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At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry.

Seiberg, Nathan (ORCID:000000033897046X)↗
DOE OSTI2023

Extending PETSc's Composable Hierarchical Solvers (Final Technical Report)

This report documents research activities conducted at CU Boulder as part of Extending PETSc’s Composable Hierarchical Solvers, which has been part of a collaboration with Argonne National Laboratory (separate award). Our work has focused on performance-portable end-to-end GPU solvers demonstrated via exemplary applications in nonlinear fluid and structural mechanics. We describe advances in algorithmic composition and analysis in the context of these applications, but the implementations are fully documented and decoupled, and in use by other projects. We believe the vertical integration achieved through collaboration with ECP’s CEED and the PSAAP center at CU was necessary to take risks with data structures and algorithms.

42 ENGINEERING↗
NASA NTRS1978

Finite element concepts in computational aerodynamics

Finite element theory was employed to establish an implicit numerical solution algorithm for the time averaged unsteady Navier-Stokes equations. Both the multidimensional and a time-split form of the algorithm were considered, the latter of particular interest for problem specification on a regular mesh. A Newton matrix iteration procedure is outlined for solving the resultant nonlinear algebraic equation systems. Multidimensional discretization procedures are discussed with emphasis on automated generation of specific nonuniform solution grids and accounting of curved surfaces. The time-split algorithm was evaluated with regards to accuracy and convergence properties for hyperbolic equations on rectangular coordinates. An overall assessment of the viability of the finite element concept for computational aerodynamics is made.

Baker, A. J.↗
NASA NTRS1979

Eigenvalues and eigenvectors for hybrid coordinate equations of motion for flexible spacecraft

The eigenvalues and eigenvectors of a system of linear time-invariant equations describing the attitude motion of flexible spacecraft in terms of hybrid coordinates are characterized in terms of literal expressions by using peculiar properties of the system parameter matrices. For the undamped case the eigenvalues are localized in terms of inertial matrices and modal parameters. A procedure for calculating the eigenvectors is proposed whereby the eigenproblem associated with the original system of dimension (2N + 6) is reduced to that of a symmetric and positive definite matrix of dimension N with the zero-damping assumption. The eigenvectors for systems of large dimension are obtained explicitly in terms of a 3x1 matrix whose elements are available from a system of three algebraic equations, which is provided.

Ohkami, Y.↗
NASA NTRS1992

A non-linearly stable implicit finite element algorithm for hypersonic aerodynamics

A generalized curvilinear coordinate Taylor weak statement implicit finite element algorithm is developed for the two-dimensional and axisymmetric compressible Navier-Stokes equations for ideal and reacting gases. For accurate hypersonic simulation, air is modeled as a mixture of five perfect gases, i.e., molecular and atomic oxygen and nitrogen as well as nitric oxide. The associated pressure is then determined via Newton solution of the classical chemical equilibrium equation system. The directional semidiscretization is achieved using an optimal metric data Galerkin finite element weak statement, on a developed 'companion conservation law system', permitting classical test and trial space definitions. Utilizing an implicit Runge-Kutta scheme, the terminal algorithm is then nonlinearly stable, and second-order accurate in space and time on arbitrary curvilinear coordinates. Subsequently, a matrix tensor product factorization procedure permits an efficient numerical linear algebra handling for large Courant numbers. For ideal- and real-gas hypersonic flows, the algorithm generates essentially nonoscillatory numerical solutions in the presence of strong detached shocks and boundary layer-inviscid flow interactions.

Iannelli, G. S.↗
DOE OSTI2025

Communication Lower Bounds and Optimal Algorithms for Symmetric Matrix Computations

In this article, we focus on the communication costs of three symmetric matrix computations: (i) multiplying a matrix with its transpose, known as a symmetric rank-k update (SYRK) (ii) adding the result of the multiplication of a matrix with the transpose of another matrix and the transpose of that result, known as a symmetric rank-2k update (SYR2K) (iii) performing matrix multiplication with a symmetric input matrix (SYMM). All three computations appear in the Level 3 Basic Linear Algebra Subroutines (BLAS) and have wide use in applications involving symmetric matrices. We establish communication lower bounds for these kernels using sequential and distributed-memory parallel computational models, and we show that our bounds are tight by presenting communication-optimal algorithms for each setting. Our lower bound proofs rely on applying a geometric inequality for symmetric computations and analytically solving constrained nonlinear optimization problems. As a result, the symmetric matrix and its corresponding computations are accessed and performed according to a triangular block partitioning scheme in the optimal algorithms.

Al Daas, Hussam [Rutherford Appleton Laboratory, D↗
NASA NTRS1994

Implementation and Assessment of Advanced Analog Vector-Matrix Processor

This paper discusses the design and implementation of an analog optical vecto-rmatrix coprocessor with a throughput of 128 Mops for a personal computer. Vector matrix calculations are inherently parallel, providing a promising domain for the use of optical calculators. However, to date, digital optical systems have proven too cumbersome to replace electronics, and analog processors have not demonstrated sufficient accuracy in large scale systems. The goal of the work described in this paper is to demonstrate a viable optical coprocessor for linear operations. The analog optical processor presented has been integrated with a personal computer to provide full functionality and is the first demonstration of an optical linear algebra processor with a throughput greater than 100 Mops. The optical vector matrix processor consists of a laser diode source, an acoustooptical modulator array to input the vector information, a liquid crystal spatial light modulator to input the matrix information, an avalanche photodiode array to read out the result vector of the vector matrix multiplication, as well as transport optics and the electronics necessary to drive the optical modulators and interface to the computer. The intent of this research is to provide a low cost, highly energy efficient coprocessor for linear operations. Measurements of the analog accuracy of the processor performing 128 Mops are presented along with an assessment of the implications for future systems. A range of noise sources, including cross-talk, source amplitude fluctuations, shot noise at the detector, and non-linearities of the optoelectronic components are measured and compared to determine the most significant source of error. The possibilities for reducing these sources of error are discussed. Also, the total error is compared with that expected from a statistical analysis of the individual components and their relation to the vector-matrix operation. The sufficiency of the measured accuracy of the processor is compared with that required for a range of typical problems. Calculations resolving alloy concentrations from spectral plume data of rocket engines are implemented on the optical processor, demonstrating its sufficiency for this problem. We also show how this technology can be easily extended to a 100 x 100 10 MHz (200 Cops) processor.

Gary, Charles K.↗
NASA NTRS1984

Acoustooptic linear algebra processors - Architectures, algorithms, and applications

Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.

Casasent, D.↗
NASA NTRS2011

Constraint Embedding Technique for Multibody System Dynamics

Multibody dynamics play a critical role in simulation testbeds for space missions. There has been a considerable interest in the development of efficient computational algorithms for solving the dynamics of multibody systems. Mass matrix factorization and inversion techniques and the O(N) class of forward dynamics algorithms developed using a spatial operator algebra stand out as important breakthrough on this front. Techniques such as these provide the efficient algorithms and methods for the application and implementation of such multibody dynamics models. However, these methods are limited only to tree-topology multibody systems. Closed-chain topology systems require different techniques that are not as efficient or as broad as those for tree-topology systems. The closed-chain forward dynamics approach consists of treating the closed-chain topology as a tree-topology system subject to additional closure constraints. The resulting forward dynamics solution consists of: (a) ignoring the closure constraints and using the O(N) algorithm to solve for the free unconstrained accelerations for the system; (b) using the tree-topology solution to compute a correction force to enforce the closure constraints; and (c) correcting the unconstrained accelerations with correction accelerations resulting from the correction forces. This constraint-embedding technique shows how to use direct embedding to eliminate local closure-loops in the system and effectively convert the system back to a tree-topology system. At this point, standard tree-topology techniques can be brought to bear on the problem. The approach uses a spatial operator algebra approach to formulating the equations of motion. The operators are block-partitioned around the local body subgroups to convert them into aggregate bodies. Mass matrix operator factorization and inversion techniques are applied to the reformulated tree-topology system. Thus in essence, the new technique allows conversion of a system with closure-constraints into an equivalent tree-topology system, and thus allows one to take advantage of the host of techniques available to the latter class of systems. This technology is highly suitable for the class of multibody systems where the closure-constraints are local, i.e., where they are confined to small groupings of bodies within the system. Important examples of such local closure-constraints are constraints associated with four-bar linkages, geared motors, differential suspensions, etc. One can eliminate these closure-constraints and convert the system into a tree-topology system by embedding the constraints directly into the system dynamics and effectively replacing the body groupings with virtual aggregate bodies. Once eliminated, one can apply the well-known results and algorithms for tree-topology systems to solve the dynamics of such closed-chain system.

Woo, Simon S.↗
NASA NTRS1981

Two-boundary grid generation for the solution of the three dimensional compressible Navier-Stokes equations

A grid generation technique called the two boundary technique is developed and applied for the solution of the three dimensional Navier-Stokes equations. The Navier-Stokes equations are transformed from a cartesian coordinate system to a computational coordinate system, and the grid generation technique provides the Jacobian matrix describing the transformation. The two boundary technique is based on algebraically defining two distinct boundaries of a flow domain and the distribution of the grid is achieved by applying functions to the uniform computational grid which redistribute the computational independent variables and consequently concentrate or disperse the grid points in the physical domain. The Navier-Stokes equations are solved using a MacCormack time-split technique. Grids and supersonic laminar flow solutions are obtained for a family of three dimensional corners and two spike-nosed bodies.

Smith, R. E.↗
NASA NTRS1996

Bit Error Probability for Maximum Likelihood Decoding of Linear Block Codes

In this paper, the bit error probability P(sub b) for maximum likelihood decoding of binary linear codes is investigated. The contribution of each information bit to P(sub b) is considered. For randomly generated codes, it is shown that the conventional approximation at high SNR P(sub b) is approximately equal to (d(sub H)/N)P(sub s), where P(sub s) represents the block error probability, holds for systematic encoding only. Also systematic encoding provides the minimum P(sub b) when the inverse mapping corresponding to the generator matrix of the code is used to retrieve the information sequence. The bit error performances corresponding to other generator matrix forms are also evaluated. Although derived for codes with a generator matrix randomly generated, these results are shown to provide good approximations for codes used in practice. Finally, for decoding methods which require a generator matrix with a particular structure such as trellis decoding or algebraic-based soft decision decoding, equivalent schemes that reduce the bit error probability are discussed.

Lin, Shu↗
NASA NTRS2006

Development of Comprehensive Reduced Kinetic Models for Supersonic Reacting Shear Layer Simulations

Large-scale simulations of multi-dimensional unsteady turbulent reacting flows with detailed chemistry and transport can be computationally extremely intensive even on distributed computing architectures. With the development of suitable reduced chemical kinetic models, the number of scalar variables to be integrated can be decreased, leading to a significant reduction in the computational time required for the simulation with limited loss of accuracy in the results. A general MATLAB-based automated mechanism reduction procedure is presented to reduce any complex starting mechanism (detailed or skeletal) with minimal human intervention. Based on the application of the quasi steady-state (QSS) approximation for certain chemical species and on the elimination of the fast reaction rates in the mechanism, several comprehensive reduced models, capable of handling different fuels such as C2H4, CH4 and H2, have been developed and thoroughly tested for several combustion problems (ignition, propagation and extinction) and physical conditions (reactant compositions, temperatures, and pressures). A key feature of the present reduction procedure is the explicit solution of the concentrations of the QSS species, needed for the evaluation of the elementary reaction rates. In contrast, previous approaches relied on an implicit solution due to the strong coupling between QSS species, requiring computationally expensive inner iterations. A novel algorithm, based on the definition of a QSS species coupling matrix, is presented to (i) introduce appropriate truncations to the QSS algebraic relations and (ii) identify the optimal sequence for the explicit solution of the concentration of the QSS species. With the automatic generation of the relevant source code, the resulting reduced models can be readily implemented into numerical codes.

Zambon, A. C.↗
DOE OSTI2023

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

Here, we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H 1 , H(curl), and H(div) and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart–Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any “black-box” preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

RE-INTEGRATE EMT Simulation Software: Graph Convolutional Network for Sparse Matrix Pattern Detection

The increasing complexity of power networks, driven by proliferation of inverters, presents analytical challenges that simplified models often fail to capture, necessitating Electromagnetic Transient (EMT) simulations. EMT models are represented as discretized differential-algebraic equations (DAEs), forming a linear system Ax = b that is computationally intensive to solve. Due to inherent sparsity of adjacency matrix A, distinct patterns emerge that, when accurately identified, enable efficient solver selection to minimize computation time. However, identifying ideal pattern is complicated by numerous reordering algorithms and limited structural insights. To address this, we introduce a Graph Convolutional Network (GCN) model for classifying sparse matrix patterns common in power system analysis. The model, achieving 96% test accuracy, is validated using PV plant models of 125 MW capacities connected to New England 39-bus transmission system (TS), and further scaled to a 4,992-bus network with 384 PV plants, yielding 191, 616 × 191, 616 sized A matrix. For all cases, the GCN model accurately identifies the matrix’s intrinsic sparse pattern, demonstrating its potential to enhance solver performance in EMT analysis.

Hossain, Md Rifat [Florida International Universit↗
DOE OSTI2024

Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in \(\boldsymbol{H}(\operatorname{\textbf{div}})\)

Here, this work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in H(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation–histopolation basis, efficient matrix-free preconditioners can be constructed for the (1, 1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the “crooked pipe” grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Compressed basis GMRES on high-performance graphics processing units

Krylov methods provide a fast and highly parallel numerical tool for the iterative solution of many large-scale sparse linear systems. To a large extent, the performance of practical realizations of these methods is constrained by the communication bandwidth in current computer architectures, motivating the investigation of sophisticated techniques to avoid, reduce, and/or hide the message-passing costs (in distributed platforms) and the memory accesses (in all architectures). This article leverages Ginkgo’s memory accessor in order to integrate a communication-reduction strategy into the (Krylov) GMRES solver that decouples the storage format (i.e., the data representation in memory) of the orthogonal basis from the arithmetic precision that is employed during the operations with that basis. Given that the execution time of the GMRES solver is largely determined by the memory accesses, the cost of the datatype transforms can be mostly hidden, resulting in the acceleration of the iterative step via a decrease in the volume of bits being retrieved from memory. Together with the special properties of the orthonormal basis (whose elements are all bounded by 1), this paves the road toward the aggressive customization of the storage format, which includes some floating-point as well as fixed-point formats with mild impact on the convergence of the iterative process. We develop a high-performance implementation of the “compressed basis GMRES” solver in the Ginkgo sparse linear algebra library using a large set of test problems from the SuiteSparse Matrix Collection. We demonstrate robustness and performance advantages on a modern NVIDIA V100 graphics processing unit (GPU) of up to 50% over the standard GMRES solver that stores all data in IEEE double-precision.

97 MATHEMATICS AND COMPUTING↗