE OSTI2023

Mathematical nuances of Gaussian process-driven autonomous experimentation

Abstract The fields of machine learning (ML) and artificial intelligence (AI) have transformed almost every aspect of science and engineering. The excitement for AI/ML methods is in large part due to their perceived novelty, as compared to traditional methods of statistics, computation, and applied mathematics. But clearly, all methods in ML have their foundations in mathematical theories, such as function approximation, uncertainty quantification, and function optimization. Autonomous experimentation is no exception; it is often formulated as a chain of off-the-shelf tools, organized in a closed loop, without emphasis on the intricacies of each algorithm involved. The uncomfortable truth is that the success of any ML endeavor, and this includes autonomous experimentation, strongly depends on the sophistication of the underlying mathematical methods and software that have to allow for enough flexibility to consider functions that are in agreement with particular physical theories. We have observed that standard off-the-shelf tools, used by many in the applied ML community, often hide the underlying complexities and therefore perform poorly. In this paper, we want to give a perspective on the intricate connections between mathematics and ML, with a focus on Gaussian process-driven autonomous experimentation. Although the Gaussian process is a powerful mathematical concept, it has to be implemented and customized correctly for optimal performance. We present several simple toy problems to explore these nuances and highlight the importance of mathematical and statistical rigor in autonomous experimentation and ML. One key takeaway is that ML is not, as many had hoped, a set of agnostic plug-and-play solvers for everyday scientific problems, but instead needs expertise and mastery to be applied successfully. Graphical abstract

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Yet Another Discriminant Analysis (YADA): A Probabilistic Model for Machine Learning Applications

This paper presents a probabilistic model for various machine learning (ML) applications. While deep learning (DL) has produced state-of-the-art results in many domains, DL models are complex and over-parameterized, which leads to high uncertainty about what the model has learned, as well as its decision process. Further, DL models are not probabilistic, making reasoning about their output challenging. In contrast, the proposed model, referred to as Yet Another Discriminate Analysis(YADA), is less complex than other methods, is based on a mathematically rigorous foundation, and can be utilized for a wide variety of ML tasks including classification, explainability, and uncertainty quantification. YADA is thus competitive in most cases with many state-of-the-art DL models. Ideally, a probabilistic model would represent the full joint probability distribution of its features, but doing so is often computationally expensive and intractable. Hence, many probabilistic models assume that the features are either normally distributed, mutually independent, or both, which can severely limit their performance. YADA is an intermediate model that (1) captures the marginal distributions of each variable and the pairwise correlations between variables and (2) explicitly maps features to the space of multivariate Gaussian variables. Numerous mathematical properties of the YADA model can be derived, thereby improving the theoretic underpinnings of ML. Validation of the model can be statistically verified on new or held-out data using native properties of YADA. However, there are some engineering and practical challenges that we enumerate to make YADA more useful.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Quasi-periodic travelling gravity–capillary waves

We present a numerical study of spatially quasi-periodic travelling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wavenumbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the travelling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least-squares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasi-periodic travelling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Nonlinear causality and strong hyperbolicity of baryon-rich Israel-Stewart hydrodynamics

We present the first set of fully-nonlinear, necessary and sufficient conditions guaranteeing causal evolution of the initial data for the Israel-Stewart hydrodynamic equations with shear and bulk viscosity coupled to a nonzero baryon current. These constraints not only provide nonlinear causality: they also (a) guarantee the existence of a locally well-posed evolution of the initial data (they enforce strong hyperbolicity) when excluding the endpoints of the bounds, (b) arise from purely algebraic constraints that make no underlying symmetry assumptions on the degrees of freedom and (c) propagate the relevant symmetries of the degrees of freedom over the entire evolution of the problem. Our work enforces a mathematically rigorous foundation for future studies of viscous relativistic hydrodynamics with baryon-rich matter including neutron star mergers and heavy-ion collisions.

Cordeiro, Ian [University of Illinois Urbana-Champ↗
NASA NTRS2006

Adjoint-Based Algorithms for Adaptation and Design Optimizations on Unstructured Grids

Schemes based on discrete adjoint algorithms present several exciting opportunities for significantly advancing the current state of the art in computational fluid dynamics. Such methods provide an extremely efficient means for obtaining discretely consistent sensitivity information for hundreds of design variables, opening the door to rigorous, automated design optimization of complex aerospace configuration using the Navier-Stokes equation. Moreover, the discrete adjoint formulation provides a mathematically rigorous foundation for mesh adaptation and systematic reduction of spatial discretization error. Error estimates are also an inherent by-product of an adjoint-based approach, valuable information that is virtually non-existent in today's large-scale CFD simulations. An overview of the adjoint-based algorithm work at NASA Langley Research Center is presented, with examples demonstrating the potential impact on complex computational problems related to design optimization as well as mesh adaptation.

Nielsen, Eric J.↗
NASA NTRS2006

The Curious Events Leading to the Theory of Shock Waves

We review the history of the development of the modern theory of shock waves. Several attempts at an early-theory quickly collapsed for lack of foundations in mathematics and thermodynamics. It is not until the works of Rankine and later Hugoniot that a full theory is established. Rankine is the first to show that within the shock a non-adiabatic process must occur. Hugoniot showed that in the absence of viscosity and heat conduction conservation of energy implies conservation of entropy in smooth regions and a jump in entropy across a shock. Even after the theory is fully developed, old notions continue to pervade the literature well into the early part of the 20th Century.

Salas, Manuel D.↗
NASA NTRS2017

An Efficient Universal Trajectory Language

The Efficient Universal Trajectory Language (EUTL) is a language for specifying and representing trajectories for Air Traffic Management (ATM) concepts such as Trajectory-Based Operations (TBO). In these concepts, the communication of a trajectory between an aircraft and ground automation is fundamental. Historically, this trajectory exchange has not been done, leading to trajectory definitions that have been centered around particular application domains and, therefore, are not well suited for TBO applications. The EUTL trajectory language has been defined in the Prototype Verification System (PVS) formal specification language, which provides an operational semantics for the EUTL language. The hope is that EUTL will provide a foundation for mathematically verified algorithms that manipulate trajectories. Additionally, the EUTL language provides well-defined methods to unambiguously determine position and velocity information between the reported trajectory points. In this paper, we present the EUTL trajectory language in mathematical detail.

Hagen, George E.↗