Search NASA⌕ Search

SEARCH · Search NASA

Results for “Iterative solvers”

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 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

High-performance equation solvers and their impact on finite element analysis

The role of equation solvers in modern structural analysis software is described. Direct and iterative equation solvers which exploit vectorization on modern high-performance computer systems are described and compared. The direct solvers are two Cholesky factorization methods. The first method utilizes a novel variable-band data storage format to achieve very high computation rates and the second method uses a sparse data storage format designed to reduce the number od operations. The iterative solvers are preconditioned conjugate gradient methods. Two different preconditioners are included; the first uses a diagonal matrix storage scheme to achieve high computation rates and the second requires a sparse data storage scheme and converges to the solution in fewer iterations that the first. The impact of using all of the equation solvers in a common structural analysis software system is demonstrated by solving several representative structural analysis problems.

Poole, Eugene L.↗
NASA NTRS1991

Blade design and analysis using a modified Euler solver

An iterative method for blade design based on Euler solver and described in an earlier paper is used to design compressor and turbine blades providing shock free transonic flows. The method shows a rapid convergence, and indicates how much the flow is sensitive to small modifications of the blade geometry, that the classical iterative use of analysis methods might not be able to define. The relationship between the required Mach number distribution and the resulting geometry is discussed. Examples show how geometrical constraints imposed upon the blade shape can be respected by using free geometrical parameters or by relaxing the required Mach number distribution. The same code is used both for the design of the required geometry and for the off-design calculations. Examples illustrate the difficulty of designing blade shapes with optimal performance also outside of the design point.

Leonard, O.↗
NASA NTRS1988

Iterative finite element solver on transputer networks

The parallelism inherent in the Conjugate Gradient method is described. The initial results of a parallel implementation on a network of twelve transputers are discussed. The high efficiencies obtained indicate that significant speedup can be obtained with larger transputer arrays if communication overhead can be kept low. To this end, a method of communication that allows large, dynamically reconfigurable transputer arrays to exchange data in log sub 4 N steps for N processors is suggested.

Danial, Albert↗
NASA NTRS1985

Development of a boundary-layer-type solver based on simultaneous iteration technique for axisymmetric separated flows

A boundary-layer-type solver is developed for the numerical solution of axisymmetric separated flows. A new fully implicit coupling scheme for the viscous and inviscid regions is demonstrated. This fully implicit coupling technique is similar to the work of Carter, Veldman, and an extension of an earlier work of Halim and Hafez. A comparison is made for the convergence rate using this new fully implicit coupling technique and the semiimplicit coupling of Halim and Hafez. Numerical results using the fully implicit coupling are obtained for laminar incompressible separated flows, including a boattail and a series of trough geometries. Also, the near-wake flow problem is considered using the present formulation. A clear conclusion of this investigation is that the present scheme using the fully implicit coupling method converges at a faster rate than the semiimplicit coupling and the partially parabolized Navier-Stokes (PPNS) procedures.

Halim, A. A. M.↗
DOE OSTI2015

Fault tolerance in an inner-outer solver: A GVR-enabled case study

Resilience is a major challenge for large-scale systems. It is particularly important for iterative linear solvers, since they take much of the time of many scientific applications. We show that single bit flip errors in the Flexible GMRES iterative linear solver can lead to high computational overhead or even failure to converge to the right answer. Informed by these results, we design and evaluate several strategies for fault tolerance in both inner and outer solvers appropriate across a range of error rates. We implement them, extending Trilinos’ solver library with the Global View Resilience (GVR) programming model, which provides multi-stream snapshots, multi-version data structures with portable and rich error checking/recovery. Lastly, experimental results validate correct execution with low performance overhead under varied error conditions.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1975

Advances in Application of Fast Semidirect Computational Methods in Transonic Flow

This paper is intended as a review and summary of the advances made in a recently developed approach for rapid numerical solution of the equations of inviscid transonic aerodynamics. The investigation has been limited to two-dimensional, steady, inviscid flow over airfoils in a subsonic free stream, with emphasis on development of a rapid computational technique, rather than on generality of application. The approach uses finite-difference algorithms called "fast direct elliptic solvers" within an iteration scheme. "Direct" means that the entire computation field is solved at once, rather than in successive traverses over the field as in a point- or line-relaxation method. Such an iterative method is referred to as "semidirect." The iterative convergence can be faster than in other relaxation methods because changes are felt simultaneously at all points in each succeeding iteration. Direct elliptic solvers and semidirect methods have restrictions, but these are gradually being removed. Direct solvers were first developed for solving Poisson's equation on a rectangle without interior boundaries. A method to treat first-order systems, a direct Cauchy-Riemann solver has also been developed. Numerical treatment of part of a system of nonlinear equations by a Poisson solver has been reported. Also Poisson solvers in semidirect methods were used for nonseparable elliptic equations. The semidirect method was extended to the solution of a problem of mixed type, where the improved Murman-Cole transonic small-disturbance difference equations were solved. A slightly supercritical flow over a biconvex airfoil was treated successfully, but the iterations did not converge for more strongly supercritical conditions In another work the addition of terms ot both sides of the difference equations stabilized the iteration for supercritical conditions with large supersonic zones. For this, the Cauchy-Riemann solver was revised to incl,ude the needed terms. Most recently, the evaluation of parameters for rapid convergence and comparisons, with Murman's line-relaxation method was described. The method was extended to full second order accuracy in a fully conservative formulation in another work.

Martin, E. Dale↗
NASA NTRS2017

Numerical Simulation of Illumination and Thermal Conditions at the Lunar Poles Using LOLA DTMs

We are interested in illumination conditions and the temperature distribution within the upper two meters of regolith near the lunar poles. Here, areas exist receiving almost constant illumination near areas in permanent shadow, which were identified as potential exploration sites for future missions. For our study a numerical simulation of the illumination and thermal environment for lunar near-polar regions is needed. Our study is based on high-resolution, twenty meters per pixel and 400 x 400 km large polar Digital Terrain Models (DTMs), which were derived from Lunar Orbiter Laser Altimeter (LOLA) data. Illumination conditions were simulated by synthetically illuminating the LOLA DTMs using the horizon method considering the Sun as an extended source. We model polar illumination for the central 50 x 50 km subset and use it as an input at each time-step (2 h) to evaluate the heating of the lunar surface and subsequent conduction in the sub-surface. At surface level we balance the incoming insolation with the subsurface conduction and radiation into space, whereas in the sub-surface we consider conduction with an additional constant radiogenic heat source at the bottom of our two-meter layer. Density is modeled as depth-dependent, the specific heat parameter as temperature-dependent and the thermal conductivity as depth- and temperature-dependent. We implemented a fully implicit finite-volume method in space and backward Euler scheme in time to solve the one-dimensional heat equation at each pixel in our 50 x 50 km DTM. Due to the non-linear dependencies of the parameters mentioned above, Newton's method is employed as the non-linear solver together with the Gauss-Seidel method as the iterative linear solver in each Newton iteration. The software is written in OpenCL and runs in parallel on the GPU cores, which allows for fast computation of large areas and long time scales.

Glaser, P.↗
DOE OSTI2019

Approximate Green’s Function Coupled Cluster A Method Employing Effective Dimension Reduction

The Green’s function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structures of molecular structures and periodic systems, especially when electrons are strongly correlated. However, in order for the GFCC method to become an approach that can be routinely used in the electronic structure calculations, more robust linear algebraic techniques and approximations need to be employed to reduce its extremely high computational overhead. In our recent studies, we showed that the GFCC equation can be numerically solved by seeking solutions of multiple linear systems in the frequency regime of interest, which can be easily distributed in a massively parallel environment, and broaden the avenue for more state-of-the-art numerical linear algebraic tools associated with multiple linear systems to be kicked in to this field. In the present pursuit of this practice, we show a successful application of model order reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires a high resolution of spectral function, instead of solving GFCC linear equation in the original dimension for every single frequency point, an easily-solvable linear system model with reduced dimension can be built upon projecting the original GFCC linear system into a subspace, from which the interpolation and extrapolation can give a reasonable approximation to the results of the original GFCC linear equations. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some select frequencies inside the frequency regime. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The approach is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems including carbon monoxide, 1,3-butadiene, benzene, and adenine molecules. As a byproduct, the high quality linear system model is also found to be able to provide high quality initial guess for the existing linear solver to improve the convergence performance.

Peng, Bo↗
DOE OSTI2020

Approximate Inverse Chain Preconditioner: Iteration Count Case Study for Spectral Support Solvers

As the growing availability of computational power slows, there has been an increasing reliance on algorithmic advances. However, faster algorithms alone will not necessarily bridge the gap in allowing computational scientists to study problems at the edge of scientific discovery in the next several decades. Often, it is necessary to simplify or precondition solvers to accelerate the study of large systems of linear equations commonly seen in a number of scientific fields. Preconditioning a problem to increase efficiency is often seen as the best approach; yet, preconditioners which are fast, smart, and efficient do not always exist. Following the progress of [1], we present a new preconditioner for symmetric diagonally dominant (SDD) systems of linear equations. These systems are common in certain PDEs, network science, and supervised learning among others. Based on spectral support graph theory, this new preconditioner builds off of the work of [2], computing and applying a V-cycle chain of approximate inverse matrices. This preconditioner approach is both algebraic in nature as well as hierarchically-constrained depending on the condition number of the system to be solved. Due to its generation of an Approximate Inverse Chain of matrices, we refer to this as the AIC preconditioner. We further accelerate the AIC preconditioner by utilizing precomputations to simplify setup and multiplications in the con-text of an iterative Krylov-subspace solver. While these iterative solvers can greatly reduce solution time, the number of iterations can grow large quickly in the absence of good preconditioners. Initial results for the AIC preconditioner have shown a very large reduction in iteration counts for SDD systems as compared to standard preconditioners such as Incomplete Cholesky (ICC) and Multigrid (MG). We further show significant reduction in iteration counts against the more advanced Combinatorial Multigrid (CMG) preconditioner. We have further developed no-fill sparsification techniques to ensure that the computational cost of applying the AIC preconditioner does not grow prohibitively large as the depth of the V-cycle grows for systems with larger condition numbers. Our numerical results have shown that these sparsifiers maintain the sparsity structure of our system while also displaying significant reductions in iteration counts.1 2

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Approximate Inverse Chain Preconditioner: Iteration Count Case Study for Spectral Support Solvers

As the growing availability of computational power slows, there has been an increasing reliance on algorithmic advances. However, faster algorithms alone will not necessarily bridge the gap in allowing computational scientists to study problems at the edge of scientific discovery in the next several decades. Often, it is necessary to simplify or precondition solvers to accelerate the study of large systems of linear equations commonly seen in a number of scientific fields. Preconditioning a problem to increase efficiency is often seen as the best approach; yet, preconditioners which are fast, smart, and efficient do not always exist. Following the progress of [1], we present a new preconditioner for symmetric diagonally dominant (SDD) systems of linear equations. These systems are common in certain PDEs, network science, and supervised learning among others. Based on spectral support graph theory, this new preconditioner builds off of the work of [2], computing and applying a V-cycle chain of approximate inverse matrices. This preconditioner approach is both algebraic in nature as well as hierarchically-constrained depending on the condition number of the system to be solved. Due to its generation of an Approximate Inverse Chain of matrices, we refer to this as the AIC preconditioner. We further accelerate the AIC preconditioner by utilizing precomputations to simplify setup and multiplications in the con-text of an iterative Krylov-subspace solver. While these iterative solvers can greatly reduce solution time, the number of iterations can grow large quickly in the absence of good preconditioners. Initial results for the AIC preconditioner have shown a very large reduction in iteration counts for SDD systems as compared to standard preconditioners such as Incomplete Cholesky (ICC) and Multigrid (MG). We further show significant reduction in iteration counts against the more advanced Combinatorial Multigrid (CMG) preconditioner. We have further developed no-fill sparsification techniques to ensure that the computational cost of applying the AIC preconditioner does not grow prohibitively large as the depth of the V-cycle grows for systems with larger condition numbers. Our numerical results have shown that these sparsifiers maintain the sparsity structure of our system while also displaying significant reductions in iteration counts.1 2

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Accelerating Lattice QCD Multigrid on GPUs Using Fine-Grained Parallelization

The past decade has witnessed a dramatic acceleration of lattice quantum chromodynamics calculations in nuclear and particle physics. This has been due to both significant progress in accelerating the iterative linear solvers using multi-grid algorithms, and due to the throughput improvements brought by GPUs. Deploying hierarchical algorithms optimally on GPUs is non-trivial owing to the lack of parallelism on the coarse grids, and as such, these advances have not proved multiplicative. Using the QUDA library, we demonstrate that by exposing all sources of parallelism that the underlying stencil problem possesses, and through appropriate mapping of this parallelism to the GPU architecture, we can achieve high efficiency even for the coarsest of grids. Results are presented for the Wilson-Clover discretization, where we demonstrate up to 10x speedup over present state-of-the-art GPU-accelerated methods on Titan. Finally, we look to the future, and consider the software implications of our findings.

Clark, M. A.↗
DOE OSTI2018

Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1992

Aerodynamic shape optimization via sensitivity analysis on decomposed computational domains

Direct and iterative method considered to be most applicable to large systems of linear equations arising in discrete sensitivity analysis are assessed. Based on a single-domain grid, computations are performed using a banded matrix solver and an iterative solver, the generalized minimum residual (GMRES) method. The banded matrix solver is found to be generally the most economical method for those applications where the number of right-hand sides is large (i.e., a large number of design variables or a large number of adjoint vectors). For systems of equations that are too large to be solved by direct methods, an approach is proposed whereby the computational domain is divided into small subdomains, and each subdomain is solved separately.

Eleshaky, Mohamed E.↗
DOE OSTI2025

Ipopt Interface to Re::Solve Linear Solver

The software provides Ipopt optimization package an interface to the Re::Solve linear solver library. Re::Solve features GPU-resident direct and iterative linear solvers that could be used to accelerate optimization computations.

Alam, Maksudul [Oak Ridge National Laboratory (ORN↗
NASA NTRS2013

Real-Time Optimization for use in a Control Allocation System to Recover from Pilot Induced Oscillations

Integration of the Control Allocation technique to recover from Pilot Induced Oscillations (CAPIO) System into the control system of a Short Takeoff and Landing Mobility Concept Vehicle simulation presents a challenge because the CAPIO formulation requires that constrained optimization problems be solved at the controller operating frequency. We present a solution that utilizes a modified version of the well-known L-BFGS-B solver. Despite the iterative nature of the solver, the method is seen to converge in real time with sufficient reliability to support three weeks of piloted runs at the NASA Ames Vertical Motion Simulator (VMS) facility. The results of the optimization are seen to be excellent in the vast majority of real-time frames. Deficiencies in the quality of the results in some frames are shown to be improvable with simple termination criteria adjustments, though more real-time optimization iterations would be required.

optimal control↗