A new method of recursive estimation in discrete linear systems
Recursive estimation equation giving algorithms for prediction, filtering, and signal smoothing in discrete linear systems
Kashyap, R. L.↗
SEARCH · Engineering Papers
Results for “Linear Systems of Equations”
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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Application of direct solvers to unstructured meshes for the Euler and Navier-Stokes equations using upwind schemes
The application of Newton iteration to inviscid and viscous airfoil calculations on unstructured meshes is examined. A cell-centered finite volume scheme is employed on an unstructured mesh consisting of triangles. Roe's flux difference splitting scheme is used to compute the inviscid fluxes. Higher order accuracy is achieved by an interpolation procedure that makes use of auxiliary gradients. The efficient solution of the sparse linear system of equations which arises upon linearization in time is addressed. Results are presented for inviscid and viscous test cases. The complications which arise due to the introduction of nonlinear limiters are addressed.
Venkatakrishnan, V.↗
Scalable Solutions to Integral Equation and Finite Element Simulations
The application of advanced computer architecture and software to a broad range of electromagnetic problems had allowed more accurate simulations of electrically larger and more complex components and systems...This paper overviews solutions to Maxwell's equations implicitly defined through systems of linear equations.
parallel↗
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.↗
Solution of structural analysis problems on a parallel computer
The problems of a blade-stiffened panel with a hole subjected to compression, and a deployable space mast subjected to tip loads, are treated through the application of FEM to model generation followed by the solution of a linear system of equations. Direct and iterative approaches to the solution of the linear systems are solved in turn; for the panel problems using varying numbers of processors, the incomplete Cholesky-conjugate gradient method was the fastest iterative method on all but two instances in which the number of processors was large.
Storaasli, Olaf↗
Greedy emulators for nuclear two-body scattering
Applications of reduced basis method emulators are increasing in low-energy nuclear physics because they enable fast and accurate sampling of high-fidelity calculations, enabling robust uncertainty quantification. Here, in this paper, we develop, implement, and test two model-driven emulators based on the (Petrov-)Galerkin projection using the prototypical test case of two-body scattering with the Minnesota potential and a more realistic local chiral potential. The high-fidelity scattering equations are solved with the matrix Numerov method, a reformulation of the popular Numerov recurrence relation for solving special second-order differential equations as a linear system of coupled equations. A novel error estimator based on reduced-space residuals is applied to an active learning approach (a greedy algorithm) to choosing training samples (“snapshots”) for the emulator and contrasted with a proper orthogonal decomposition (POD) approach. Both approaches allow for computationally efficient offline-online decompositions, but the greedy approach requires many fewer snapshot calculations. These developments set the groundwork for emulating scattering observables based on chiral nucleon-nucleon and three-nucleon interactions and optical models, where computational speed-ups are necessary for Bayesian uncertainty quantification. Our emulators and error estimators are widely applicable to linear systems.
Bayesian methods↗
Aeroelastic instabilities of hingeless helicopter blades.
In this study the stability boundaries of a hingeless helicopter blade are studied using a system of coupled flap-lag-pitch equations of motion. Divergence and flutter boundaries for the linearized system of equations are presented. Flap-lag, flap-pitch and coupled flap-lag-pitch instabilities in hovering flight are studied. The effect of the torsional degree of freedom on the flap-lag type of instability is investigated. Similarly the effect of lag on the flap pitch type of instability is also considered. Results illustrating these effects, together with the effect of various important blade parameters on blade stability are presented.
Friedmann, P.↗
Dynamic model for an instrument pointing system
The instrument pointing system concept discussed in the present paper was developed for Spacelab missions. The system is mounted on soft shockmounts to minimize Shuttle disturbances (such as man motion, thruster firings, etc.) and to reduce the effect of large center of mass offsets between the instrument and the pointing system gimbal axes. The instrument pointing system is soft mounted to the Spacelab pallet located in the Orbiter payload bay. The nonlinear model, incorporating the dynamic equations of motion, the controller equations and the definition of sensor dynamics and gimbal bearing friction, is given, along with a system of linearized equations of motion of the Space Shuttle Orbiter and instrument pointing system (linearized model).
Howell, J. T.↗
Multigrid and Krylov Subspace Methods for the Discrete Stokes Equations
Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations, a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper, we compare the performance of four such methods: variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual, and multigrid methods, for solving several two-dimensional model problems. The results indicate that where it is applicable, multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being both independent of iteration parameters and widely applicable.
Elman, Howard C.↗
Should Pruning be a Pre-Processor of any Linear System?
There are many real-world problems whose mathematical models turn out to be linear systems Ax = b, where A is an m x n matrix. Each equation of the linear system is an information. An information, in a physical problem, such as 4 mangoes, 6 bananas, and 5 oranges cost $10, is mathematically modeled as an equation 4x(sub 1) + 6x(sub 2) + 5x(sub 3) = 10 , where x(sub 1), x(sub 2), x(sub 3) are each cost of one mango, that of one banana, and that of one orange, respectively. All the information put together in a specified context, constitutes the physical problem and need not be all distinct. Some of these could be redundant, which cannot be readily identified by inspection. The resulting mathematical model will thus have equations corresponding to this redundant information and hence are linearly dependent and thus superfluous. Consequently, these equations once identified should be better pruned in the process of solving the system. The benefits are (i) less computation and hence less error and consequently a better quality of solution and (ii) reduced storage requirements. In literature, the pruning concept is not in vogue so far although it is most desirable. It is assumed that at least one information, i.e. one equation is known to be correct and which will be our first equation. In a numerical linear system, the system could be slightly inconsistent or inconsistent of varying degree. If the system is too inconsistent, then we should fall back on to the physical problem (PP), check the correctness of the PP derived from the material universe, modify it, if necessary, and then check the corresponding mathematical model (MM) and correct it. In nature/material universe, inconsistency is completely nonexistent. If the MM becomes inconsistent, it could be due to error introduced by the concerned measuring device and/or due to assumptions made on the PP to obtain an MM which is relatively easily solvable or simply due to human error. No measuring device can usually measure a quantity with an accuracy greater that 0.005% or, equivalently with a relative error less than 0.005%. Hence measurement error is unavoidable in a numerical linear system when the quantities are continuous (or even discrete with extremely large number). Assumptions, though not desirable, are usually made when we find the problem sufficiently difficult to be solved within the available means/tools/resources and hence distort the PP and the corresponding MM. The . error thus introduced in the system could (not always necessarily though) make the system somewhat inconsistent. If the inconsistency (contradiction) is too much then one should definitely not proceed to solve the system in terms of getting a least-squares solution or the minimum-norm least-squares solution. All these solutions will be invariably of no real-world use. If, on the other hand, inconsistency is reasonably low, i.e. the system is near-consistent or, equivalently, has near-linearly-dependent rows, then the foregoing solutions are useful. Pruning in such a near-consistent system should be performed based on the desired accuracy and on the definition of near-linear dependence. In this article, we discuss pruning over various kinds of linear systems and strongly suggest its use as a pre-processor or as a part of an algorithm. Ideally pruning should (i) be a part of the solution process (algorithm) of the system, (ii) reduce both computational error and complexity of the process, and (iii) take into account the numerical zero defined in the context. These are precisely what we achieve through our proposed O(mn2) algorithm presented in Matlab, that uses a subprogram of solving a single linear equation and that has embedded in it the pruning.
Sen, Syamal K.↗
Planar Cubics Through a Point in a Direction
It is shown that the planar cubics through three points and the associated tangent directions can be found by solving a cubic equation and a 2 x 2 system of linear equations. The result is combined with a previous published scheme to produce a better curve-fitting method.
Chou, J. J.↗
Simulation of an inductively coupled plasma with a two-dimensional Darwin particle-in-cell code
A two-dimensional particle-in-cell code for the simulation of low-frequency electromagnetic processes in laboratory plasmas has been developed. The code uses the Darwin method omitting the electromagnetic wave propagation. The Darwin method separates the electric field into solenoidal and irrotational parts. The solenoidal electric field is calculated with a new algorithm based on the equation for the electric field vorticity. The system of linear equations in the new algorithm is readily solved using a standard iterative method. The irrotational electric field is the electrostatic field calculated with the direct implicit algorithm. The code is verified by reproducing the two-stream instability, electron electromagnetic waves, and shear Alfvén waves. The code is applied to simulate an inductively coupled plasma with the driving current flowing around the plasma region. In this simulation, a ring of dense plasma forms at the initial stage but then the density becomes maximal in the center and decays monotonically toward the walls. The skin effect is in the transitional mode between local and non-local, and the electron velocity distribution function is non-Maxwellian.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Sub-optimal control of non-linear single input systems.
Suboptimal control generation for nonlinear single input systems by linear combination of states and repeated linearization of equations system
Aggarwal, J. K.↗
A General Method for Solving Systems of Non-Linear Equations
The method of steepest descent is modified so that accelerated convergence is achieved near a root. It is assumed that the function of interest can be approximated near a root by a quadratic form. An eigenvector of the quadratic form is found by evaluating the function and its gradient at an arbitrary point and another suitably selected point. The terminal point of the eigenvector is chosen to lie on the line segment joining the two points. The terminal point found lies on an axis of the quadratic form. The selection of a suitable step size at this point leads directly to the root in the direction of steepest descent in a single step. Newton's root finding method not infrequently diverges if the starting point is far from the root. However, the current method in these regions merely reverts to the method of steepest descent with an adaptive step size. The current method's performance should match that of the Levenberg-Marquardt root finding method since they both share the ability to converge from a starting point far from the root and both exhibit quadratic convergence near a root. The Levenberg-Marquardt method requires storage for coefficients of linear equations. The current method which does not require the solution of linear equations requires more time for additional function and gradient evaluations. The classic trade off of time for space separates the two methods.
Nachtsheim, Philip R.↗
Iterative solution of large, sparse linear systems on a static data flow architecture - Performance studies
The applicability of static data flow architectures to the iterative solution of sparse linear systems of equations is investigated. An analytic performance model of a static data flow computation is developed. This model includes both spatial parallelism, concurrent execution in multiple PE's, and pipelining, the streaming of data from array memories through the PE's. The performance model is used to analyze a row partitioned iterative algorithm for solving sparse linear systems of algebraic equations. Based on this analysis, design parameters for the static data flow architecture as a function of matrix sparsity and dimension are proposed.
Reed, D. A.↗
Optimal filtering and filter stability of linear stochastic delay systems
Optimal filtering equations are obtained for very general linear stochastic delay systems. Stability of the optimal filter is studied in the case where there are no delays in the observations. Using the duality between linear filtering and control, asymptotic stability of the optimal filter is proved. Finally, the cascade of the optimal filter and the deterministic optimal quadratic control system is shown to be asymptotically stable as well.
Kwong, R. H.-S.↗
Tensor-GMRES method for large sparse systems of nonlinear equations
This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.
Feng, Dan↗
Limitations of Fault-Tolerant Quantum Linear System Solvers for Quantum Power Flow
Quantum computers hold promise for solving problems intractable for classical computers, especially those with high time or space complexity. Practical quantum advantage can be said to exist for such problems when the end-to-end time for solving such a problem using a classical algorithm exceeds that required by a quantum algorithm. Reducing the power flow (PF) problem into a linear system of equations allows for the formulation of quantum PF (QPF) algorithms, which are based on solving methods for quantum linear systems such as the Harrow-Hassidim-Lloyd (HHL) algorithm. Speedup from using QPF algorithms is often claimed to be exponential when compared to classical PF solved by state-of-the-art algorithms. Here, we investigate the potential for practical quantum advantage in solving QPF compared to classical methods on gate-based quantum computers. Notably, this paper does not present a new QPF solving algorithm but scrutinizes the end-to-end complexity of the QPF approach, providing a nuanced evaluation of the purported quantum speedup in this problem. Our analysis establishes a best-case bound for the HHL-based quantum power flow complexity, conclusively demonstrating that the HHL-based method has higher runtime complexity compared to the classical algorithm for solving the direct current power flow (DCPF) and fast decoupled load flow (FDLF) problem. Notably, our analysis and conclusions can be extended to any quantum linear system solver with rigorous performance guarantees, based on the known complexity lower bounds for this problem. Additionally, we establish that for potential practical quantum advantage (PQA) to exist it is necessary to consider DCPF-type problems with a very narrow range of condition number values and readout requirements.
29 ENERGY PLANNING, POLICY, AND ECONOMY↗