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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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At least 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

Numerical Study of Multigrid Methods with Various Smoothers for the Elliptical Grid Generation Equations

A robust solver for the elliptic grid generation equations is sought via a numerical study. The system of PDEs is discretized with finite differences, and multigrid methods are applied to the resulting nonlinear algebraic equations. Multigrid iterations are compared with respect to the robustness and efficiency. Different smoothers are tried to improve the convergence of iterations. The methods are applied to four 2D grid generation problems over a wide range of grid distortions. The results of the study help to select smoothing schemes and the overall multigrid procedures for elliptic grid generation.

Golik, W. L.↗
NASA NTRS1977

Linear systems with structure group and their feedback invariants

A general method described by Hermann and Martin (1976) for the study of the feedback invariants of linear systems is considered. It is shown that this method, which makes use of ideas of topology and algebraic geometry, is very useful in the investigation of feedback problems for which the classical methods are not suitable. The transfer function as a curve in the Grassmanian is examined. The general concepts studied in the context of specific systems and applications are organized in terms of the theory of Lie groups and algebraic geometry. Attention is given to linear systems which have a structure group, linear mechanical systems, and feedback invariants. The investigation shows that Lie group techniques are powerful and useful tools for analysis of the feedback structure of linear systems.

Martin, C.↗
DOE OSTI2025

Analyzing the Quantum Approximate Optimization Algorithm: Ansätze, Symmetries, and Lie Algebras

The quantum approximate optimization algorithm (QAOA) has been proposed as a method to obtain approximate solutions for combinatorial optimization tasks. In this work, we study the underlying algebraic properties of three QAOA ansätze for the maximum-cut problem on connected graphs, while focusing on the generated Lie algebras as well as their invariant subspaces. Specifically, we analyze the standard QAOA ansatz as well as the orbit and multiangle ansätze. We are able to fully characterize the Lie algebras of the multiangle ansatz across arbitrary connected graphs, finding that they only fall into one of just six families. Aside from the cycle and path graphs, the Lie dimensions for every graph are exponentially large in the system size, meaning that multiangle ansätze are extremely prone to exhibiting barren plateaus. Then, a similar quasi-graph-independent Lie-algebraic characterization beyond the multiangle ansatz is impeded as the circuit exhibits additional “hidden” symmetries besides those naturally arising from a certain parity-superselection operator and all automorphisms of the considered graph. Disregarding the “hidden” symmetries, we can upper bound the dimensions of the orbit and the standard Lie algebras, and the dimensions of the associated invariant subspaces are determined via explicit character formulas. To finish, we conjecture that (for most graphs) the standard Lie algebras have only components that are either exponential or that grow, at most, polynomially with the system size. This would imply that the QAOA is either prone to barren plateaus or classically simulable. More generally, our work provides a symmetry framework and tools to analyze any desired variational quantum algorithm.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1978

Extending reliability: Transformational tailoring of abstract mathematical software

Methods for automatically constructing concrete executable programs from an abstract prototype program by applying transformations based on theorems of matrix algebra and on algebraic properties of programming languages are described. These methods provide a user with highly efficient programs tailored to his environment while maintaining the advantages of high reliability and low cost associated with routines from the best mathematical software libraries. Also, the transformations which produce such programs represent a formal codification of rules for writing linear algebra programs.

Boyle, J. M.↗
NASA NTRS1996

A Full Multi-Grid Method for the Solution of the Cell Vertex Finite Volume Cauchy-Riemann Equations

The system of inhomogeneous Cauchy-Riemann equations defined on a square domain and subject to Dirichlet boundary conditions is considered. This problem is discretised by using the cell vertex finite volume method on quadrilateral meshes. The resulting algebraic problem is overdetermined and the solution is defined in a least squares sense. By this approach a consistent algebraic problem is obtained which differs from the original one by O(h(exp 2)) perturbations of the right-hand side. A suitable cell-based convergent smoothing iteration is presented which is naturally linked to the least squares formulation. Hence, a standard multi-grid algorithm is reported which combines the given smoother and cell-based transfer operators. Some remarkable reduction properties of these operators are shown. A full multi-grid method is discussed which solves the discrete problem to the level of truncation error by employing one multi-grid cycle at each current level of discretisation. Experiments and applications of the full multi-grid scheme are presented.

Borzi, A.↗
NASA NTRS1979

On the overlapping plate method in astrometry

A method is proposed for the simplification of the algorithms of Eichhorn and Russell (1976) for obtaining the solutions of astrometric problems using the overlapping plate method. The method provides the parameter vector (consisting of 'star constants' and plate constants) as well as the covariance matrix of the parameters. A considerable reduction in the amount of matrix algebra required is obtained with the present method.

Jefferys, W. H.↗
DOE OSTI2024

U(1) fields from qubits: An approach via D-theory algebra

A new quantum link microstructure was proposed for the lattice quantum chromodynamics (QCD) Hamiltonian, replacing the Wilson gauge links with a bilinear of fermionic qubits, later generalized to D-theory. This formalism provides a general framework for building lattice field theory algorithms for quantum computing. We focus mostly on the simplest case of a quantum rotor for a single compact U(1) field. We also make some progress for non-Abelian setups, making it clear that the ideas developed in the U(1) case extend to other groups. These in turn are building blocks for 1 + 0 -dimensional ( 1 + 0 -D) matrix models, 1 + 1 -D sigma models and non-Abelian gauge theories in 2 + 1 and 3 + 1 dimensions. By introducing multiple flavors for the U(1) field, where the flavor symmetry is gauged, we can efficiently approach the infinite-dimensional Hilbert space of the quantum O(2) rotor with increasing flavors. The emphasis of the method is on preserving the symplectic algebra exchanging fermionic qubits by sigma matrices (or hard bosons) and developing a formal strategy capable of generalization to a SU ( 3 ) field for lattice QCD and other non-Abelian 1 + 1 -D sigma models or 3 + 1 -D gauge theories. For U(1), we discuss briefly the qubit algorithms for the study of the discrete 1 + 1 -D sine-Gordon equation. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
DOE OSTI2026

A projection method for particle resampling

Particle discretizations of partial differential equations are advantageous for high-dimensional kinetic models in phase-space due to their better scalability than continuum approaches with respect to dimension. Complex processes collectively referred to as particle noise hamper long time simulations with particle methods. One approach to address this problem is particle mesh adaptivity, or remapping, known as particle resampling and remeshing. Here, this work introduces a resampling method that projects particles to and from a (finite element) function space. The method is simple, using standard sparse linear algebra and finite element techniques, and it preserves all moments up to the order of a polynomial represented exactly by the continuum function space. It is distinguished from most other mesh-based methods in that new particle positions and number are decoupled from the mesh, allowing particle and continuum meshes to be adapted relatively independently. While this work is developed with structured particle and continuum phase-space grids on 1X + 1V Vlasov-Poisson models of Landau damping and two-stream instability, the method is well-suited to unstructured grids. Stable long time dynamics are demonstrated up to time T = 500. Reproducibility artifacts and data are publicly available.

Kinetic methods↗
DOE OSTI2021

Asynchronous Iterative Solvers for Extreme-Scale Computing

The Asynchronous Iterative Solvers for Extreme-Scale Computing (AsyncIS) project aims to explore more efficient numerical algorithms by decreasing their overhead. AsyncIS does this by replacing the outer Krylov subspace solver with an asynchronous optimized Schwarz method, thereby removing the global synchronization and bulk synchronous operations typically used in numerical codes. AsyncIS—a U.S. Department of Energy (DOE)-funded collaboration between Georgia Tech, the University of Tennessee, Knoxville, Temple University, and Sandia National Laboratories—also focuses on the development and optimization of asynchronous preconditioners (i.e., preconditioners that are generated and/or applied in an asynchronous fashion). The novel preconditioning algorithms that provide fine-grained parallelism enable preconditioned Krylov solvers to run efficiently on large-scale distributed systems and manycore accelerators like GPUs.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1991

Linear and circular digital spectral analysis of serial data

Two methods of digital spectral analysis of unevenly sampled data are developed and illustrated here. One method uses a linear function of time (or space), the other uses circular functions. The circular method turns out to be essentially equivalent to a least-squares sine-wave analysis. The linear, anharmonic method uses only the field of real numbers and elementary algebraic operations, and hence it can be made computationally very fast and accurate. Both methods are very general, properly handling all kinds of time series ranging from simple series consisting only of the times of events to complicated series consisting of pulses with long duty cycles. The two methods are here applied to the analysis of annual mean relative sunspot numbers.

Stothers, Richard B.↗
DOE OSTI2022

PyAMG: Algebraic Multigrid Solvers in Python

PyAMG is a Python package of algebraic multigrid (AMG) solvers and supporting tools for approximating the solution to large, sparse linear systems of algebraic equations, Ax = b, where A is an n × n sparse matrix. Sparse linear systems arise in a range of problems in science, from fluid flows to solid mechanics to data analysis. While the direct solvers available in SciPy’s sparse linear algebra package (scipy.sparse.linalg) are highly efficient, in many cases iterative methods are preferred due to overall complexity. However, the iterative methods in SciPy, such as CG and GMRES, often require an efficient preconditioner in order to achieve a lower complexity. Preconditioning is a powerful tool whereby the conditioning of the linear system and convergence rate of the iterative method are both dramatically improved. PyAMG constructs multigrid solvers for use as a preconditioner in this setting. A summary of multigrid and algebraic multigrid solvers can be found in Olson (2015a), in Olson (2015b), and in Falgout (2006); a detailed description can be found in Briggs et al. (2000) and Trottenberg et al. (2001).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Automated label flows for excited states of correlation functions in lattice gauge theory

Extracting excited states from lattice gauge theory correlation functions can be achieved through $χ^2$ minimization fits or algebraic approaches such as the variational method and Prony's method. Performing any kind of error analysis often leads to overlapping confidence regions of model parameters, even when the spectrum is not particularly dense. To correctly estimate errors, one must beware of mislabeling the states. In this work, we provide an algorithm that we call automated label flows which consistently and systematically identifies a deterministic labeling of states. This is a black-box approach in the sense that it gives a sensible set of labels without user guidance. Further, as an example, we pair one black box method with another, analyzing a lattice correlation function from real data using automated label flows in the context of Prony's method.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1966

Matrix methods and automation in structural engineering

The partial differential equations of motion of shell structures subject to arbitrary loads may be extremely difficult or even impossible to solve. Since the digital computer is now an available tool for the structural engineer, current research has been directed toward methods which involve matrix formulation of large systems of algebraic equations and matrix solutions for discrete elements rather than the solution of the partial differential equations. These methods require rapid and accurate computer solutions. Recognizing the accuracy problems inherent in working with large matrices, a comprehensive survey of available computer programs was performed for large matrix inversion and for eigenvalue and eigenvector solutions under a Research Grant from the National Aeronautics and Space Administration and the techniques are developed in this paper for using these programs most efficiently for structural applications. The contents of the paper include: automation of matrix compilation, methods of very large matrix inversion and solution of simultaneous equation, and techniques for finding eigenvalues and vectors. In addition, a finite element stiffness matrix approach developed at the Denver Research Institute for both plates and shells subject to arbitrary dynamic loads is described as it was instrumented with complete automation on the digital computer.

Anita S West↗
NASA NTRS1973

Plane elasto-plastic analysis of v-notched plate under bending by boundary integral equation method

A method of solution is presented, which, when applied to the elasto-plastic analysis of plates having a v-notch on one edge and subjected to pure bending, will produce stress and strain fields in much greater detail than presently available. Application of the boundary integral equation method results in two coupled Fredholm-type integral equations, subject to prescribed boundary conditions. These equations are replaced by a system of simultaneous algebraic equations and solved by a successive approximation method employing Prandtl-Reuss incremental plasticity relations. The method is first applied to number of elasto-static problems and the results compared with available solutions. Good agreement is obtained in all cases. The elasto-plastic analysis provides detailed stress and strain distributions for several cases of plates with various notch angles and notch depths. A strain hardening material is assumed and both plane strain and plane stress conditions are considered.

Rzasnicki, W.↗
NASA NTRS1975

Temperature fields of an absorptive medium in a radiating system of arbitrary configuration (the spatial problem)

A generalized zonal method based on systems of linear algebraic equations is used to determine the temperature fields in an absorptive grey medium filling a closed radiation system that consists of three boundary zones, of which one is adiabatic and the other two are isothermal. The example calculation considers the case of a solenoidal radiation field of local radiative equilibrium.

Surinov, Y. A.↗
NASA NTRS1974

Nonlinear finite-element analysis of laminated composite shells

Two aspects of the geometrically nonlinear analysis of laminated composite shells are considered in this paper. The first pertains to the simplifications and computational advantages resulting from the use of group-theoretic methods in conjunction with computerized symbolic (algebraic) manipulation in the element development. The second aspect pertains to the combined effects of shear deformation and nonlinearity on the accuracy and convergence of two-dimensional, shear-flexible stiffness finite element models.

Noor, A. K.↗
NASA NTRS1975

Computer-developed construction of analytic expressions for the coordinates and partial derivatives of Jupiter's Galilean satellites

A method for improving Sampson's (1910, 1912, 1921) original work in developing series expressions for accurate coordinates of the Galilean satellites is discussed. The method, which utilizes computer-based algebraic manipulation software, was developed to reconstruct Sampson's theory, remove existing errors, introduce neglected effects, and provide analytic expressions for the coordinates as well as for the partial derivatives with respect to orbital parameters, Jupiter's mass and oblateness, the satellite masses, and Jupiter's pole and rotation period. The software system, capable of handling Poisson series with up to 73 polynomial variables and 28 trigonometric arguments, is described. The preliminary solution is presented, and procedures are outlined for calculating perturbations and eliminating auxiliary parameters.

Lieske, J. H.↗