Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “residual minimization”

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 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

A biconjugate gradient type algorithm on massively parallel architectures

The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine.

Freund, Roland W.↗
NASA NTRS1990

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2

It is shown how the look-ahead Lanczos process (combined with a quasi-minimal residual QMR) approach) can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included.

Freund, Roland W.↗
NASA NTRS1989

Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices

We consider conjugate gradient type methods for the solution of large sparse linear system Ax equals b with complex symmetric coefficient matrices A equals A(T). Such linear systems arise in important applications, such as the numerical solution of the complex Helmholtz equation. Furthermore, most complex non-Hermitian linear systems which occur in practice are actually complex symmetric. We investigate conjugate gradient type iterations which are based on a variant of the nonsymmetric Lanczos algorithm for complex symmetric matrices. We propose a new approach with iterates defined by a quasi-minimal residual property. The resulting algorithm presents several advantages over the standard biconjugate gradient method. We also include some remarks on the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.

Freund, Roland↗
NASA NTRS1992

Upper bounds for convergence rates of vector extrapolation methods on linear systems with initial iterations

The application of the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) to a vector sequence obtained by the linear iterative technique x(sub j) + 1 = Ax(sub j) = b,j = 1,2,..., is considered. Both methods produce a two dimensional array of approximations s(sub n,k) to the solution of the system (I - A)x = b. Here, s(sub n,k) is obtained from the vectors x(sub j), n is less than or equal to j is less than or equal to n + k + 1. It was observed in an earlier publication by the first author that the sequence s(sub n,k), k = 1,2,..., for n greater than 0, but fixed, possesses better convergence properties than the sequence s(sub 0,k), k = 1,2,.... A detailed theoretical explanation for this phenomenon is provided in the present work. This explanation is heavily based on approximations by incomplete polynomials. It is demonstrated by numerical examples when the matrix A is sparse that cycling with s(sub n,k) for n greater than 0, but fixed, produces better convergence rates and costs less computationally than cycling with s(sub 0,k). It is also illustrated numerically with a convection-diffusion problem that the former may produce excellent results where the latter may fail completely. As has been shown in an earlier publication, the results produced by s(sub 0,k) are identical to the corresponding results obtained by applying the Arnoldi method or generalized minimal residual scheme (GMRES) to the system (I - A)x = b.

Sidi, Avram↗
NASA NTRS1992

Development of iterative techniques for the solution of unsteady compressible viscous flows

A research effort was initiated at Georgia Tech in February 1991 on the development of efficient techniques for the computation of 2-D and 3-D unsteady compressible flow problems. It was found that in 2-D unsteady viscous flow applications, the generalized minimal residual (GMRES) scheme was able to significantly improve the accuracy and stability characteristics of an existing 2-D ADI (Alternating Direction Implicit) time marching scheme. That is, the GMRES/ADI combination allowed 10 to 20 times larger time steps compared to an ADI scheme. Because the GMRES algorithm requires 5 to 10 times the CPU work compared to the ADI scheme, the combined GMRES/ADI scheme yields a net factor of 2 savings in CPU cost. During the past year, we also experimented with GMRES/multigrid/ADI combination. The purpose of this combination was to compute the low frequency components of the change in the flow properties from one time step to the next on a coarse grid. This strategy reduces the memory requirements of the GMRES method roughly by a factor of 4-8 for steady flow problems.

Sankar, Lakshmi N.↗
NASA NTRS1991

A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids

A solution procedure for solving nonlinear time-marching problems is presented. The nonsymmetric systems of equations arising from a Newton-type linearization of these time-marching problems are solved using an iterative strategy based on the generalized minimal residual (GMRES) algorithm. Matrix-free techniques leading to reduction in storage are presented. Incorporation of a linesearch algorithm in the Newton-GMRES scheme is discussed. An automatic time-increment control strategy is developed to increase the stability of the time-marching process. High-speed flow computations demonstrate the effectiveness of these algorithms.

Johan, Zdenek↗
NASA NTRS1991

Krylov Subspace Methods for Complex Non-Hermitian Linear Systems

We consider Krylov subspace methods for the solution of large sparse linear systems Ax = b with complex non-Hermitian coefficient matrices. Such linear systems arise in important applications, such as inverse scattering, numerical solution of time-dependent Schrodinger equations, underwater acoustics, eddy current computations, numerical computations in quantum chromodynamics, and numerical conformal mapping. Typically, the resulting coefficient matrices A exhibit special structures, such as complex symmetry, or they are shifted Hermitian matrices. In this paper, we first describe a Krylov subspace approach with iterates defined by a quasi-minimal residual property, the QMR method, for solving general complex non-Hermitian linear systems. Then, we study special Krylov subspace methods designed for the two families of complex symmetric respectively shifted Hermitian linear systems. We also include some results concerning the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.

Freund, Roland W.↗
NASA NTRS1992

Implementation details of the coupled QMR algorithm

The original quasi-minimal residual method (QMR) relies on the three-term look-ahead Lanczos process, to generate basis vectors for the underlying Krylov subspaces. However, empirical observations indicate that, in finite precision arithmetic, three-term vector recurrences are less robust than mathematically equivalent coupled two-term recurrences. Therefore, we recently proposed a new implementation of the QMR method based on a coupled two-term look-ahead Lanczos procedure. In this paper, we describe implementation details of this coupled QMR algorithm, and we present results of numerical experiments.

Freund, Roland W.↗
NASA NTRS1993

A comparison of two methods for solving 3-D unsteady compressible viscous flows

Numerical solutions of 3D unsteady compressible flows for several steady and unsteady cases are obtained using two procedures - a noniterative Alternating Direction Implicit (ADI) scheme and a Generalized Minimal RESidual (GMRES) method. Results obtained are compared with each other and also with experimental data. It is found that the GMRES procedure can provide significant speedups compared to the noniterative ADI procedure.

Hixon, Ray↗
NASA NTRS1994

Discrete sensitivity derivatives of the Navier-Stokes equations with a parallel Krylov solver

This paper solves an 'incremental' form of the sensitivity equations derived by differentiating the discretized thin-layer Navier Stokes equations with respect to certain design variables of interest. The equations are solved with a parallel, preconditioned Generalized Minimal RESidual (GMRES) solver on a distributed-memory architecture. The 'serial' sensitivity analysis code is parallelized by using the Single Program Multiple Data (SPMD) programming model, domain decomposition techniques, and message-passing tools. Sensitivity derivatives are computed for low and high Reynolds number flows over a NACA 1406 airfoil on a 32-processor Intel Hypercube, and found to be identical to those computed on a single-processor Cray Y-MP. It is estimated that the parallel sensitivity analysis code has to be run on 40-50 processors of the Intel Hypercube in order to match the single-processor processing time of a Cray Y-MP.

Ajmani, Kumud↗
NASA NTRS1992

An implementation of the QMR method based on coupled two-term recurrences

The authors have proposed a new Krylov subspace iteration, the quasi-minimal residual algorithm (QMR), for solving non-Hermitian linear systems. In the original implementation of the QMR method, the Lanczos process with look-ahead is used to generate basis vectors for the underlying Krylov subspaces. In the Lanczos algorithm, these basis vectors are computed by means of three-term recurrences. It has been observed that, in finite precision arithmetic, vector iterations based on three-term recursions are usually less robust than mathematically equivalent coupled two-term vector recurrences. This paper presents a look-ahead algorithm that constructs the Lanczos basis vectors by means of coupled two-term recursions. Implementation details are given, and the look-ahead strategy is described. A new implementation of the QMR method, based on this coupled two-term algorithm, is described. A simplified version of the QMR algorithm without look-ahead is also presented, and the special case of QMR for complex symmetric linear systems is considered. Results of numerical experiments comparing the original and the new implementations of the QMR method are reported.

Freund, Roland W.↗
NASA NTRS1996

Assembling Precise Truss Structures With Minimal Stresses

Improved method of assembling precise truss structures involves use of simple devices. Tapered pins that fit in tapered holes indicate deviations from prescribed lengths. Method both helps to ensure precision of finished structures and minimizes residual stresses within structures.

Sword, Lee F.↗
NASA NTRS2001

The NOAA-NASA CZCS Reanalysis Effort

Satellite observations of global ocean chlorophyll span over two decades. However, incompatibilities between processing algorithms prevent us from quantifying natural variability. We applied a comprehensive reanalysis to the Coastal Zone Color Scanner (CZCS) archive, called the NOAA-NASA CZCS Reanalysis (NCR) Effort. NCR consisted of 1) algorithm improvement (AI), where CZCS processing algorithms were improved using modernized atmospheric correction and bio-optical algorithms, and 2) blending, where in situ data were incorporated into the CZCS AI to minimize residual errors. The results indicated major improvement over the previously available CZCS archive. Global spatial and seasonal patterns of NCR chlorophyll indicated remarkable correspondence with modern sensors, suggesting compatibility. The NCR permits quantitative analyses of interannual and interdecadal trends in global ocean chlorophyll.

Gregg, Watson W.↗
NASA NTRS2002

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

Jothiprasad, Giridhar↗
NASA NTRS1997

Implicit/Multigrid Algorithms for Incompressible Turbulent Flows on Unstructured Grids

An implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described. The foundation of the code is a backward Euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned Generalized Minimal Residual (GMRES) technique. For the GMRES calculations, several techniques are investigated for forming the matrix-vector product. Convergence acceleration is achieved through a multigrid scheme that uses non-nested coarse grids that are generated using a technique described in the present paper. Convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four-element airfoil. Viscous results, which are compared with experimental data, include the turbulent flow over a NACA 4412 airfoil, a three-element airfoil for which Mach number effects are investigated, and three-dimensional flow over a wing with a partial-span flap.

Anderson, W. Kyle↗
NASA NTRS2004

Improved Edge Performance in MRF

The fabrication of large segmented optics requires a polishing process that can correct the figure of a surface to within a short distance from its edges-typically, a few millimeters. The work here is to develop QED's Magnetorheological Finishing (MRF) precision polishing process to minimize residual edge effects.

Shorey, Aric↗
NASA NTRS2010

Miniature Piezoelectric Shaker for Distribution of Unconsolidated Samples to Instrument Cells

The planned Mars Science Laboratory mission requires inlet funnels for channeling unconsolidated powdered samples from the sampling and sieving mechanisms into instrument test cells, which are required to reduce cross-contamination of the samples and to minimize residue left in the funnels after each sample transport. To these ends, a solid-state shaking mechanism has been created that requires low power and is lightweight, but is sturdy enough to survive launch vibration. The funnel mechanism is driven by asymmetrically mounted, piezoelectric flexure actuators that are out of the load path so that they do not support the funnel mass. Each actuator is a titanium, flextensional piezoelectric device driven by a piezoelectric stack. The stack has Invar endcaps with a half-spherical recess. The Invar is used to counteract the change in stress as the actuators are cooled to Mars ambient temperatures. A ball screw is threaded through the actuator frame into the recess to apply pre-stress, and to trap the piezoelectric stack and endcaps in flexure. During the vibration cycle of the flextensional actuator frame, the compression in the piezoelectric stack may decrease to the point that it is unstressed; however, because the ball joint cannot pull, tension in the piezoelectric stack cannot be produced. The actuators are offset at 120 . In this flight design, redundancy is required, so three actuators are used though only one is needed to assist in the movement. The funnel is supported at three contact points offset to the hexapod support contacts. The actuator surface that does not contact the ring is free to expand. Two other configurations can be used to mechanically tune the vibration. The free end can be designed to drive a fixed mass, or can be used to drive a free mass to excite impacts (see figure). Tests on this funnel mechanism show a high density of resonance modes between 1 and 20 kHz. A subset of these between 9 and 12 kHz was used to drive the CheMin actuators at 7 V peak to peak. These actuators could be driven by a single resonance, or swept through a frequency range to decrease the possibility that a portion of the funnel surface was not coincident with a nodal line (line of no displacement). The frequency of actuation can be electrically controlled and monitored and can also be mechanically tuned by the addition of tuning mass on the free end of the actuator. The devices are solid-state and can be designed with no macroscopically moving parts. This design has been tested in a vacuum at both Mars and Earth ambient temperatures ranging from 30 to 25 C

Sherrit, Stewart↗
NASA NTRS2002

Higher Order Bases in a 2D Hybrid BEM/FEM Formulation

The advantages of using higher order, interpolatory basis functions are examined in the analysis of transverse electric (TE) plane wave scattering by homogeneous, dielectric cylinders. A boundary-element/finite-element (BEM/FEM) hybrid formulation is employed in which the interior dielectric region is modeled with the vector Helmholtz equation, and a radiation boundary condition is supplied by an Electric Field Integral Equation (EFIE). An efficient method of handling the singular self-term arising in the EFIE is presented. The iterative solution of the partially dense system of equations is obtained using the Quasi-Minimal Residual (QMR) algorithm with an Incomplete LU Threshold (ILUT) preconditioner. Numerical results are shown for the case of an incident wave impinging upon a square dielectric cylinder. The convergence of the solution is shown versus the number of unknowns as a function of the completeness order of the basis functions.

Fink, Patrick W.↗