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Choosing the Forcing Terms in an Inexact Newton Method
 SIAM J. SCI. COMPUT
, 1994
"... An inexact Newton method is a generalization of Newton's method for solving F(x) = 0, F:/ /, in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition ]lF(x) + F'(x)s]l _< /]lF(xk)]l for a "forcing term" / [0,1). I ..."
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Cited by 161 (6 self)
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An inexact Newton method is a generalization of Newton's method for solving F(x) = 0, F:/ /, in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition ]lF(x) + F'(x)s]l _< /]lF(xk)]l for a "forcing term" / [0,1). In typical applications, the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well. Promising choices of the forcing terms arc given, their local convergence properties are analyzed, and their practical performance is shown on a representative set of test problems.
BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems
 SIAM J. SCI. COMPUT
, 1999
"... We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically e ..."
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Cited by 54 (29 self)
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We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed. The advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM, and the dualthresholdbased ILUT preconditioners. In particular, tests with some convectiondiffusion problems show that it may be possible to obtain convergence that is nearly independent of the Reynolds number as well as of the grid size.
NITSOL: A NEWTON ITERATIVE SOLVER FOR NONLINEAR SYSTEMS
, 1998
"... We introduce a welldeveloped Newton iterative (truncated Newton) algorithm for solving largescale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fort ..."
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Cited by 52 (8 self)
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We introduce a welldeveloped Newton iterative (truncated Newton) algorithm for solving largescale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fortran solver called NITSOL that is robust yet easy to use and provides a number of useful options and features. The structure offers the user great flexibility in addressing problem specicity through preconditioning and other means and allows easy adaptation to parallel environments. Features and capabilities are illustrated in numerical experiments.
GMRES on (Nearly) Singular Systems
 SIAM J. Matrix Anal. Appl
, 1994
"... . We consider the behavior of the gmres method for solving a linear system Ax = b when A is singular or nearly so, i.e., illconditioned. The (near) singularity of A may or may not affect the performance of gmres, depending on the nature of the system and the initial approximate solution. For singu ..."
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Cited by 48 (4 self)
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. We consider the behavior of the gmres method for solving a linear system Ax = b when A is singular or nearly so, i.e., illconditioned. The (near) singularity of A may or may not affect the performance of gmres, depending on the nature of the system and the initial approximate solution. For singular A, we give conditions under which the gmres iterates converge safely to a leastsquares solution or to the pseudoinverse solution. These results also apply to any residual minimizing Krylov subspace method that is mathematically equivalent to gmres. A practical procedure is outlined for efficiently and reliably detecting singularity or illconditioning when it becomes a threat to the performance of gmres. Key words. gmres method, residual minimizing methods, Krylov subspace methods, iterative linear algebra methods, singular or illconditioned linear systems AMS(MOS) subject classifications. 65F10 1. Introduction. The generalized minimal residual (gmres) method of Saad and Schultz [1...
Transition in pipe flow: the saddle structure on the boundary of turbulence
 J. FLUID MECH. SUBMITTED
, 2008
"... The laminar–turbulent boundary Σ is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow appear to be chaotic and show recurring evidence of coherent structures. A general numer ..."
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Cited by 26 (6 self)
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The laminar–turbulent boundary Σ is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow appear to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognizing approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on Σ are organized around only a few travelling waves and their heteroclinic connections. If the flow is suitably constrained to a subspace with a discrete rotational symmetry, it is possible to find locally attracting travelling waves embedded within Σ. Four new types of travelling waves were found using this approach.
On The Choice Of Subspace For Iterative Methods For Linear Discrete IllPosed Problems
 Int. J. Appl. Math. Comput. Sci
, 2001
"... . Many iterative methods for the solution of linear discrete illposed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that it may be possible to determine a meaningful approximate solution with less computational work ..."
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Cited by 25 (20 self)
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. Many iterative methods for the solution of linear discrete illposed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that it may be possible to determine a meaningful approximate solution with less computational work when this requirement is not imposed. Key words. Minimal residual method, conjugate gradient method, linear illposed problems. 1. Introduction. This paper is concerned with the design of iterative methods for the computation of approximate solutions of linear systems of equations Ax = b, A # R mn , x # R n , b # R m , (1.1) with a large matrix A of illdetermined rank. Thus, A has many "tiny" singular values of di#erent orders of magnitude. In particular, A is severely illconditioned. Some of the singular values of A may be vanishing. We allow m # n or m < n. The righthand side vector b is not required to be in the range of A. Linear systems of equations of the fo...
Preconditioned Krylov Subspace Methods for Solving Nonsymmetric Matrices from CFD Applications
 Comput. Methods Appl. Mech. Engrg
, 1999
"... We conduct experimental study on the behavior of several preconditioned iterative methods to solve nonsymmetric matrices arising from computational fluid dynamics (CFD) applications. The preconditioned iterative methods consist of Krylov subspace accelerators and a powerful general purpose multil ..."
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Cited by 22 (13 self)
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We conduct experimental study on the behavior of several preconditioned iterative methods to solve nonsymmetric matrices arising from computational fluid dynamics (CFD) applications. The preconditioned iterative methods consist of Krylov subspace accelerators and a powerful general purpose multilevel block ILU (BILUM) preconditioner. The BILUM preconditioner and an enhanced version of it are slightly modified versions of the originally proposed preconditioners. They will be used in combination with different Krylov subspace methods. We choose to test three popular transposefree Krylov subspace methods: BiCGSTAB, GMRES and TFQMR. Numerical experiments, using several sets of test matrices arising from various relevant CFD applications, are reported. Key words: Multilevel preconditioner, Krylov subspace methods, nonsymmetric matrices, CFD applications. AMS subject classifications: 65F10, 65F50, 65N06, 65N55. 1 Introduction A challenging problem in computational fluid dynamics (...
Using mixed precision for sparse matrix computations to enhance the performance while achieving 64bit accuracy
 ACM Trans. Math. Softw
"... By using a combination of 32bit and 64bit floating point arithmetic the performance of many sparse linear algebra algorithms can be significantly enhanced while maintaining the 64bit accuracy of the resulting solution. These ideas can be applied to sparse multifrontal and supernodal direct techni ..."
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Cited by 20 (1 self)
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By using a combination of 32bit and 64bit floating point arithmetic the performance of many sparse linear algebra algorithms can be significantly enhanced while maintaining the 64bit accuracy of the resulting solution. These ideas can be applied to sparse multifrontal and supernodal direct techniques and sparse iterative techniques such as Krylov subspace methods. The approach presented here can apply not only to conventional processors but also to exotic technologies such as
An Inexact Newton Method for Fully Coupled Solution of the Navier–Stokes Equations with Heat and Mass Transport
 JOURNAL OF COMPUTATIONAL PHYSICS 137, 155–185 (1997)
, 1997
"... The solution of the governing steady transport equations for momentum, heat and mass transfer in flowing fluids can be very difficult. These difficulties arise from the nonlinear, coupled, nonsymmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. ..."
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Cited by 19 (1 self)
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The solution of the governing steady transport equations for momentum, heat and mass transfer in flowing fluids can be very difficult. These difficulties arise from the nonlinear, coupled, nonsymmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this manuscript we focus on evaluating a proposed nonlinear solution method based on an inexact Newton method with backtracking. In this context we use a particular spatial discretization based on a pressure stabilized PetrovGalerkin finite element formulation of the low Mach number Navier–Stokes equations with heat and mass transport. Our discussion considers computational efficiency, robustness and some implementation issues related to the proposed nonlinear solution scheme. Computational results are presented for several challenging CFD benchmark problems as well as two large scale 3D flow simulations.
Fullchip harmonic balance
 Proceedings of the IEEE Custom Integrated Circuits Conference
, 1997
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