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NASA NTRS1996

Implicit Extrapolation Methods for Variable Coefficient Problems

Implicit extrapolation methods for the solution of partial differential equations are based on applying the extrapolation principle indirectly. Multigrid tau-extrapolation is a special case of this idea. In the context of multilevel finite element methods, an algorithm of this type can be used to raise the approximation order, even when the meshes are nonuniform or locally refined. Here previous results are generalized to the variable coefficient case and thus become applicable for nonlinear problems. The implicit extrapolation multigrid algorithm converges to the solution of a higher order finite element system. This is obtained without explicitly constructing higher order stiffness matrices but by applying extrapolation in a natural form within the algorithm. The algorithm requires only a small change of a basic low order multigrid method.

Jung, M.↗

NASA NTRS1990

Improving stability and strength characteristics of framed structures with nonlinear behavior

In this paper an optimal design procedure is introduced to improve the overall performance of nonlinear framed structures. The design methodology presented here is a multiple-objective optimization procedure whose objective functions involve the buckling eigenvalues and eigenvectors of the structure. A constant volume with bounds on the design variables is used in conjunction with an optimality criterion approach. The method provides a general tool for solving complex design problems and generally leads to structures with better limit strength and stability. Many algorithms have been developed to improve the limit strength of structures. In most applications geometrically linear analysis is employed with the consequence that overall strength of the design is overestimated. Directly optimizing the limit load of the structure would require a full nonlinear analysis at each iteration which would be prohibitively expensive. The objective of this paper is to develop an algorithm that can improve the limit-load of geometrically nonlinear framed structures while avoiding the nonlinear analysis. One of the novelties of the new design methodology is its ability to efficiently model and design structures under multiple loading conditions. These loading conditions can be different factored loads or any kind of loads that can be applied to the structure simultaneously or independently. Attention is focused on optimal design of space framed structures. Three-dimensional design problems are more complicated to carry out, but they yield insight into real behavior of the structure and can help avoiding some of the problems that might appear in planar design procedure such as the need for out-of-plane buckling constraint. Although researchers in the field of structural engineering generally agree that optimum design of three-dimension building frames especially in the seismic regions would be beneficial, methods have been slow to emerge. Most of the research in this area has dealt with the optimization of truss and plane frame structures.

Pezeshk, Shahram↗

NASA NTRS1998

Attitude Estimation Signal Processing: A First Report on Possible Algorithms and Their Utility

In this brief effort, time has been of the essence. The data had to be acquired from APL/Lincoln Labs, stored, and sorted out to obtain the pertinent streams. This has been a significant part of this effort and hardware and software problems have been addressed with the appropriate solutions to accomplish this part of the task. Passed this, some basic and important algorithms are utilized to improve the performance of the attitude estimation systems. These algorithms are an essential part of the signal processing for the attitude estimation problem as they are utilized to reduce the amount of the additive/multiplicative noise that in general may or may not change its structure and probability density function, pdf, in time. These algorithms are not currently utilized in the processing of the data, at least, we are not aware of their use in this attitude estimation problem. Some of these algorithms, like the variable thresholding, are new conjectures, but one would expect that someone somewhere must have utilized this kind of scheme before. The variable thresholding idea is a straightforward scheme to use in case of a slowly varying pdf, or statistical moments of the unwanted random process. The algorithms here are kept simple but yet effective for processing the data and removing the unwanted noise. For the most part, these algorithms can be arranged so that their consecutive and orderly execution would complement the preceding algorithm and improve the overall performance of the signal processing chain.

Riasati, Vahid R.↗

DOE OSTI2021

A parallel evolutionary multiple-try metropolis Markov chain Monte Carlo algorithm for sampling spatial partitions

We develop an Evolutionary Markov Chain Monte Carlo (EMCMC) algorithm for sampling spatial partitions that lie within a large, complex, and constrained spatial state space. Our algorithm combines the advantages of evolutionary algorithms (EAs) as optimization heuristics for state space traversal and the theoretical convergence properties of Markov Chain Monte Carlo algorithms for sampling from unknown distributions. Local optimality information that is identified via a directed search by our optimization heuristic is used to adaptively update a Markov chain in a promising direction within the framework of a Multiple-Try Metropolis Markov Chain model that incorporates a generalized Metropolis-Hastings ratio. We further expand the reach of our EMCMC algorithm by harnessing the computational power afforded by massively parallel computing architecture through the integration of a parallel EA framework that guides Markov chains running in parallel.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1997

Computation of Reacting Flows in Combustion Processes

The main objective of this research was to develop an efficient three-dimensional computer code for chemically reacting flows. The main computer code developed is ALLSPD-3D. The ALLSPD-3D computer program is developed for the calculation of three-dimensional, chemically reacting flows with sprays. The ALL-SPD code employs a coupled, strongly implicit solution procedure for turbulent spray combustion flows. A stochastic droplet model and an efficient method for treatment of the spray source terms in the gas-phase equations are used to calculate the evaporating liquid sprays. The chemistry treatment in the code is general enough that an arbitrary number of reaction and species can be defined by the users. Also, it is written in generalized curvilinear coordinates with both multi-block and flexible internal blockage capabilities to handle complex geometries. In addition, for general industrial combustion applications, the code provides both dilution and transpiration cooling capabilities. The ALLSPD algorithm, which employs the preconditioning and eigenvalue rescaling techniques, is capable of providing efficient solution for flows with a wide range of Mach numbers. Although written for three-dimensional flows in general, the code can be used for two-dimensional and axisymmetric flow computations as well. The code is written in such a way that it can be run in various computer platforms (supercomputers, workstations and parallel processors) and the GUI (Graphical User Interface) should provide a user-friendly tool in setting up and running the code.

Keith, Theo G., Jr.↗

NASA NTRS2013

Surface Ocean pCO2 Seasonality and Sea-Air CO2 Flux Estimates for the North American East Coast

Underway and in situ observations of surface ocean pCO2, combined with satellite data, were used to develop pCO2 regional algorithms to analyze the seasonal and interannual variability of surface ocean pCO2 and sea-air CO2 flux for five physically and biologically distinct regions of the eastern North American continental shelf: the South Atlantic Bight (SAB), the Mid-Atlantic Bight (MAB), the Gulf of Maine (GoM), Nantucket Shoals and Georges Bank (NS+GB), and the Scotian Shelf (SS). Temperature and dissolved inorganic carbon variability are the most influential factors driving the seasonality of pCO2. Estimates of the sea-air CO2 flux were derived from the available pCO2 data, as well as from the pCO2 reconstructed by the algorithm. Two different gas exchange parameterizations were used. The SS, GB+NS, MAB, and SAB regions are net sinks of atmospheric CO2 while the GoM is a weak source. The estimates vary depending on the use of surface ocean pCO2 from the data or algorithm, as well as with the use of the two different gas exchange parameterizations. Most of the regional estimates are in general agreement with previous studies when the range of uncertainty and interannual variability are taken into account. According to the algorithm, the average annual uptake of atmospheric CO2 by eastern North American continental shelf waters is found to be between 3.4 and 5.4 Tg C/yr (areal average of 0.7 to 1.0 mol CO2 /sq m/yr) over the period 2003-2010.

interannual variability↗

DOE OSTI2021

Identifying and tracking bubbles and drops in simulations: A toolbox for obtaining sizes, lineages, and breakup and coalescence statistics

Knowledge of bubble and drop size distributions in two-phase flows is important for characterizing a wide range of phenomena, including combustor ignition, sonar communication, and cloud formation. The physical mechanisms driving the background flow also drive the time evolution of these distributions. Accurate and robust identification and tracking algorithms for the dispersed phase are necessary to reliably measure this evolution and thereby quantify the underlying mechanisms in interface-resolving flow simulations. The identification of individual bubbles and drops traditionally relies on an algorithm used to identify connected regions. This traditional algorithm can be sensitive to the presence of spurious structures. A cost-effective refinement is proposed to maximize volume accuracy while minimizing the identification of spurious bubbles and drops. An accurate identification scheme is crucial for distinguishing bubble and drop pairs with large size ratios. The identified bubbles and drops need to be tracked in time to obtain breakup and coalescence statistics that characterize the evolution of the size distribution, including breakup and coalescence frequencies, and the probability distributions of parent and child bubble and drop sizes. An algorithm based on mass conservation is proposed to construct bubble and drop lineages using simulation snapshots that are not necessarily from consecutive time steps. These lineages are then used to detect breakup and coalescence events, and obtain the desired statistics. Accurate identification of large-size-ratio bubble and drop pairs enables accurate detection of breakup and coalescence events over a large size range. Accurate detection of successive breakup and coalescence events requires that the snapshot interval be an order of magnitude smaller than the characteristic breakup and coalescence times to capture these successive events while minimizing the identification of repeated confounding events. Together, these algorithms serve as a toolbox for detailed analysis of two-phase simulations, and enable insights into the mechanisms behind bubble and drop formation and evolution in flows of practical importance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1995

Control Algorithms For Kinematically Redundant Manipulators

Report presents improved algorithms for controlling kinematically redundant robotic manipulators, producing unique configuration for each end-effector location. Furthermore, they generally produce configurations in which no interference between robot boom and payload or base structures. Report presents comparison between various other robot-kinematic algorithms.

Wegner, David R.↗

NASA NTRS1981

Two-Dimensional Inlet Simulation Using a Diagonal Implicit Algorithm

A modification of an implicit approximate-factorization finite-difference algorithm applied to the two-dimensional Euler and Navier-Stokes equations in general curvilinear coordinates is presented for supersonic freestream flow about and through inlets. The modification transforms the coupled system of equations Into an uncoupled diagonal form which requires less computation work. For steady-state applications the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. Solutions are given for inviscid and laminar flow about a two-dimensional wedge inlet configuration. Comparisons are made between computed results and exact theory.

Chaussee, D.S.↗

NASA NTRS2014

Learning Extended Finite State Machines

We present an active learning algorithm for inferring extended finite state machines (EFSM)s, combining data flow and control behavior. Key to our learning technique is a novel learning model based on so-called tree queries. The learning algorithm uses the tree queries to infer symbolic data constraints on parameters, e.g., sequence numbers, time stamps, identifiers, or even simple arithmetic. We describe sufficient conditions for the properties that the symbolic constraints provided by a tree query in general must have to be usable in our learning model. We have evaluated our algorithm in a black-box scenario, where tree queries are realized through (black-box) testing. Our case studies include connection establishment in TCP and a priority queue from the Java Class Library.

NASA NTRS2002

Navigation Algorithms for the SeaWiFS Mission

The navigation algorithms for the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) were designed to meet the requirement of 1-pixel accuracy-a standard deviation (sigma) of 2. The objective has been to extract the best possible accuracy from the spacecraft telemetry and avoid the need for costly manual renavigation or geometric rectification. The requirement is addressed by postprocessing of both the Global Positioning System (GPS) receiver and Attitude Control System (ACS) data in the spacecraft telemetry stream. The navigation algorithms described are separated into four areas: orbit processing, attitude sensor processing, attitude determination, and final navigation processing. There has been substantial modification during the mission of the attitude determination and attitude sensor processing algorithms. For the former, the basic approach was completely changed during the first year of the mission, from a single-frame deterministic method to a Kalman smoother. This was done for several reasons: a) to improve the overall accuracy of the attitude determination, particularly near the sub-solar point; b) to reduce discontinuities; c) to support the single-ACS-string spacecraft operation that was started after the first mission year, which causes gaps in attitude sensor coverage; and d) to handle data quality problems (which became evident after launch) in the direct-broadcast data. The changes to the attitude sensor processing algorithms primarily involved the development of a model for the Earth horizon height, also needed for single-string operation; the incorporation of improved sensor calibration data; and improved data quality checking and smoothing to handle the data quality issues. The attitude sensor alignments have also been revised multiple times, generally in conjunction with the other changes. The orbit and final navigation processing algorithms have remained largely unchanged during the mission, aside from refinements to data quality checking. Although further improvements are certainly possible, future evolution of the algorithms is expected to be limited to refinements of the methods presented here, and no substantial changes are anticipated.

Hooker, Stanford B.↗

NASA NTRS1993

Irrelevance Reasoning in Knowledge Based Systems

This dissertation considers the problem of reasoning about irrelevance of knowledge in a principled and efficient manner. Specifically, it is concerned with two key problems: (1) developing algorithms for automatically deciding what parts of a knowledge base are irrelevant to a query and (2) the utility of relevance reasoning. The dissertation describes a novel tool, the query-tree, for reasoning about irrelevance. Based on the query-tree, we develop several algorithms for deciding what formulas are irrelevant to a query. Our general framework sheds new light on the problem of detecting independence of queries from updates. We present new results that significantly extend previous work in this area. The framework also provides a setting in which to investigate the connection between the notion of irrelevance and the creation of abstractions. We propose a new approach to research on reasoning with abstractions, in which we investigate the properties of an abstraction by considering the irrelevance claims on which it is based. We demonstrate the potential of the approach for the cases of abstraction of predicates and projection of predicate arguments. Finally, we describe an application of relevance reasoning to the domain of modeling physical devices.

INFERENCE↗

NASA NTRS1987

Spectral multigrid methods for the solution of homogeneous turbulence problems

New three-dimensional spectral multigrid algorithms are analyzed and implemented to solve the variable coefficient Helmholtz equation. Periodicity is assumed in all three directions which leads to a Fourier collocation representation. Convergence rates are theoretically predicted and confirmed through numerical tests. Residual averaging results in a spectral radius of 0.2 for the variable coefficient Poisson equation. In general, non-stationary Richardson must be used for the Helmholtz equation. The algorithms developed are applied to the large-eddy simulation of incompressible isotropic turbulence.

Erlebacher, G.↗

NASA NTRS1988

Spectral multigrid methods for the solution of homogeneous turbulence problems

Erlebacher, G.↗

NASA NTRS1993

Computational mechanics analysis tools for parallel-vector supercomputers

Computational algorithms for structural analysis on parallel-vector supercomputers are reviewed. These parallel algorithms, developed by the authors, are for the assembly of structural equations, 'out-of-core' strategies for linear equation solution, massively distributed-memory equation solution, unsymmetric equation solution, general eigen-solution, geometrically nonlinear finite element analysis, design sensitivity analysis for structural dynamics, optimization algorithm and domain decomposition. The source code for many of these algorithms is available from NASA Langley.

Storaasli, O. O.↗

DOE OSTI2025

Exponentially Reduced Circuit Depths Using Trotter Error Mitigation

Product formulas are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance, which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by the use of these formulas. This work provides a rigorous, general analysis of these techniques for computing time-evolved observables, simplifying the interpolation algorithm in the process, and shows that extrapolation generically improves the performance of product formulas for this task. We demonstrate that, to achieve error 𝜖 in a simulation of time 𝑇 using a 𝑝 ⁢th-order product formula with extrapolation, circuit depths of 𝑂⁡(𝑇 1+1/𝑝 ⁢polylog (1/𝜖)) are sufficient—an exponential improvement in the precision over product formulas alone. Furthermore, we prove that these algorithms achieve commutator scaling, and improve the 𝑇 complexity for the interpolation algorithm. By relaxing the requirement of performing exact Chebyshev interpolation, our simplified algorithm eliminates the need for fractional implementations of Trotter steps, reducing computational overhead. Finally, we show these techniques can be combined with the classical shadows method to estimate many time-evolved local observables. Taken together, our findings provide the strongest evidence yet for the utility of Trotter error-mitigation techniques in algorithmic applications.

quantum algorithms & computation↗

NASA NTRS2014

Phase Transitions in Planning Problems: Design and Analysis of Parameterized Families of Hard Planning Problems

There are two common ways to evaluate algorithms: performance on benchmark problems derived from real applications and analysis of performance on parametrized families of problems. The two approaches complement each other, each having its advantages and disadvantages. The planning community has concentrated on the first approach, with few ways of generating parametrized families of hard problems known prior to this work. Our group's main interest is in comparing approaches to solving planning problems using a novel type of computational device - a quantum annealer - to existing state-of-the-art planning algorithms. Because only small-scale quantum annealers are available, we must compare on small problem sizes. Small problems are primarily useful for comparison only if they are instances of parametrized families of problems for which scaling analysis can be done. In this technical report, we discuss our approach to the generation of hard planning problems from classes of well-studied NP-complete problems that map naturally to planning problems or to aspects of planning problems that many practical planning problems share. These problem classes exhibit a phase transition between easy-to-solve and easy-to-show-unsolvable planning problems. The parametrized families of hard planning problems lie at the phase transition. The exponential scaling of hardness with problem size is apparent in these families even at very small problem sizes, thus enabling us to characterize even very small problems as hard. The families we developed will prove generally useful to the planning community in analyzing the performance of planning algorithms, providing a complementary approach to existing evaluation methods. We illustrate the hardness of these problems and their scaling with results on four state-of-the-art planners, observing significant differences between these planners on these problem families. Finally, we describe two general, and quite different, mappings of planning problems to QUBOs, the form of input required for a quantum annealing machine such as the D-Wave II.

Problems↗

NASA NTRS2002

Using Clustering to Establish Climate Regimes from PCM Output

A multivariate statistical clustering technique--based on the k-means algorithm of Hartigan has been used to extract patterns of climatological significance from 200 years of general circulation model (GCM) output. Originally developed and implemented on a Beowulf-style parallel computer constructed by Hoffman and Hargrove from surplus commodity desktop PCs, the high performance parallel clustering algorithm was previously applied to the derivation of ecoregions from map stacks of 9 and 25 geophysical conditions or variables for the conterminous U.S. at a resolution of 1 sq km. Now applied both across space and through time, the clustering technique yields temporally-varying climate regimes predicted by transient runs of the Parallel Climate Model (PCM). Using a business-as-usual (BAU) scenario and clustering four fields of significance to the global water cycle (surface temperature, precipitation, soil moisture, and snow depth) from 1871 through 2098, the authors' analysis shows an increase in spatial area occupied by the cluster or climate regime which typifies desert regions (i.e., an increase in desertification) and a decrease in the spatial area occupied by the climate regime typifying winter-time high latitude perma-frost regions. The patterns of cluster changes have been analyzed to understand the predicted variability in the water cycle on global and continental scales. In addition, representative climate regimes were determined by taking three 10-year averages of the fields 100 years apart for northern hemisphere winter (December, January, and February) and summer (June, July, and August). The result is global maps of typical seasonal climate regimes for 100 years in the past, for the present, and for 100 years into the future. Using three-dimensional data or phase space representations of these climate regimes (i.e., the cluster centroids), the authors demonstrate the portion of this phase space occupied by the land surface at all points in space and time. Any single spot on the globe will exist in one of these climate regimes at any single point in time. By incrementing time, that same spot will trace out a trajectory or orbit between and among these climate regimes (or atmospheric states) in phase (or state) space. When a geographic region enters a state it never previously visited, a climatic change is said to have occurred. Tracing out the entire trajectory of a single spot on the globe yields a 'manifold' in state space representing the shape of its predicted climate occupancy. This sort of analysis enables a researcher to more easily grasp the multivariate behavior of the climate system.

Oglesby, Robert↗