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NASA NTRS1992

Domain decomposition: A bridge between nature and parallel computers

Domain decomposition is an intuitive organizing principle for a partial differential equation (PDE) computation, both physically and architecturally. However, its significance extends beyond the readily apparent issues of geometry and discretization, on one hand, and of modular software and distributed hardware, on the other. Engineering and computer science aspects are bridged by an old but recently enriched mathematical theory that offers the subject not only unity, but also tools for analysis and generalization. Domain decomposition induces function-space and operator decompositions with valuable properties. Function-space bases and operator splittings that are not derived from domain decompositions generally lack one or more of these properties. The evolution of domain decomposition methods for elliptically dominated problems has linked two major algorithmic developments of the last 15 years: multilevel and Krylov methods. Domain decomposition methods may be considered descendants of both classes with an inheritance from each: they are nearly optimal and at the same time efficiently parallelizable. Many computationally driven application areas are ripe for these developments. A progression is made from a mathematically informal motivation for domain decomposition methods to a specific focus on fluid dynamics applications. To be introductory rather than comprehensive, simple examples are provided while convergence proofs and algorithmic details are left to the original references; however, an attempt is made to convey their most salient features, especially where this leads to algorithmic insight.

Keyes, David E.↗

NASA NTRS2012

High Resolution Observations of Drop Size Distribution for GPM Ground Validation

During the Mid-latitude Continental Convective Cloud Experiment (MC3E), NASA's GPM GV Disdrometer and Radar Observations of Precipitation (DROP) Facility deployed an array of disdrometers and rain gauges in northern Oklahoma to sample, with high resolution, the drop size distribution for use in development of precipitation retrieval algorithms for the GPM core satellites. The DROP Facility instruments deployed during MC3E consisted of 16 autonomous Parsivel units, 5 two-dimensional video disdrometers (2dvds), a vertically pointing K band radar, and 32 tipping bucket rain gauges. There were several rainfall events during MC3E in which rain drops exceeding 6 mm in diameter were recorded. The disdrometer array revealed large rain drops with diameters exceeding 6 mm and 8 mm during two separate stratiform and convective rainfall events, respectively. The NPOL radar, which was scanning in high resolution RHI mode (every 40 sec) over the disdrometer array during the stratiform event, indicated a 1 km thick bright band with a differential reflectivity column of 2-3 dB extending below the melting layer to the surface where the large drops were recorded by the 2dvds. These large drops are important for GPM since they can have a great impact upon satellite precipitation retrieval, especially near the ground and below heavy convective rainfall cores where satellites have had problems depicting the rainfall.

Gatlin, Patrick N.↗

DOE OSTI2026

Collaborative Research: Enabling multi-scale studies of magnetic reconnection with interpretable data-driven models

The development of accurate reduced descriptions and improved closures for magnetic reconnection is an important and a long‐standing challenge in plasma physics. The four‐fluid approach, and associated closures, that were investigated have the potential to improve the accuracy of plasma fluid models, capturing physical effects which would otherwise require a kinetic description. If successful, this approach could have an important impact for the modeling of laboratory and space plasmas. The major goals of this project were to develop new machine learning (ML) tools based on sparse and symbolic regression techniques, and to extract interpretable and generalizable reduced models (e.g., in the form of partial differential equations - PDEs) from data generated by first principles plasma simulations. Preserving interpretability of such data‐driven models is key to addressing the long‐standing theoretical and numerical challenges. Prior proof‐of‐principle studies have demonstrated the enormous potential of this approach, by recovering the well‐established hierarchy of plasma equations (from Vlasov to MHD) from data produced by particle‐in‐cell (PIC) simulations. Our goal in this project was to extend and apply these new tools to construct better kinetic closures for magnetic reconnection; to derive better models of particle injection and acceleration by this fundamental plasma process; and to use this understanding to accelerate the development of multi‐scale plasma algorithms. While our immediate focus was on the problem of magnetic reconnection, the tools that were will developed are general and applicable to other areas of plasma physics, and more broadly to many‐body phenomena. We anticipate that the development of these multi‐scale models will have a significant impact across different areas of plasma science, from fusion to space and astrophysical plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1978

A comparison of the Method of Lines to finite difference techniques in solving time-dependent partial differential equations

Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (1) Burger's equation over a finite space domain by a forward time central space explicit method, and (2) the stream function - vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to 'set up' time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.

Kurtz, L. A.↗

NASA NTRS1990

Dual-shaped offset reflector antenna designs from solutions of the geometrical optics first-order partial differential equations

In obtaining solutions to the first-order nonlinear partial differential equations (PDEs) for synthesizing offset dual-shaped reflectors, it is found that previously observed computational problems can be avoided if the integration of the PDEs is started from an inner projected perimeter and integrated outward rather than starting from an outer projected perimeter and integrating inward. This procedure, however, introduces a new parameter, the main reflector inner perimeter radius p(o), when given a subreflector inner angle 0(o). Furthermore, a desired outer projected perimeter (e.g., a circle) is no longer guaranteed. Stability of the integration is maintained if some of the initial parameters are determined first from an approximate solution to the PDEs. A one-, two-, or three-parameter optimization algorithm can then be used to obtain a best set of parameters yielding a close fit to the desired projected outer rim. Good low cross-polarization mapping functions are also obtained. These methods are illustrated by synthesis of a high-gain offset-shaped Cassegrainian antenna and a low-noise offset-shaped Gregorian antenna.

Galindo-Israel, V.↗

NASA NTRS2000

Sliding Mode Control of the X-33 with an Engine Failure

Ascent flight control of the X-3 is performed using two XRS-2200 linear aerospike engines. in addition to aerosurfaces. The baseline control algorithms are PID with gain scheduling. Flight control using an innovative method. Sliding Mode Control. is presented for nominal and engine failed modes of flight. An easy to implement, robust controller. requiring no reconfiguration or gain scheduling is demonstrated through high fidelity flight simulations. The proposed sliding mode controller utilizes a two-loop structure and provides robust. de-coupled tracking of both orientation angle command profiles and angular rate command profiles in the presence of engine failure, bounded external disturbances (wind gusts) and uncertain matrix of inertia. Sliding mode control causes the angular rate and orientation angle tracking error dynamics to be constrained to linear, de-coupled, homogeneous, and vector valued differential equations with desired eigenvalues. Conditions that restrict engine failures to robustness domain of the sliding mode controller are derived. Overall stability of a two-loop flight control system is assessed. Simulation results show that the designed controller provides robust, accurate, de-coupled tracking of the orientation angle command profiles in the presence of external disturbances and vehicle inertia uncertainties, as well as the single engine failed case. The designed robust controller will significantly reduce the time and cost associated with flying new trajectory profiles or orbits, with new payloads, and with modified vehicles

Shtessel, Yuri B.↗

NASA NTRS1981

Numerical algorithms for finite element computations on arrays of microprocessors

The development of a multicolored successive over relaxation (SOR) program for the finite element machine is discussed. The multicolored SOR method uses a generalization of the classical Red/Black grid point ordering for the SOR method. These multicolored orderings have the advantage of allowing the SOR method to be implemented as a Jacobi method, which is ideal for arrays of processors, but still enjoy the greater rate of convergence of the SOR method. The program solves a general second order self adjoint elliptic problem on a square region with Dirichlet boundary conditions, discretized by quadratic elements on triangular regions. For this general problem and discretization, six colors are necessary for the multicolored method to operate efficiently. The specific problem that was solved using the six color program was Poisson's equation; for Poisson's equation, three colors are necessary but six may be used. In general, the number of colors needed is a function of the differential equation, the region and boundary conditions, and the particular finite element used for the discretization.

Ortega, J. M.↗

NASA NTRS1990

Digital halftoning methods for selectively partitioning error into achromatic and chromatic channels

A method is described for reducing the visibility of artifacts arising in the display of quantized color images on CRT displays. The method is based on the differential spatial sensitivity of the human visual system to chromatic and achromatic modulations. Because the visual system has the highest spatial and temporal acuity for the luminance component of an image, a technique which will reduce luminance artifacts at the expense of introducing high-frequency chromatic errors is sought. A method based on controlling the correlations between the quantization errors in the individual phosphor images is explored. The luminance component is greatest when the phosphor errors are positively correlated, and is minimized when the phosphor errors are negatively correlated. The greatest effect of the correlation is obtained when the intensity quantization step sizes of the individual phosphors have equal luminances. For the ordered dither algorithm, a version of the method can be implemented by simply inverting the matrix of thresholds for one of the color components.

Mulligan, Jeffrey B.↗

NASA NTRS2003

Efficient Processing of Data for Locating Lightning Strikes

Two algorithms have been devised to increase the efficiency of processing of data in lightning detection and ranging (LDAR) systems so as to enable the accurate location of lightning strikes in real time. In LDAR, the location of a lightning strike is calculated by solving equations for the differences among the times of arrival (DTOAs) of the lightning signals at multiple antennas as functions of the locations of the antennas and the speed of light. The most difficult part of the problem is computing the DTOAs from digitized versions of the signals received by the various antennas. One way (a time-domain approach) to determine the DTOAs is to compute cross-correlations among variously differentially delayed replicas of the digitized signals and to select, as the DTOAs, those differential delays that yield the maximum correlations. Another way (a frequency-domain approach) to determine the DTOAs involves the computation of cross-correlations among Fourier transforms of variously differentially phased replicas of the digitized signals, along with utilization of the relationship among phase difference, time delay, and frequency.

Medelius, Pedro J.↗

DOE OSTI2024

Reduced-dimension Bayesian optimization for model calibration of transient vapor compression cycles

Development and calibration of first-principles dynamic models of vapor compression cycles (VCCs) is of critical importance for applications that include control design and fault detection and diagnostics. Nevertheless, the inherent complexity of models that are represented by large systems of differential–algebraic equations leads to significant challenges for model calibration processes that utilize classical gradient-based methods. Bayesian optimization (BO) is a sample-efficient and gradient-free approach using a probabilistic surrogate model and optimal search over a feasible parameter space. Despite the benefits of BO in reducing computational costs, challenges remain in dealing with a high-dimensional calibration task resulting from a large set of parameters that have significant impacts on system behavior and need to be calibrated simultaneously. This paper presents a reduced-dimension BO framework for calibrating transient VCCs models where the calibration space is projected to a low-dimensional subspace for accelerating convergence of the solution algorithm and consequently reducing the number of transient simulations. The proposed approach was demonstrated via two case studies associated with different VCC applications where 10 parameters were calibrated in each case using laboratory measurements. The reduced-dimension BO framework only required 1 / 8 th of the iterations associated with a standard BO method that deals with high-dimensional calibration parameters for converged solutions and yielded comparable accuracy. Furthermore, both calibrated models revealed significant accuracy improvements compared to uncalibrated models.

Ma, Jiacheng↗

DOE OSTI2021

Bringing discrete-time Langevin splitting methods into agreement with thermodynamics

In light of the recently published complete set of statistically correct Grønbech–Jensen (GJ) methods for discrete-time thermodynamics, we revise a differential operator splitting method for the Langevin equation in order to comply with the basic GJ thermodynamic sampling features, namely, the Boltzmann distribution and Einstein diffusion, in linear systems. This revision, which is based on the introduction of time scaling along with flexibility of a discrete-time velocity attenuation parameter, provides a direct link between the ABO splitting formalism and the GJ methods. This link brings about the conclusion that any GJ method has at least weak second order accuracy in the applied time step. It further helps identify a novel half-step velocity, which simultaneously produces both correct kinetic statistics and correct transport measures for any of the statistically sound GJ methods. Explicit algorithmic expressions are given for the integration of the new half-step velocity into the GJ set of methods. Finally, numerical simulations, including quantum-based molecular dynamics (QMD) using the QMD suite Los Alamos Transferable Tight-Binding for Energetics, highlight the discussed properties of the algorithms as well as exhibit the direct application of robust, time-step-independent stochastic integrators to QMD.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1992

Airborne gravimetry, altimetry, and GPS navigation errors

Proper interpretation of airborne gravimetry and altimetry requires good knowledge of aircraft trajectory. Recent advances in precise navigation with differential GPS have made it possible to measure gravity from the air with accuracies of a few milligals, and to obtain altimeter profiles of terrain or sea surface correct to one decimeter. These developments are opening otherwise inaccessible regions to detailed geophysical mapping. Navigation with GPS presents some problems that grow worse with increasing distance from a fixed receiver: the effect of errors in tropospheric refraction correction, GPS ephemerides, and the coordinates of the fixed receivers. Ionospheric refraction and orbit error complicate ambiguity resolution. Optimal navigation should treat all error sources as unknowns, together with the instantaneous vehicle position. To do so, fast and reliable numerical techniques are needed: efficient and stable Kalman filter-smoother algorithms, together with data compression and, sometimes, the use of simplified dynamics.

Colombo, Oscar L.↗

DOE OSTI2022

The Schwarz Alternating Method for the Seamless Coupling of Nonlinear Reduced Order Models and Full Order Models

Projection-based model order reduction allows for the parsimonious representation of full order models (FOMs), typically obtained through the discretization of a set of partial differential equations (PDEs) using conventional techniques (e.g., finite element, finite volume, finite difference methods) where the discretization may contain a very large number of degrees of freedom. As a result of this more compact representation, the resulting projection-based reduced order models (ROMs) can achieve considerable computational speedups, which are especially useful in real-time or multi-query analyses. One known deficiency of projection-based ROMs is that they can suffer from a lack of robustness, stability and accuracy, especially in the predictive regime, which ultimately limits their useful application. Another research gap that has prevented the widespread adoption of ROMs within the modeling and simulation community is the lack of theoretical and algorithmic foundations necessary for the “plug-and-play” integration of these models into existing multi-scale and multi-physics frameworks. This paper describes a new methodology that has the potential to address both of the aforementioned deficiencies by coupling projection-based ROMs with each other as well as with conventional FOMs by means of the Schwarz alternating method [41]. Leveraging recent work that adapted the Schwarz alternating method to enable consistent and concurrent multiscale coupling of finite element FOMs in solid mechanics [35, 36], we present a new extension of the Schwarz framework that enables FOM-ROM and ROM-ROM coupling, following a domain decomposition of the physical geometry on which a PDE is posed. In order to maintain efficiency and achieve computation speed-ups, we employ hyper-reduction via the Energy-Conserving Sampling and Weighting (ECSW) approach [13]. We evaluate the proposed coupling approach in the reproductive as well as in the predictive regime on a canonical test case that involves the dynamic propagation of a traveling wave in a nonlinear hyper-elastic material.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1982

Geometrically derived difference formulae for the numerical integration of trajectory problems

An initial value problem for the autonomous system of ordinary differential equations dy/dt = f(y), where y is a vector, is considered. In a number of practical applications the interest lies in obtaining the curve traced by the solution y. These applications include the computation of trajectories in mechanical problems. The term 'trajectory problem' is employed to refer to these cases. Lambert and McLeod (1979) have introduced a method involving local rotation of the axes in the y-plane for the two-dimensional case. The present investigation continues the study of difference schemes specifically derived for trajectory problems. A simple geometrical way of constructing such methods is presented, and the local accuracy of the schemes is investigated. A circularly exact, fixed-step predictor-corrector algorithm is defined, and a variable-step version of a circularly exact algorithm is presented.

Mcleod, R. J. Y.↗

DOE OSTI2020

Unfolding quantum computer readout noise

Abstract In the current era of noisy intermediate-scale quantum computers, noisy qubits can result in biased results for early quantum algorithm applications. This is a significant challenge for interpreting results from quantum computer simulations for quantum chemistry, nuclear physics, high energy physics (HEP), and other emerging scientific applications. An important class of qubit errors are readout errors. The most basic method to correct readout errors is matrix inversion, using a response matrix built from simple operations to probe the rate of transitions from known initial quantum states to readout outcomes. One challenge with inverting matrices with large off-diagonal components is that the results are sensitive to statistical fluctuations. This challenge is familiar to HEP, where prior-independent regularized matrix inversion techniques (“unfolding”) have been developed for years to correct for acceptance and detector effects, when performing differential cross section measurements. We study one such method, known as iterative Bayesian unfolding, as a potential tool for correcting readout errors from universal gate-based quantum computers. This method is shown to avoid pathologies from commonly used matrix inversion and least squares methods.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRSDate not supplied

A Generalized, Compactly-Supported Correlation Function for Data Assimilation Applications

Correlation functions play an essential role in modern data assimilation, where they are used to model covariances given a set of tunable parameters or applied as tapering functions to localize covariances in ensemble-based schemes. One of the most widely-used correlation functions in data assimilation is the Gaspari and Cohn (1999) piecewise-rational, compactly-supported parametric correlation function (hereafter referred to as GC99). The GC99 correlation function is useful due to its tunable cut-off parameter c and Gaussian-like shape achieved when the parameter a is set to one-half. These properties are attractive for tapering functions in data assimilation applications. However, the GC99 correlation function is homogeneous over Euclidean 3-space and isotropic when restricted to the sphere, properties that may be less than ideal for some geophysical applications. GC99 is also compactly-supported on a sphere of fixed radius, which requires tuning of the cut-off parameter c that can depend on the specific application. This work presents a generalization of the GC99 correlation function that allows the cut-off parameter c and shape parameter a to vary over space to gain more flexibility in shape while maintaining its compact support property. The function, which we call the Generalized Gaspari Cohn (GenGC) correlation function, introduces inhomogeneity in Euclidean 3-space and anisotropy when restricted to the sphere by allowing both parameters c and a to vary, as functions, over the spatial domain. The GC99 correlation function is a special case of GenGC where the functions c and a are held constant, as fixed parameters rather than functions. The GenGC correlation function also generalizes the follow-on to the work of Gaspari and Cohn (1999) presented in Gaspari et al. (2006), which allowed a to vary while keeping c fixed. We illustrate through simple one- and two-dimensional examples the variety of inhomogeneous and anisotropic correlation functions GenGC can produce by varying c and a over space, and suggest applications where they may be useful in data assimilation, such as covariance modeling or localization. In particular, we describe how the GenGC correlation function can be used to construct covariances using correlation length and variance fields derived from dynamics. For example, the correlation length field for advective dynamics is governed by a partial differential equation (PDE) in N spatial dimensions, where N is the number of space dimensions of the state. Correlation length fields can be determined from this PDE and used with GenGC to construct the corresponding correlations. We can then approximate the full covariance by rescaling by the variance, which also satisfies a PDE in N spatial dimensions for advective dynamics. Thus we can approximate the full covariance without solving the covariance PDE, which is in 2N spatial dimensions, by solving just two PDEs each in only N spatial dimensions. This approach to evolving the correlation length and variance fields, then reconstructing the correlations using GenGC, is suggested as an alternative to current methods of covariance modeling in data assimilation algorithms.

GC99↗

DOE OSTI2020

A conservative phase-space moving-grid strategy for a 1D-2V Vlasov–Fokker–Planck Solver

In this work, we develop a conservative configuration- and velocity-space (i.e., phase-space) moving-grid strategy for the Vlasov–Fokker–Planck (VFP) equation in a planar geometry. The velocity-space grid is normalized and shifted in terms of the thermal speed and the bulk-fluid velocity, respectively. The configuration-space grid is moved according to a mesh-motion-partial-differential equation (MMPDE), which equidistributes a monitor function that is inversely proportional to the gradient-length scales of the macroscopic plasma quantities. The resulting inertial terms in the transformed VFP equations are discretized to ensure the discrete conservation of mass, momentum, and energy. To satisfy the discrete conservation theorems in the presence of phase-space mesh motion, we employ the method of discrete nonlinear constraints – explored in previous studies – but the underlying symmetries are determined in a much more efficient manner than before. The conservative grid-adaptivity strategy provides an efficient scheme that resolves important physical structures in the phase-space while controlling the computational complexity at all times. We demonstrate the favorable features of the proposed algorithm through a set of test cases of increasing complexity. The problems test independent components of the algorithms, as well as the integrated capability on settings relevant to inertial confinement fusion.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Scalable algorithms for physics-informed neural and graph networks

Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.

42 ENGINEERING↗