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Results for “Newton-GMRES”

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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6 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

Practical Aspects of Krylov Subspace Iterative Methods in CFD

Implementation issues associated with the application of Krylov subspace iterative methods, such as Newton-GMRES, are presented within the framework of practical computational fluid dynamic (CFD) applications. This paper categorizes, evaluates, and contrasts the major ingredients (function evaluations, matrix-vector products, and preconditioners) of Newton-GMRES Krylov subspace methods in terms of their effect on the local linear and global nonlinear convergence, memory requirements, and accuracy. The discussion focuses on Newton-GMRES in both a structured multi-zone incompressible Navier-Stokes solver and an unstructured mesh finite-volume Navier-Stokes solver. Approximate versus exact matrix-vector products, effective preconditioners, and other pertinent issues are addressed.

Pulliam, Thomas H.↗
DOE OSTI2025

Convergence Analysis of the Alternating Anderson–Picard Method for Nonlinear Fixed-Point Problems

Anderson acceleration (AA) has been widely used to solve nonlinear fixed-point problems due to its rapid convergence. This work focuses on a variant of AA in which multiple Picard iterations are performed between each AA step, referred to as the Alternating Anderson–Picard (AAP) method. Furthermore, despite introducing more “slow” Picard iterations, this method has been shown to be efficient and even more robust in both linear and nonlinear cases. However, there is a lack of theoretical analysis for AAP in the nonlinear case. In this paper, we address this gap by establishing the equivalence between AAP and a multisecant-GMRES method that uses GMRES to solve a multisecant linear system at each iteration. From this perspective, we show that AAP “converges” to the Newton-GMRES method. Specifically, as the residual approaches zero, the multisecant matrix, the approximate Jacobian inverse, the search direction, and the optimization gain of AAP converge to their counterparts in the Newton-GMRES method. These connections provide insights for analyzing the asymptotic convergence properties of AAP. Consequently, we show that AAP is locally 𝑞-linear convergent and provide an upper bound for the convergence factor of AAP. To validate the theoretical results, numerical examples are provided.

Anderson acceleration↗
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 NTRS1994

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↗
NASA NTRS1998

Multigrid Strategies for Viscous Flow Solvers on Anisotropic Unstructured Meshes

Unstructured multigrid techniques for relieving the stiffness associated with high-Reynolds number viscous flow simulations on extremely stretched grids are investigated. One approach consists of employing a semi-coarsening or directional-coarsening technique, based on the directions of strong coupling within the mesh, in order to construct more optimal coarse grid levels. An alternate approach is developed which employs directional implicit smoothing with regular fully coarsened multigrid levels. The directional implicit smoothing is obtained by constructing implicit lines in the unstructured mesh based on the directions of strong coupling. Both approaches yield large increases in convergence rates over the traditional explicit full-coarsening multigrid algorithm. However, maximum benefits are achieved by combining the two approaches in a coupled manner into a single algorithm. An order of magnitude increase in convergence rate over the traditional explicit full-coarsening algorithm is demonstrated, and convergence rates for high-Reynolds number viscous flows which are independent of the grid aspect ratio are obtained. Further acceleration is provided by incorporating low-Mach-number preconditioning techniques, and a Newton-GMRES strategy which employs the multigrid scheme as a preconditioner. The compounding effects of these various techniques on speed of convergence is documented through several example test cases.

Movriplis, Dimitri J.↗
DOE OSTI2025

FedOSAA: Improving Federated Learning with One-Step Anderson Acceleration

Federated learning (FL) is a distributed machine learning approach that enables multiple local clients and a central server to collaboratively train a model while keeping the data on their own devices. First-order methods, particularly those incorporating variance reduction techniques, are the most widely used FL algorithms due to their simple implementation and stable performance. However, these methods tend to be slow and require a large number of communication rounds to reach the global minimizer. We propose FedOSAA, a novel approach that preserves the simplicity of first-order methods while achieving the rapid convergence typically associated with second-order methods. Our approach applies one Anderson acceleration (AA) step following classical local updates based on first-order methods with variance reduction, such as FedSVRG and SCAFFOLD, during local training. This AA step is able to leverage curvature information from the history points and gives a new update that approximates the Newton-GMRES direction, thereby significantly improving the convergence. We establish a local linear convergence rate to the global minimizer of FedOSAA for smooth and strongly convex loss functions. Numerical comparisons show that FedOSAA substantially improves the communication and computation efficiency of the original first-order methods, achieving performance comparable to second-order methods like GIANT.

Feng, Xue [University of California, Davis]↗