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Results for “iterative linear solution”

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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At least 505 records · Page 28
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Nonlinear system guidance in the presence of transmission zero dynamics

An iterative procedure is proposed for computing the commanded state trajectories and controls that guide a possibly multiaxis, time-varying, nonlinear system with transmission zero dynamics through a given arbitrary sequence of control points. The procedure is initialized by the system inverse with the transmission zero effects nulled out. Then the 'steady state' solution of the perturbation model with the transmission zero dynamics intact is computed and used to correct the initial zero-free solution. Both time domain and frequency domain methods are presented for computing the steady state solutions of the possibly nonminimum phase transmission zero dynamics. The procedure is illustrated by means of linear and nonlinear examples.

Meyer, G.↗
NASA NTRS1993

A noniterative improvement of Guyan reduction

In determining the natural modes and frequencies of a linear elastic structure, Guyan reduction is often used to reduce the size of the mass and stiffness matrices and the solution of the reduced system is obtained first. The reduced system modes are then expanded to the size of the original system by using a static transformation linking the retained degrees of freedom to the omitted degrees of freedom. In the present paper, the transformation matrix of Guyan reduction is modified to include additional terms from a series accounting for the inertial effects. However, the inertial terms are dependent on the unknown frequencies. A practical approximation is employed to compute the inertial terms without any iteration. This new transformation is implemented in NASTRAN using a DMAP sequence alter. Numerical examples using a cantilever beam illustrate the necessary condition for allowing a large number of additional terms in the proposed series correction of Guyan reduction. A practical example of a large model of the Plasma Motor Generator module to be flown on a Delta launch vehicle is also presented.

Ganesan, N.↗
DOE OSTI2016

Solution of monotone complementarity and general convex programming problems using a modified potential reduction interior point method

We present a homogeneous algorithm equipped with a modified potential function for the monotone complementarity problem. We show that this potential function is reduced by at least a constant amount if a scaled Lipschitz condition (SLC) is satisfied. A practical algorithm based on this potential function is implemented in a software package named iOptimize. The implementation in iOptimize maintains global linear and polynomial time convergence properties, while achieving practical performance. It either successfully solves the problem, or concludes that the SLC is not satisfied. When compared with the mature software package MOSEK (barrier solver version 6.0.0.106), iOptimize solves convex quadratic programming problems, convex quadratically constrained quadratic programming problems, and general convex programming problems in fewer iterations. Moreover, several problems for which MOSEK fails are solved to optimality. In addition, we also find that iOptimize detects infeasibility more reliably than the general nonlinear solvers Ipopt (version 3.9.2) and Knitro (version 8.0).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Analysis of cell-based diffusion acceleration for the slice balance approach

In this work, we perform analysis on the use of cell-based diffusion acceleration methodologies to accelerate the convergence of transport solutions discretized with the slice balance approach (SBA) on unstructured polygonal grids.We investigated both linear diffusion synthetic acceleration (DSA) and non linear diffusion acceleration (NDA), including its partial-current variant (pNDA). DSA and NDA were both shown to diverge for intermediate ranges of mesh optical thicknesses. However, pNDA and Krylov methods like GMRES and Broyden stabilized the acceleration schemes, including problems with degenerate cells formed by mesh refinement. (author)

42 ENGINEERING↗
DOE OSTI2022

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

Fully implicit Runge--Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2 x 2 systems. Under quite general assumptions, it is proven that the preconditioned 2 x 2 operator's condition number is bounded by a small constant close to one, independent of the spatial discretization, spatial mesh, and time step, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier--Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier--Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK, while in all cases fourth-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard Singly diagonally implicit Runge--Kutta methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics

We introduce a novel Newton–Krylov (NK)-based fully implicit algorithm for solving fluid flows in a wide range of flow conditions—from variable density nearly incompressible to supersonic shock dynamics. The key enabling feature of our all-speed solver is the ability to efficiently solve conservation laws by choosing a set of independent variables that produce a well-conditioned Jacobian matrix for the linear iterations of the global nonlinear iterative solver. In particular, instead of choosing to discretize the conservative variables (density, momentum, total energy), which is traditionally used in Eulerian high-speed compressible fluid dynamics, we demonstrate superior performance by discretizing the primitive variables—pressure–velocity–temperature in the very low-Mach flow limits or density–velocity–temperature/entropy in the shock dynamics range. Moreover, our method allows us to avoid direct inversion of the mass matrix in discrete time derivatives, which is usually an additional source for stiffness, especially pronounced when going to very high-order schemes with non-orthogonal basis functions. Here, we show robust solutions obtained for discontinuous finite element discretization up to seventh-order accuracy. Another important aspect of the solution algorithm is the Advection Upstream Splitting Method (AUSM), adopted to compute numerical fluxes within our reconstructed discontinuous Galerkin (rDG) spatial discretization scheme. The use of the low-Mach modification of the hyperbolic flux operator is found to be necessary for enabling robust simulations of very stiff liquids and metals for Mach numbers below M=10 -5 , which is well known to be very computationally challenging for compressible solvers. We demonstrate that our fully implicit rDG-NK solver with the AUSM + -up flux treatment produces efficient and high-resolution numerical solutions at all speeds, ranging from vanishing Mach numbers to transonic and supersonic, without substantial modifications of the solution procedures. (At high speed, we add limiting and use a simpler preconditioning of the Krylov solver.) Numerical examples include nearly incompressible constant-property flow past a backward-facing step with heat transfer, low-Mach variable-property channel flow of water at supercritical state, phase change and melt pool dynamics for laser spot welding and selective laser melting in additive manufacturing, and Mach 3 flow in a wind tunnel with a step.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1987

Efficient solutions to the Euler equations for supersonic flow with embedded subsonic regions

A line Gauss-Seidel (LGS) relaxation algorithm in conjunction with a one-parameter family of upwind discretizations of the Euler equations in two dimensions is described. Convergence of the basic algorithm to the steady state is quadratic for fully supersonic flows and is linear for other flows. This is in contrast to the block alternating direction implicit methods (either central or upwind differenced) and the upwind biased relaxation schemes, all of which converge linearly, independent of the flow regime. Moreover, the algorithm presented herein is easily coupled with methods to detect regions of subsonic flow embedded in supersonic flow. This allows marching by lines in the supersonic regions, converging each line quadratically, and iterating in the subsonic regions, and yields a very efficient iteration strategy. Numerical results are presented for two-dimensional supersonic and transonic flows containing oblique and normal shock waves which confirm the efficiency of the iteration strategy.

Walters, Robert W.↗
DOE OSTI2022

Deblurring galaxy images with Tikhonov regularization on magnitude domain

Abstract We propose a regularization-based deblurring method that works efficiently for galaxy images. The spatial resolution of a ground-based telescope is generally limited by seeing conditions and is much worse than space-based telescopes. This circumstance has generated considerable research interest in the restoration of spatial resolution. Since image deblurring is a typical inverse problem and often ill-posed, solutions tend to be unstable. To obtain a stable solution, much research has adopted regularization-based methods for image deblurring, but the regularization term is not necessarily appropriate for galaxy images. Although galaxies have an exponential or Sérsic profile, the conventional regularization assumes the image profiles to behave linearly in space. The significant deviation between the assumption and real situations leads to blurring of the images and smoothing out the detailed structures. Clearly, regularization on logarithmic domain, i.e., magnitude domain, should provide a more appropriate assumption, which we explore in this study. We formulate a problem of deblurring galaxy images by an objective function with a Tikhonov regularization term on a magnitude domain. We introduce an iterative algorithm minimizing the objective function with a primal–dual splitting method. We investigate the feasibility of the proposed method using simulation and observation images. In the simulation, we blur galaxy images with a realistic point spread function and add both Gaussian and Poisson noise. For the evaluation with the observed images, we use galaxy images taken by the Subaru HSC-SSP. Both of these evaluations show that our method successfully recovers the spatial resolution of the deblurred images and significantly outperforms the conventional methods. The code is publicly available from the GitHub 〈https://github.com/kzmurata-astro/PSFdeconv_amag〉.

Murata, Kazumi↗
NASA NTRS1992

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

The development of efficient iterative solution methods for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations is discussed. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. In this work, another approach based on the classical conjugate gradient method, known as the Generalized Minimum Residual (GMRES) algorithm is investigated. The GMRES algorithm has been used in the past by a number of researchers for solving steady viscous and inviscid flow problems. Here, we investigate the suitability of this algorithm for solving the system of non-linear equations that arise in unsteady Navier-Stokes solvers at each time step.

Sankar, Lakshmi N.↗
DOE OSTI2019

Sparse, iterative simulation methods for one-dimensional laminar flames

Sparse, iterative simulation methods for one-dimensional laminar flames are proposed. The resulting steady and unsteady flame solvers exploit approximate Jacobians to greatly reduce the computational cost associated with matrix operations. The constant, non-unity Lewis number assumption is introduced to further reduce the computational cost. The solvers are also parallelized to reduce time-to-solution on distributed memory computer systems. Computed laminar flame speeds and species profiles for a range of chemical mechanisms (from 10 to 2878 species) are compared against a well-validated commercial code and are found to be consistent within solver tolerances. The computation times of both the unsteady and steady solutions increase only linearly with the number of species, which is a significant improvement over the quadratic or cubic scaling of existing steady-state flame solvers. For the largest mechanism tested, the steady-state flame solver is two orders of magnitude faster than commonly-used codes. The use of an approximate Jacobian is shown to reduce the rate of convergence for the steady-state solver, but does not significantly affect the domain of convergence. The steady-state solver with approximate Jacobian is thus well suited for computationally efficient laminar flame speed sweeps with large kinetic mechanisms.

42 ENGINEERING↗
NASA NTRS1988

Totally parallel multilevel algorithms

Four totally parallel algorithms for the solution of a sparse linear system have common characteristics which become quite apparent when they are implemented on a highly parallel hypercube such as the CM2. These four algorithms are Parallel Superconvergent Multigrid (PSMG) of Frederickson and McBryan, Robust Multigrid (RMG) of Hackbusch, the FFT based Spectral Algorithm, and Parallel Cyclic Reduction. In fact, all four can be formulated as particular cases of the same totally parallel multilevel algorithm, which are referred to as TPMA. In certain cases the spectral radius of TPMA is zero, and it is recognized to be a direct algorithm. In many other cases the spectral radius, although not zero, is small enough that a single iteration per timestep keeps the local error within the required tolerance.

Frederickson, Paul O.↗
DOE OSTI2017

Next generation extended Lagrangian first principles molecular dynamics

Extended Lagrangian Born-Oppenheimer molecular dynamics [Phys. Rev. Lett., 100, 123004 (2008)] is formulated for general Hohenberg-Kohn density functional theory and compared to the extended Lagrangian framework of rst principles molecular dynamics by Car and Parrinello [Phys. Rev. Lett. 55, 2471 (1985)]. It is shown how extended Lagrangian Born-Oppenheimer molecular dynamics overcomes several shortcomings of regular, direct Born-Oppenheimer molecular dynamics, while improving or maintaining important features of Car-Parrinello simulations. The accuracy of the electronic degrees of freedom in extended Lagrangian Born-Oppenheimer molecular dynamics, with respect to the exact Born-Oppenheimer solution, is of second order in the size of the integration time step and of fourth order in the potential energy surface. Improved stability over recent formulations of extended Lagrangian Born-Oppenheimer molecular dynamics is achieved by generalizing the theory to finite temperature ensembles, using fractional occupation numbers in the calculation of the inner-product kernel of the extended harmonic oscillator that appears as a preconditioner in the electronic equations of motion. Materials systems that normally exhibit slow self-consistent field convergence can be simulated using integration time steps of the same order as in direct Born- Oppenheimer molecular dynamics, but without the requirement of an iterative, non-linear electronic ground state optimization prior to the force evaluations and without a systematic drift in the total energy. In combination with proposed low-rank and on-the- y updates of the kernel, this formulation provides an efficient and general framework for quantum based Born-Oppenheimer molecular dynamics simulations.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2020

Repair and Resource Scheduling in Unbalanced Distribution Systems Using Neighborhood Search

—This paper proposes an optimization strategy to assist utility operators to recover power distribution systems after large outages. Specifically, a mixed-integer linear programming (MILP) model is developed for co-optimizing crews, resources, and network operations. The MILP model coordinates damage isolation, network reconfiguration, distributed generator re-dispatch, and crew/resource logistics. In addition, a framework for integrating different types of photovoltaic (PV) systems in the restoration process is developed. We consider two different types of crews, namely, line crews for damage repair and tree crews for obstacle removal. We also model the repair resource logistic constraints. Furthermore, a new algorithm is developed for solving the distribution system repair and restoration problem (DSRRP). The algorithm starts by solving DSRRP using an assignmentbased method, then a neighborhood search method is designed to iteratively improve the solution. The proposed method is validated on modified IEEE 123- and 8500-bus distribution test systems.

Outage management↗
DOE OSTI2024

Modifying the Asynchronous Jacobi Method for Data Corruption Resilience

Moving scientific computation from high-performance computing (HPC) and cloud computing (CC) environments to devices on the edge, i.e., physically near instruments of interest, has received tremendous interest in recent years. Such edge computing environments can operate on data in situ, offering enticing benefits over data aggregation to HPC and CC facilities that include avoiding costs of transmission, increased data privacy, and real-time data analysis. Because of the inherent unreliability of edge computing environments, new fault-tolerant approaches must be developed before the benefits of edge computing can be realized. Motivated by algorithm-based fault tolerance, a variant of the asynchronous Jacobi (ASJ) method is developed that achieves resilience to data corruption by rejecting solution approximations from neighbor devices according to a bound derived from convergence theory. Numerical results on a two-dimensional Poisson problem show that the new rejection criterion, along with a novel approximation to the shortest path length on which the criterion depends, restores convergence for the ASJ variant in the presence of certain types data corruption. Numerical results are obtained for when the singular values in the analytic bound are approximated. Additional linear systems are also explored, one with a more dense sparsity pattern and one that includes advection. All results indicate that successful resilience to data corruption depends on whether the bound tightens fast enough to reject corrupted data before the iteration evolution deviates significantly from that predicted by the convergence theory defining the bound. This observation generalizes to future work on algorithm-based fault tolerance for other asynchronous algorithms, including upcoming approaches that leverage Krylov subspaces.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1976

An improved method for the prediction of completely three-dimensional aerodynamic load distributions of configurations with leading edge vortex separation

The application of a higher-order subsonic potential flow panel method to the solution of three-dimensional flow about wing and wing-body combinations with leading-edge vortex separation is presented. The governing equations are the linear flow differential equation and nonlinear boundary conditions which require that the flow be parallel to the wing and body surfaces and that the free vortex sheet, springing from the leading and trailing edges, be aligned with the local flow and support no pressure jump. The vortex core is modeled as a simple line vortex which receives vorticity from the free sheet through a connecting sheet. The Kutta condition is imposed on all appropriate edges of the wing. This set of nonlinear equations is solved by an iterative procedure. The Goethert rule accounts for compressibility. The method has been programmed for the CDC 6600. Delta wings, gothic wings, arrow wings, cambered wings, and wing with body have been analyzed. Initial studies involving variations of panel density, vortex sheet sizing, Jacobian update, and initial geometry demonstrate that the present method generally exhibits good convergence characteristics.

Rubbert, p. E.↗
DOE OSTI2021

A Multi-Objective Bayesian Optimization Approach Using the Weighted Tchebycheff Method

Abstract Bayesian optimization (BO) is a low-cost global optimization tool for expensive black-box objective functions, where we learn from prior evaluated designs, update a posterior surrogate Gaussian process model, and select new designs for future evaluation using an acquisition function. This research focuses upon developing a BO model with multiple black-box objective functions. In the standard multi-objective (MO) optimization problem, the weighted Tchebycheff method is efficiently used to find both convex and non-convex Pareto frontiers. This approach requires knowledge of utopia values before we start optimization. However, in the BO framework, since the functions are expensive to evaluate, it is very expensive to obtain the utopia values as a prior knowledge. Therefore, in this paper, we develop a MO-BO framework where we calibrate with multiple linear regression (MLR) models to estimate the utopia value for each objective as a function of design input variables; the models are updated iteratively with sampled training data from the proposed MO-BO. These iteratively estimated mean utopia values are used to formulate the weighted Tchebycheff MO acquisition function. The proposed approach is implemented in two numerical test examples and one engineering design problem of optimizing thin tube geometries under constant loading of temperature and pressure, with minimizing the risk of creep-fatigue failure and design cost, along with risk-based and manufacturing constraints. Finally, the model accuracy with frequentist, Bayesian and without MLR-based calibration are compared to true Pareto solutions.

Engineering↗
DOE OSTI2023

Iterated Gauss-Seidel GMRES

The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869] is an iterative method for approximately solving linear systems Ax = b, with initial guess x0 and residual r0 = b Ax0. The algorithm employs the Arnoldi process to generate the Krylov basis vectors (the columns of Vk ). It is well known that this process can be viewed as a QR factorization of the matrix Bk = [r0, AVk] at each iteration. Despite an O (..epsilon..)..kappa.. (Bk ) loss of orthogonality, for unit roundoff ..epsilon..and condition number ..kappa.. , the modified Gram-Schmidt formulation was shown to be backward stable in the seminal paper by Paige et al. [SIAM J. Matrix Anal.Appl., 28 (2006), pp. 264-284]. We present an iterated Gauss-Seidel formulation of the GMRES algorithm (IGS-GMRES) based on the ideas of Ruhe [Linear Algebra Appl., 52 (1983), pp. 591-601] and Swirydowicz et al. [Numer. Linear Algebra Appl., 28 (2020), pp. 1-20]. IGS-GMRES maintains orthogonality to the level O (..epsilon..)..kappa.. (Bk ) or O (..epsilon..), depending on the choice of one or two iterations; for two Gauss-Seidel iterations, the computed Krylov basis vectors remain orthogonal to working accuracy and the smallest singular value of Vk remains close to one. The resulting GMRES method is thus backward stable. We show that IGS-GMRES can be implemented with only a single synchronization point per iteration, making it relevant to large-scale parallel computing environments. We also demonstrate that, unlike MGS-GMRES, in IGS-GMRES the relative Arnoldi residual corresponding to the computed approximate solution no longer stagnates above machine precision even for highly nonnormal systems.

Arnoldi-QR↗
NASA NTRS1994

Global Asymptotic Behavior of Iterative Implicit Schemes

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Yee, H. C.↗