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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 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

The analysis of convolutional codes via the extended Smith algorithm

Convolutional codes have been the central part of most error-control systems in deep-space communication for many years. Almost all such applications, however, have used the restricted class of (n,1), also known as 'rate 1/n,' convolutional codes. The more general class of (n,k) convolutional codes contains many potentially useful codes, but their algebraic theory is difficult and has proved to be a stumbling block in the evolution of convolutional coding systems. In this article, the situation is improved by describing a set of practical algorithms for computing certain basic things about a convolutional code (among them the degree, the Forney indices, a minimal generator matrix, and a parity-check matrix), which are usually needed before a system using the code can be built. The approach is based on the classic Forney theory for convolutional codes, together with the extended Smith algorithm for polynomial matrices, which is introduced in this article.

Mceliece, R. J.↗
DOE OSTI2024

Scalable learning of potentials to predict time-dependent Hartree–Fock dynamics

We propose a framework to learn the time-dependent Hartree–Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Although the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange–correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to 60 × 60 × 60 × 60. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1972

Elimination on sparse symmetric systems of a special structure.

Consideration of the problem of finding a permutation of rows and columns and an algorithm for solving ordered systems of linear algebraic equations with sparse matrices having a certain regular structure. Two approaches to the solution of this problem, in which the sparsity is used to some extent, are outlined. One of them is a very general approach where optimal (or nearly optimal) ordering is sought and the algorithm for solving the ordered system treats the matrix element by element to perform only necessary operations. The other approach involves the use of band matrices. After comparing these two approaches, a third approach is then suggested which involves the use of pipe matrices, and a means of ordering the rows and columns to obtain this type of matrix is presented. Examples of matrices reordered by the proposed procedure are cited.

Segethova, J.↗
DOE OSTI2024

Efficient Unitary Designs from Random Sums and Permutations

A unitary k-design is an ensemble of unitaries that matches the first k moments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ(k poly (n)) many gates. The k dependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension (N) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-N expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension N.

algebra↗
NASA NTRS1996

A Note on Multigrid Theory for Non-nested Grids and/or Quadrature

We provide a unified theory for multilevel and multigrid methods when the usual assumptions are not present. For example, we do not assume that the solution spaces or the grids are nested. Further, we do not assume that there is an algebraic relationship between the linear algebra problems on different levels. What we provide is a computationally useful theory for adaptively changing levels. Theory is provided for multilevel correction schemes, nested iteration schemes, and one way (i.e., coarse to fine grid with no correction iterations) schemes. We include examples showing the applicability of this theory: finite element examples using quadrature in the matrix assembly and finite volume examples with non-nested grids. Our theory applies directly to other discretizations as well.

Douglas, C. C.↗
NASA NTRS1996

Research in Computational Aeroscience Applications Implemented on Advanced Parallel Computing Systems

Improving the numerical linear algebra routines for use in new Navier-Stokes codes, specifically Tim Barth's unstructured grid code, with spin-offs to TRANAIR is reported. A fast distance calculation routine for Navier-Stokes codes using the new one-equation turbulence models is written. The primary focus of this work was devoted to improving matrix-iterative methods. New algorithms have been developed which activate the full potential of classical Cray-class computers as well as distributed-memory parallel computers.

Wigton, Larry↗
NASA NTRS1989

Algebraic multigrid methods applied to problems in computational structural mechanics

The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

Mccormick, Steve↗
NASA NTRS1994

CFD analyses for advanced pump design

As one of the activities of the NASA/MSFC Pump Stage Technology Team, the present effort was focused on using CFD in the design and analysis of high performance rocket engine pumps. Under this effort, a three-dimensional Navier-Stokes code was used for various inducer and impeller flow field calculations. An existing algebraic grid generation procedure was-extended to allow for nonzero blade thickness, splitter blades, and hub/shroud cavities upstream or downstream of the (main) blades. This resulted in a fast, robust inducer/impeller geometry/grid generation package. Problems associated with running a compressible flow code to simulate an incompressible flow were resolved; related aspects of the numerical algorithm (viz., the matrix preconditioning, the artificial dissipation, and the treatment of low Mach number flows) were addressed. As shown by the calculations performed under the present effort, the resulting code, in conjunction with the grid generation package, is an effective tool for the rapid solution of three-dimensional viscous inducer and impeller flows.

Dejong, F. J.↗
NASA NTRS1990

Domain decomposition methods in aerodynamics

Compressible Euler equations are solved for two-dimensional problems by a preconditioned conjugate gradient-like technique. An approximate Riemann solver is used to compute the numerical fluxes to second order accuracy in space. Two ways to achieve parallelism are tested, one which makes use of parallelism inherent in triangular solves and the other which employs domain decomposition techniques. The vectorization/parallelism in triangular solves is realized by the use of a recording technique called wavefront ordering. This process involves the interpretation of the triangular matrix as a directed graph and the analysis of the data dependencies. It is noted that the factorization can also be done in parallel with the wave front ordering. The performances of two ways of partitioning the domain, strips and slabs, are compared. Results on Cray YMP are reported for an inviscid transonic test case. The performances of linear algebra kernels are also reported.

Venkatakrishnan, V.↗
DOE OSTI2025

A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints

Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.

PDE-constrained optimization↗
DOE OSTI2026

Chiral spin liquid and quantum phase transition in the triangular-lattice Hofstadter-Hubbard model

Recent advances in moiré engineering motivate the study of lattice models of strongly correlated electrons subjected to substantial orbital magnetic flux. We analyze the triangular-lattice Hofstadter-Hubbard model at one-quarter flux quantum per plaquette and a density of one electron per site, where a chiral spin liquid phase may exist between weak-coupling integer quantum Hall and strong-coupling 120° antiferromagnetic phases. Here, we use matrix product state methods and analytical arguments to investigate this model compactified to cylinders of finite circumference. We uncover a glide particle-hole symmetry operation which, we argue, is spontaneously broken at the quantum Hall to spin liquid transition on odd-circumference cylinders. We numerically verify the spontaneous symmetry breaking and further demonstrate that this transition is associated with algebraic long-range correlations of various spin-singlet, charge-neutral operators. For even-circumference cylinders, the transition becomes a crossover associated with a large correlation length that grows substantially with circumference. Our findings suggest that in the two-dimensional limit, the transition to a chiral spin liquid phase is continuous and features critical fluctuations of the current.

Divic, Stefan [University of Pennsylvania, Philade↗
DOE OSTI2025

Classical eikonal from Magnus expansion

In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. We exploit a Hopf algebra structure behind the Magnus expansion to develop a fast algorithm which can compute the tree coefficients up to the 12th order (over half a million trees) in less than an hour. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries in the worldline formalism. We demonstrate the methods by computing the 3PM eikonal and find agreement with previous results based on amplitude methods. Importantly, the Magnus expansion yields a finite eikonal, while the naïve eikonal based on the time-symmetric propagator is infrared-divergent from 3PM on.

Black Holes↗
NASA NTRS1985

Comparison of Two Algebraic Methods for Curve/curve Intersection

Most geometric modeling systems use either polynomial or rational functions to represent geometry. In such systems most computational problems can be formulated as systems of polynomials in one or more variables. Classical elimination theory can be used to solve such systems. Here Cayley's method of elimination is summarized and it is shown how it can best be used to solve the curve/curve intersection problem. Cayley's method was found to be a more straightforward approach. Furthermore, it is computationally simpler, since the elements of the Cayley matrix are one variable instead of two variable polynomials. Researchers implemented and tested both methods and found Cayley's to be more efficient. Six pairs of curves, representing mixtures of lines, circles, and cubic arcs were used. Several examples had multiple intersection points. For all six cases Cayley's required less CPU time than the other method. The average time ratio of method 1 to method 2 was 3.13:1, the least difference was 2.33:1, and the most dramatic was 6.25:1. Both of the above methods can be extended to solve the surface/surface intersection problem.

Demontaudouin, Y.↗
NASA NTRS2004

Symmetric normal mixtures

We consider mixture density estimation under the symmetry constraint x = Az for an orthogonal matrix A. This distributional constraint implies a corresponding constraint on the mixture parameters. Focusing on the gaussian case, we derive an expectation-maximization (EM) algorithm to enforce the constraint and show results for modeling of image feature vectors.

symmetry constraint↗
DOE OSTI2024

Batched sparse direct solver design and evaluation in SuperLU_DIST

Over the course of interactions with various application teams, the need for batched sparse linear algebra functions has emerged in order to make more efficient use of the GPUs for many small and sparse linear algebra problems. In this paper, we present our recent work on a batched sparse direct solver for GPUs. The sparse LU factorization is computed by the levels of the elimination tree, leveraging the batched dense operations at each level and a new batched Scatter GPU kernel. The sparse triangular solve is computed by the level sets of the directed acyclic graph (DAG) of the triangular matrix. Batched operations overcome the large overhead associated with launching many small kernels. For medium sized matrix batches with not-so-small bandwidth, using an NVIDIA A100 GPU, our new batched sparse direct solver is orders of magnitude faster than a batched banded solver and uses less than one-tenth of the memory.

Boukaram, Wajih↗
NASA NTRS1982

Subsonic 3-D surface panel method for rapid analysis of multiple geometry perturbations

For any baseline aircraft configuration, a matrix of partial derivatives of surface velocity potential with respect to geometry coordinates is calculated. Linear extrapolation is then used to analyze the subcritical potential flow corresponding to a series of arbitrary small geometry perturbations. Each perturbation analysis is more than an order of magnitude more efficient than a conventional panel method solution because no influence coefficients are calculated and no large system of linear algebraic equations is solved. Wing and wing-fuselage examples are presented to demonstrate that the predicted pressure distributions are nearly exact for large changes to wing camber, thickness, and leading edge radius.

Bristow, D. R.↗
NASA NTRS1988

Recursive mass matrix factorization and inversion: An operator approach to open- and closed-chain multibody dynamics

This report advances a linear operator approach for analyzing the dynamics of systems of joint-connected rigid bodies.It is established that the mass matrix M for such a system can be factored as M=(I+H phi L)D(I+H phi L) sup T. This yields an immediate inversion M sup -1=(I-H psi L) sup T D sup -1 (I-H psi L), where H and phi are given by known link geometric parameters, and L, psi and D are obtained recursively by a spatial discrete-step Kalman filter and by the corresponding Riccati equation associated with this filter. The factors (I+H phi L) and (I-H psi L) are lower triangular matrices which are inverses of each other, and D is a diagonal matrix. This factorization and inversion of the mass matrix leads to recursive algortihms for forward dynamics based on spatially recursive filtering and smoothing. The primary motivation for advancing the operator approach is to provide a better means to formulate, analyze and understand spatial recursions in multibody dynamics. This is achieved because the linear operator notation allows manipulation of the equations of motion using a very high-level analytical framework (a spatial operator algebra) that is easy to understand and use. Detailed lower-level recursive algorithms can readily be obtained for inspection from the expressions involving spatial operators. The report consists of two main sections. In Part 1, the problem of serial chain manipulators is analyzed and solved. Extensions to a closed-chain system formed by multiple manipulators moving a common task object are contained in Part 2. To retain ease of exposition in the report, only these two types of multibody systems are considered. However, the same methods can be easily applied to arbitrary multibody systems formed by a collection of joint-connected regid bodies.

Rodriguez, G.↗
NASA NTRS1979

Concerning an application of the method of least squares with a variable weight matrix

An estimate of a state vector for a physical system when the weight matrix in the method of least squares is a function of this vector is considered. An iterative procedure is proposed for calculating the desired estimate. Conditions for the existence and uniqueness of the limit of this procedure are obtained, and a domain is found which contains the limit estimate. A second method for calculating the desired estimate which reduces to the solution of a system of algebraic equations is proposed. The question of applying Newton's method of tangents to solving the given system of algebraic equations is considered and conditions for the convergence of the modified Newton's method are obtained. Certain properties of the estimate obtained are presented together with an example.

Sukhanov, A. A.↗