Search NASA⌕ Search

SEARCH · Search NASA

Results for “Symmetric positive definite”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Model Reduction by Generalized Falk Method for Efficient Field-Circuit Simulations

The Generalized Falk Method (GFM) for coordinate transformation, together with two model-reduction strategies based on this method, are presented for efficient coupled field-circuit simulations. Each model-reduction strategy is based on a decision to retain specific linearly-independent vectors, called trial vectors, to construct a vector basis for coordinate transformation. The reduced-order models are guaranteed to be stable and passive since the GFM is a congruence transformation of originally symmetric positive definite systems. We also show that, unlike the Pad´e-via-Lanczos (PVL) method, the GFM does not generate unstable positive poles while reducing the order of circuit problems. Further, the proposed GFM is also faster when compared to methods of the type Lanczos (or Krylov) that are already widely used in circuit simulations for electrothermal and electromagnetic problems. The concept of response participation factors is introduced for the selection of the trial vectors in the proposed model-reduction methods. Further, we present methods to develop simple equivalent circuit networks for the field component of the overall field-circuit system. The implementation of these equivalent circuit networks in circuit simulators is discussed. With the proposed model-reduction strategies, significant improvement on the efficiency of the generalized Falk method is illustrated for coupled field-circuit problems.

42 ENGINEERING↗
NASA NTRS1970

Sorting and ordering sparse linear systems

Transformations of sparse linear systems by row-column permutations are considered and various algorithms are constructed to transform arbitrary symmetric positive definite sparse matrices, as well as matrices in band form, doubly bordered band form, and doubly bordered block diagonal form.

Tewarson, R. P.↗
NASA NTRS1976

Multivariate normal integration

Monte Carlo program evaluates integrals over rectangular regions for dimensions less than six and over elliptical regions in bivariate case. Program gives positive definite symmetric variance/covariance matrix factorization and calculates reciprocal of lower triangular matrix and product of diagonal elements of triangular matrix.

Falls, L. W.↗
NASA NTRS1976

Recursive partitioned inversion of large (1500 x 1500) symmetric matrices

A recursive algorithm was designed to invert large, dense, symmetric, positive definite matrices using small amounts of computer core, i.e., a small fraction of the core needed to store the complete matrix. The described algorithm is a generalized Gaussian elimination technique. Other algorithms are also discussed for the Cholesky decomposition and step inversion techniques. The purpose of the inversion algorithm is to solve large linear systems of normal equations generated by working geodetic problems. The algorithm was incorporated into a computer program called SOLVE. In the past the SOLVE program has been used in obtaining solutions published as the Goddard earth models.

Putney, B. H.↗
NASA NTRS1976

Computation and sensitivity considerations of the Ricatti equation in spacecraft CMG steering laws

The steady-state matrix Ricatti equation is studied with respect to sensitivity and computational burden for a linear regulator controller in spacecraft. The class of systems studied use control moment gyros (CMGs) as the torque driver, assuming that the CMGs are free of gimbal stops. Different solution techniques are evaluated relative to computation time and accuracy to determine which solution method is most acceptable with respect to the prescribed requirements. It is shown that it is not feasible to eliminate off-diagonal Ricatti equation terms in the relevent updata equation due to their high sensitivity, that Potter's algebraic technique requires the least computation time of the methods studied, and that the simple Euler method yields the best results when accuracy and computation time are equally weighted. If an optimal linear regulator formulation is employed, a full Ricatti equation solution of n(n + 1)2 elements of the symmetric positive definite gain matrix which is the solution to the matrix Ricatti equation must be computed at each update cycle.

Colburn, B. K.↗
NASA NTRS1978

A split band-Cholesky equation solving strategy for finite element analysis of transient field problems

The paper describes the split-Cholesky strategy for banded matrices arising from the large systems of equations in certain fluid mechanics problems. The basic idea is that for a banded matrix the computation can be carried out in pieces, with only a small portion of the matrix residing in core. Mesh considerations are discussed by demonstrating the manner in which the assembly of finite element equations proceeds for linear trial functions on a triangular mesh. The FORTRAN code which implements the out-of-core decomposition strategy for banded symmetric positive definite matrices (mass matrices) of a coupled initial value problem is given.

Cooke, C. H.↗
NASA NTRS1982

Quadratic integer programming for large scale banded matrices

This paper is concerned with the integer quadratic program where the variables are constrained to belong to a given set of discrete values. This quadratic integer program is shown to be equivalent to a problem of finding the shortest path in a particular directed graph called a trellis when the matrix is a positive-definite symmetric banded matrix. An efficient procedure for solving this shortest path problem is presented which allows the solution of the integer quadratic program. This method is particularly effective when the half-bandwidth of the matrix is significantly smaller than its dimension.

Yan, T. Y.↗
NASA NTRS1984

Parallel triangularization of substructured finite element problems

Much of the computational effort of the finite element process involves the solution of a system of linear equations. The coefficient matrix of this system, known as the global stiffness matrix, is symmetric, positive definite, and generally sparse. An important technique for reducing the time required to solve this system is substructuring or matrix partitioning. Substructuring is based on the idea of dividing a structure into pieces, each of which can then be analyzed relatively indepenently. As a result of this division, each point in the finite element discretization is either interior to a substructure or on a boundary between substructures. Contributions to the global stiffness matrix from connections between boundary points from the K(bb) matrix are reported. The triangularization of a general K(bb) matrix on a parallel machine is specifically discussed.

Leuze, M. R.↗
NASA NTRS1986

Multi-color incomplete Cholesky conjugate gradient methods for vector computers

In this research, we are concerned with the solution on vector computers of linear systems of equations, Ax = b, where A is a larger, sparse symmetric positive definite matrix. We solve the system using an iterative method, the incomplete Cholesky conjugate gradient method (ICCG). We apply a multi-color strategy to obtain p-color matrices for which a block-oriented ICCG method is implemented on the CYBER 205. (A p-colored matrix is a matrix which can be partitioned into a pXp block matrix where the diagonal blocks are diagonal matrices). This algorithm, which is based on a no-fill strategy, achieves O(N/p) length vector operations in both the decomposition of A and in the forward and back solves necessary at each iteration of the method. We discuss the natural ordering of the unknowns as an ordering that minimizes the number of diagonals in the matrix and define multi-color orderings in terms of disjoint sets of the unknowns. We give necessary and sufficient conditions to determine which multi-color orderings of the unknowns correpond to p-color matrices. A performance model is given which is used both to predict execution time for ICCG methods and also to compare an ICCG method to conjugate gradient without preconditioning or another ICCG method. Results are given from runs on the CYBER 205 at NASA's Langley Research Center for four model problems.

Poole, E. L.↗
NASA NTRS1989

A language comparison for scientific computing on MIMD architectures

Choleski's method for solving banded symmetric, positive definite systems is implemented on a multiprocessor computer using three FORTRAN based parallel programming languages, the Force, PISCES and Concurrent FORTRAN. The capabilities of the language for expressing parallelism and their user friendliness are discussed, including readability of the code, debugging assistance offered, and expressiveness of the languages. The performance of the different implementations is compared. It is argued that PISCES, using the Force for medium-grained parallelism, is the appropriate choice for programming Choleski's method on the multiprocessor computer, Flex/32.

Jones, Mark T.↗
NASA NTRS1989

Data traffic reduction schemes for Cholesky factorization on asynchronous multiprocessor systems

Communication requirements of Cholesky factorization of dense and sparse symmetric, positive definite matrices are analyzed. The communication requirement is characterized by the data traffic generated on multiprocessor systems with local and shared memory. Lower bound proofs are given to show that when the load is uniformly distributed the data traffic associated with factoring an n x n dense matrix using n to the alpha power (alpha less than or equal 2) processors is omega(n to the 2 + alpha/2 power). For n x n sparse matrices representing a square root of n x square root of n regular grid graph the data traffic is shown to be omega(n to the 1 + alpha/2 power), alpha less than or equal 1. Partitioning schemes that are variations of block assignment scheme are described and it is shown that the data traffic generated by these schemes are asymptotically optimal. The schemes allow efficient use of up to O(n to the 2nd power) processors in the dense case and up to O(n) processors in the sparse case before the total data traffic reaches the maximum value of O(n to the 3rd power) and O(n to the 3/2 power), respectively. It is shown that the block based partitioning schemes allow a better utilization of the data accessed from shared memory and thus reduce the data traffic than those based on column-wise wrap around assignment schemes.

Naik, Vijay K.↗
NASA NTRS1989

Parallel algorithms and architecture for computation of manipulator forward dynamics

Parallel computation of manipulator forward dynamics is investigated. Considering three classes of algorithms for the solution of the problem, that is, the O(n), the O(n exp 2), and the O(n exp 3) algorithms, parallelism in the problem is analyzed. It is shown that the problem belongs to the class of NC and that the time and processors bounds are of O(log2/2n) and O(n exp 4), respectively. However, the fastest stable parallel algorithms achieve the computation time of O(n) and can be derived by parallelization of the O(n exp 3) serial algorithms. Parallel computation of the O(n exp 3) algorithms requires the development of parallel algorithms for a set of fundamentally different problems, that is, the Newton-Euler formulation, the computation of the inertia matrix, decomposition of the symmetric, positive definite matrix, and the solution of triangular systems. Parallel algorithms for this set of problems are developed which can be efficiently implemented on a unique architecture, a triangular array of n(n+2)/2 processors with a simple nearest-neighbor interconnection. This architecture is particularly suitable for VLSI and WSI implementations. The developed parallel algorithm, compared to the best serial O(n) algorithm, achieves an asymptotic speedup of more than two orders-of-magnitude in the computation the forward dynamics.

Fijany, Amir↗
NASA NTRS1989

Efficient ICCG on a shared memory multiprocessor

Different approaches are discussed for exploiting parallelism in the ICCG (Incomplete Cholesky Conjugate Gradient) method for solving large sparse symmetric positive definite systems of equations on a shared memory parallel computer. Techniques for efficiently solving triangular systems and computing sparse matrix-vector products are explored. Three methods for scheduling the tasks in solving triangular systems are implemented on the Sequent Balance 21000. Sample problems that are representative of a large class of problems solved using iterative methods are used. We show that a static analysis to determine data dependences in the triangular solve can greatly improve its parallel efficiency. We also show that ignoring symmetry and storing the whole matrix can reduce solution time substantially.

Hammond, Steven W.↗
NASA NTRS1993

Large-scale computation of incompressible viscous flow by least-squares finite element method

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.

Jiang, Bo-Nan↗
NASA NTRS1993

Accuracy of least-squares methods for the Navier-Stokes equations

Recently there has been substantial interest in least-squares finite element methods for velocity-vorticity-pressure formulations of the incompressible Navier-Stokes equations. The main cause for this interest is the fact that algorithms for the resulting discrete equations can be devised which require the solution of only symmetric, positive definite systems of algebraic equations. On the other hand, it is well-documented that methods using the vorticity as a primary variable often yield very poor approximations. Thus, here we study the accuracy of these methods through a series of computational experiments, and also comment on theoretical error estimates. It is found, despite the failure of standard methods for deriving error estimates, that computational evidence suggests that these methods are, at the least, nearly optimally accurate. Thus, in addition to the desirable matrix properties yielded by least-squares methods, one also obtains accurate approximations.

Bochev, Pavel B.↗
NASA NTRS1993

A least-squares finite element method for 3D incompressible Navier-Stokes equations

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Jiang, Bo-Nan↗
NASA NTRS1993

Accuracy of least-squares methods for the Navier-Stokes equations

Recently there has been substantial interest in least-squares finite element methods for velocity-vorticity-pressure formulations of the incompressible Navier-Stokes equations. The main cause for this interest is the fact that algorithms for the resulting discrete equations can be devised which require the solution of only symmetric, positive definite systems of algebraic equations. On the other hand, it is well-documented that methods using the vorticity as a primary variable often yield very poor approximations. Thus, here we study the accuracy of these methods through a series of computational experiments, and also comment on theoretical error estimates. It is found, despite the failure of standard methods for deriving error estimates, that computational evidence suggests that these methods are, at the least, nearly optimally accurate. Thus, in addition to the desirable matrix properties yielded by least-squares methods, one also obtains accurate approximations.

Bochev, Pavel B.↗