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QMR: a Quasi-Minimal Residual Method for Non-Hermitian Linear Systems
, 1991
"... ... In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from t ..."
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Cited by 395 (26 self)
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... In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.
A Flexible Inner-Outer Preconditioned GMRES Algorithm
, 1993
"... We present a variant of the GMRES lgorithm which l]ows changes in the prcconditioning at every step. There arc many possible applications o the new lgorithm some o which arc briefly discussed. In particular, a result o the flexibility o the new variant is that any iterative method can bc used as a p ..."
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Cited by 358 (30 self)
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We present a variant of the GMRES lgorithm which l]ows changes in the prcconditioning at every step. There arc many possible applications o the new lgorithm some o which arc briefly discussed. In particular, a result o the flexibility o the new variant is that any iterative method can bc used as a prcconditioncr. For example, the standard GMRES lgorithm itself can bc used as a prcconditioncr, as can CGNR (or CGNE) the conjugate gradient method applied to the normal equations. However, the more appealing utilization o the method is in conjunction with relaxation techniques, possibly multi-level techniques. The possibility o changing prcconditioncrs may bc exploited to develop efficient iterative methods and to enhance robustness. A cw numcricM experiments arc reported to illustrate this act.
PETSc users manual
- ANL-95/11 - Revision 2.1.0, Argonne National Laboratory
, 2001
"... tract W-31-109-Eng-38. 2 This manual describes the use of PETSc for the numerical solution of partial differential equations and related problems on high-performance computers. The Portable, Extensible Toolkit for Scientific Compu-tation (PETSc) is a suite of data structures and routines that provid ..."
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Cited by 282 (20 self)
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tract W-31-109-Eng-38. 2 This manual describes the use of PETSc for the numerical solution of partial differential equations and related problems on high-performance computers. The Portable, Extensible Toolkit for Scientific Compu-tation (PETSc) is a suite of data structures and routines that provide the building blocks for the implemen-tation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all message-passing communication. PETSc includes an expanding suite of parallel linear, nonlinear equation solvers and time integrators that may be used in application codes written in Fortran, C, and C++. PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper
ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods.
, 1997
"... this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the ..."
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Cited by 218 (18 self)
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this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the software. 1.7 Dependence on LAPACK and BLAS
FastHenry: A Multipole-Accelerated 3-D Inductance Extraction Program
, 1993
"... ... based on mesh analysis can be combined with a GMRES-style iterative matrix solution technique to make a reasonably fast 3-D frequency dependent inductance and resistance extraction algorithm. Unfortunately, both the computation time and memory re- quired for that approach grow faster than n 2, w ..."
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Cited by 216 (48 self)
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... based on mesh analysis can be combined with a GMRES-style iterative matrix solution technique to make a reasonably fast 3-D frequency dependent inductance and resistance extraction algorithm. Unfortunately, both the computation time and memory re- quired for that approach grow faster than n 2, where n is the number of volume-filaments. In this paper, we show that it is possible to use multipole-acceleration to reduce both required memory and computation time to nearly order n. Results from examples are given to demonstrate that the multipole acceleration can reduce required computation time and memory by more than an order of magnitude for realistic packaging problems.
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
- J. Comput. Phys
"... Jacobian-free Newton-Krylov (JFNK) methods are synergistic combinations of Newton-type methods for superlinearly convergent solution of nonlinear equa-tions and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobian-vector product, which ..."
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Cited by 204 (6 self)
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Jacobian-free Newton-Krylov (JFNK) methods are synergistic combinations of Newton-type methods for superlinearly convergent solution of nonlinear equa-tions and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobian-vector product, which may be probed approximately without forming and storing the elements of the true Jacobian, through a variety of means. Various approximations to the Jacobian matrix may still be required for preconditioning the resulting Krylov iteration. As with Krylov methods for linear problems, successful application of the JFNK method to any given problem is dependent on adequate preconditioning. JFNK has potential for application throughout problems governed by nonlinear partial dierential equations and integro-dierential equations. In this survey article we place JFNK in context with other nonlinear solution algorithms for both bound-ary value problems (BVPs) and initial value problems (IVPs). We provide an overview of the mechanics of JFNK and attempt to illustrate the wide variety of preconditioning options available. It is emphasized that JFNK can be wrapped (as an accelerator) around another nonlinear xed point method (interpreted as a preconditioning process, potentially with signicant code reuse). The aim of this article is not to trace fully the evolution of JFNK, nor to provide proofs of accuracy or optimal convergence for all of the constituent methods, but rather to present the reader with a perspective on how JFNK may be applicable to problems of physical interest and to provide sources of further practical information. A review paper solicited by the Editor-in-Chief of the Journal of Computational
Preconditioning techniques for large linear systems: A survey
- J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 192 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
SUNDIALS: Suite of Nonlinear and Differential/ Algebraic Equation Solvers
- ACM Trans. Math. Software
, 2005
"... SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordi-nary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVOD ..."
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Cited by 162 (6 self)
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SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordi-nary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suit-able for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of differ-ent preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.
Aerodynamic Design Optimization On Unstructured Meshes Using the Navier-Stokes Equations
- AIAA J
, 1999
"... A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discret ..."
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Cited by 150 (4 self)
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A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these diculties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-
