Simultaneous solution of non-linear algebraic equations Technical memorandum no. 163
Digital computer program for nonlinear equation sets characteristic of filter circuit analysis
NONLINEAR EQUATION↗
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Computer subroutine ISUDS accurately solves large system of simultaneous linear algebraic equations
Computer program, an Iterative Scheme Using a Direct Solution, obtains double precision accuracy using a single-precision coefficient matrix. ISUDS solves a system of equations written in matrix form as AX equals B, where A is a square non-singular coefficient matrix, X is a vector, and B is a vector.
Collier, G.↗
Leapfrog variants of iterative methods for linear algebra equations
Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.
Saylor, Paul E.↗
Computational strategies and improvements in the linear algebraic variational approach to rearrangement scattering
The computational steps in calculating quantum mechanical reactive scattering amplitudes by the L2 generalized Newton variational principle are discussed with emphasis on computational strategies and recent improvements that make the calculations more efficient. Special emphasis is placed on quadrature techniques, storage management strategies, use of symmetry, and boundary conditions. It is concluded that an efficient implementation of these procedures provides a powerful algorithm for the accurate solution of the Schroedinger equation for rearrangements.
Schwenke, David W.↗
A high-speed linear algebra library with automatic parallelism
Parallel or distributed processing is key to getting highest performance workstations. However, designing and implementing efficient parallel algorithms is difficult and error-prone. It is even more difficult to write code that is both portable to and efficient on many different computers. Finally, it is harder still to satisfy the above requirements and include the reliability and ease of use required of commercial software intended for use in a production environment. As a result, the application of parallel processing technology to commercial software has been extremely small even though there are numerous computationally demanding programs that would significantly benefit from application of parallel processing. This paper describes DSSLIB, which is a library of subroutines that perform many of the time-consuming computations in engineering and scientific software. DSSLIB combines the high efficiency and speed of parallel computation with a serial programming model that eliminates many undesirable side-effects of typical parallel code. The result is a simple way to incorporate the power of parallel processing into commercial software without compromising maintainability, reliability, or ease of use. This gives significant advantages over less powerful non-parallel entries in the market.
Boucher, Michael L.↗
Accuracy requirements of optical linear algebra processors in adaptive optics imaging systems
A ground-based adaptive optics imaging telescope system attempts to improve image quality by detecting and correcting for atmospherically induced wavefront aberrations. The required control computations during each cycle will take a finite amount of time. Longer time delays result in larger values of residual wavefront error variance since the atmosphere continues to change during that time. Thus an optical processor may be well-suited for this task. This paper presents a study of the accuracy requirements in a general optical processor that will make it competitive with, or superior to, a conventional digital computer for the adaptive optics application. An optimization of the adaptive optics correction algorithm with respect to an optical processor's degree of accuracy is also briefly discussed.
Downie, John D.↗
Improved Linear Algebra Methods for Redshift Computation from Limited Spectrum Data - II
Given photometric broadband measurements of a galaxy, Gaussian processes may be used with a training set to solve the regression problem of approximating the redshift of this galaxy. However, in practice solving the traditional Gaussian processes equation is too slow and requires too much memory. We employed several methods to avoid this difficulty using algebraic manipulation and low-rank approximation, and were able to quickly approximate the redshifts in our testing data within 17 percent of the known true values using limited computational resources. The accuracy of one method, the V Formulation, is comparable to the accuracy of the best methods currently used for this problem.
Foster, Leslie↗
Simulating Image Coaddition with the Nancy Grace Roman Space Telescope. III. Software Improvements and New Linear Algebra Strategies
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Kaili 开力 Cao 曹↗
Variational quantum and neural quantum states algorithms for the linear complementarity problem
Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems—and whether quantum-inspired classical algorithms can match their performance—remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid-body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modelling certain physical systems.
neural quantum states↗
On recent advances and future research directions for computational fluid dynamics
This paper highlights some recent accomplishments regarding CFD numerical algorithm constructions for generation of discrete approximate solutions to classes of Reynolds-averaged Navier-Stokes equations. Following an overview of turbulent closure modeling, and development of appropriate conservation law systems, a Taylor weak-statement semi-discrete approximate solution algorithm is developed. Various forms for completion to the final linear algebra statement are cited, as are a range of candidate numerical linear algebra solution procedures. This development sequence emphasizes the key building blocks of a CFD RNS algorithm, including solution trial and test spaces, integration procedure and added numerical stability mechanisms. A range of numerical results are discussed focusing on key topics guiding future research directions.
Baker, A. J.↗
Improved Evaluation of Large Network Matrices for Linear Power Flow Within Optimization Problems
This work presents methods for evaluating the Power Transfer Distribution Factor (PTDF) and Line Outage Distribution Factor (LODF) matrices by employing sparse linear algebra for large-scale computing applications. These matrices play a critical role in many power system applications, such as the Unit Commitment Problem (UC), pre- and post-contingency power flow analysis, and transmission expansion. These matrices are typically dense, which means they require a significant amount of time and memory to be computed for large networks. However, by analyzing the structure of the matrices and their computation method, it is possible to use reduced memory methods based on sparse matrix operations. This paper shows that sparse linear algebra algorithms are faster and require less memory and time than traditional dense approaches. Additionally, we explore the effect of matrix sparsification by eliminating trailing digits on power flow calculations.
large scale↗
Lie algebras and linear differential equations.
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Brockett, R. W.↗
Performance of direct and iterative algorithms on an optical systolic processor
The frequency-multiplexed optical linear algebra processor (OLAP) is treated in detail with attention to its performance in the solution of systems of linear algebraic equations (LAEs). General guidelines suitable for most OLAPs, including digital-optical processors, are advanced concerning system and component error source models, guidelines for appropriate use of direct and iterative algorithms, the dominant error sources, and the effect of multiple simultaneous error sources. Specific results are advanced on the quantitative performance of both direct and iterative algorithms in the solution of systems of LAEs and in the solution of nonlinear matrix equations. Acoustic attenuation is found to dominate iterative algorithms and detector noise to dominate direct algorithms. The effect of multiple spatial errors is found to be additive. A theoretical expression for the amount of acoustic attenuation allowed is advanced and verified. Simulations and experimental data are included.
Ghosh, A. K.↗
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↗
A Contextually-Aware Sensitivity Analysis to Guide the Design of Randomized Least Squares Solvers in Applications
Our work on the DOE-sponsored project “A Contextually-Aware Sensitivity Analysis to Guide the Design of Randomized Least Squares Solvers in Applications,” was an effort to address critical challenges in nu merical computing and its applications to optimization. The increasing demand for robust and scalable solutions to large-scale linear algebra problems has highlighted the limitations of traditional approaches, particularly in heterogeneous and extreme-scale computing environments. Randomized Numerical Linear Algebra (RandNLA) offers a promising framework to address these challenges, and this proposal builds on this foundation by introducing innovations in sensitivity analysis and computational adaptability.
97 MATHEMATICS AND COMPUTING↗
Developing Information Power Grid Based Algorithms and Software
This was an exploratory study to enhance our understanding of problems involved in developing large scale applications in a heterogeneous distributed environment. It is likely that the large scale applications of the future will be built by coupling specialized computational modules together. For example, efforts now exist to couple ocean and atmospheric prediction codes to simulate a more complete climate system. These two applications differ in many respects. They have different grids, the data is in different unit systems and the algorithms for inte,-rating in time are different. In addition the code for each application is likely to have been developed on different architectures and tend to have poor performance when run on an architecture for which the code was not designed, if it runs at all. Architectural differences may also induce differences in data representation which effect precision and convergence criteria as well as data transfer issues. In order to couple such dissimilar codes some form of translation must be present. This translation should be able to handle interpolation from one grid to another as well as construction of the correct data field in the correct units from available data. Even if a code is to be developed from scratch, a modular approach will likely be followed in that standard scientific packages will be used to do the more mundane tasks such as linear algebra or Fourier transform operations. This approach allows the developers to concentrate on their science rather than becoming experts in linear algebra or signal processing. Problems associated with this development approach include difficulties associated with data extraction and translation from one module to another, module performance on different nodal architectures, and others. In addition to these data and software issues there exists operational issues such as platform stability and resource management.
Dongarra, Jack↗
Efficient matrix partitioning for optical computing
Techniques for partitioning optical linear algebra problems to make them amenable to solution using optical processors programmed with simple algorithms are explored. Generalized methods for splitting a linear algebra matrix into a series of submatrices are reviewed, showing that simple forms can be pipelined smoothly and that parallel accumulation can be achieved by beam combining on detectors or by summing electronically. The techniques offer simplified bookkeeping, algorithmic independence, and high efficiency. The computational speed will depend on the number of multiplier-accumulators devoted to the task.
Caulfield, H. J.↗
ORACLS: A system for linear-quadratic-Gaussian control law design
A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.
Armstrong, E. S.↗