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Jacobianfree NewtonKrylov methods: a survey of approaches and applications
 J. Comput. Phys
"... Jacobianfree NewtonKrylov (JFNK) methods are synergistic combinations of Newtontype methods for superlinearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobianvector product, which ..."
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Cited by 204 (6 self)
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Jacobianfree NewtonKrylov (JFNK) methods are synergistic combinations of Newtontype methods for superlinearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations. The link between the two methods is the Jacobianvector 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 integrodierential equations. In this survey article we place JFNK in context with other nonlinear solution algorithms for both boundary 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 EditorinChief of the Journal of Computational
Parallel Optimisation Algorithms for Multilevel Mesh Partitioning
 Parallel Comput
, 2000
"... Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent optimisation problems in interface regions. The next, alternating optimisation, is ..."
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Cited by 55 (14 self)
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Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent optimisation problems in interface regions. The next, alternating optimisation, is a restriction of this technique in which mesh entities are only allowed to migrate between subdomains in one direction. The third treats the gain as a potential field and uses the concept of relative gain for selecting appropriate vertices to migrate. The results are compared and seen to produce very high global quality partitions, very rapidly. The results are also compared with another partitioning tool and shown to be of higher quality although taking longer to compute. 2000 Elsevier Science B.V. All rights reserved.
Globalized Newton–Krylov–Schwarz algorithms and software for parallel implicit CFD
 Int. J. High Perform. Comput. Appl
"... Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is essential. The pseudotransient matrixfree NewtonKrylovSchwarz ( Y NKS) algorithmic ..."
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Cited by 46 (18 self)
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Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is essential. The pseudotransient matrixfree NewtonKrylovSchwarz ( Y NKS) algorithmic framework is presented as a widely applicable answer. This article shows that for the classical problem of threedimensional transonic Euler flow about an M6 wing, Y NKS can simultaneously deliver globalized, asymptotically rapid convergence through adaptive pseudotransient continuation and Newton’s method; reasonable parallelizability for an implicit method through deferred synchronization and favorable communicationtocomputation scaling in the Krylov linear solver; and high per processor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner. Two discouraging features of Y NKS methods are their sensitivity to the coding of the underlying PDE discretization and the large number of parameters that must be selected to govern convergence. The authors therefore distill several recommendations from their experience and reading of the literature on various algorithmic components of Y NKS, and they describe a freely available MPIbased portable parallel software implementation of the solver employed here. 1
A minimum overlap restricted additive Schwarz preconditioner and applications in 3D flow simulations
 Contemporary Mathematics
, 1998
"... Numerical simulations of unsteady threedimensional compressible flow problems require the solution of large, sparse, nonlinear systems of equations arising from the discretization of Euler or NavierStokes equations on unstructured, possibly dynamic, meshes. In this ..."
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Cited by 27 (2 self)
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Numerical simulations of unsteady threedimensional compressible flow problems require the solution of large, sparse, nonlinear systems of equations arising from the discretization of Euler or NavierStokes equations on unstructured, possibly dynamic, meshes. In this
A Parallel DynamicMesh Lagrangian Method For Simulation Of Flows With Dynamic Interfaces
, 2000
"... Many important phenomena in science and engineering, including our motivating problem of microstructural blood flow, can be modeled as flows with dynamic interfaces. The major challenge faced in simulating such flows is resolving the interfacial motion. Lagrangian methods are ideally suited for such ..."
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Cited by 9 (2 self)
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Many important phenomena in science and engineering, including our motivating problem of microstructural blood flow, can be modeled as flows with dynamic interfaces. The major challenge faced in simulating such flows is resolving the interfacial motion. Lagrangian methods are ideally suited for such problems, since interfaces are naturally represented and propagated. However, the material description of motion results in dynamic meshes, which become hopelessly distorted unless they are regularly regenerated. Lagrangian methods are particularly challenging on parallel computers, because scalable dynamic mesh methods remain elusive. Here, we present a parallel dynamic mesh Lagrangian method for flows with dynamic interfaces. We take an aggressive approach to dynamic meshing by triangulating the propagating grid points at every timestep using a scalable parallel Delaunay algorithm. Contrary to conventional wisdom, we show that the costs of the geometric components (triangulation, coarsening, refinement, and partitioning) can be made small relative to the flow solver.
Some observations on the l 2 convergence of the additive Schwarz preconditioned GMRES method
 Numer. Linear Algebra Appl. 9
, 2002
"... Abstract. Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence theory is available only in the energy ..."
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Cited by 4 (0 self)
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Abstract. Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence theory is available only in the energy norm (or the equivalent Sobolev H 1 norm). Very little progress has been made in the theoretical understanding of the l 2 behavior of this very successful algorithm. To add to the difficulty in developing a full l 2 theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l 2 can not be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the EisenstatElmanSchultz theory [11], has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the SaadSchultz theory [18], is bounded from both above and below by constants multiplied by h −1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l 2 convergence theory and in other areas of domain decomposition methods. Key words. Domain decomposition, additive Schwarz preconditioner, Krylov subspace iterative method, finite elements, eigenvalue, eigenmatrix AMS(MOS) subject classifications. 65N30, 65F10 1. Introduction. Additive Schwarz (AS
A Parallel NewtonKrylov Method for Rotarywing Flow field Calculations
 AIM Journal
, 1999
"... The use of Krylov subspace iterative metbods for tile implicit solution of rotarywing ftowtields on parallel com· puters is explored. A NewtonKrylov scheme is proposed that couples conjul!ategradlentlike Iterative methods within the baseline structuredgrid EulerlNavierStokes flow solver, trans ..."
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Cited by 2 (2 self)
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The use of Krylov subspace iterative metbods for tile implicit solution of rotarywing ftowtields on parallel com· puters is explored. A NewtonKrylov scheme is proposed that couples conjul!ategradlentlike Iterative methods within the baseline structuredgrid EulerlNavierStokes flow solver, transonic unsteady rotor Naviel'Stokes. Two Krylov methods are studied, generalized minimum residual and orthogonal sostep orthomin. Preconditioning is performed with a parallelized form of the 10weI'upper symmetric GaussSeldel operator. The scheme is imple mented on the IBM SP2 multiprocessor and applied to threedimensional computations of a rotor In forward flight. The NewtonKrylov scheme is found to be more robust and to attain a higher level ofUme accuracy In implicit time stepping, increasing tbe allowable time step. The method yields approximately a 20 % reduction In solution time with the same level of accuracy In timeaccurate calculations but requires more memory than do more traditional implicit techniqes. T Introduction methods make them well suited for CPO calculations on largescale HE accurate numerical simulation of the aerodynamics and massively parallel petaflop computer architectures. the aeroacoustics of rotarywing aircraft is acomplex and chal In this paper, we investigate the performance of Krylov subspace lenging problem. Threedimensional unsteady EulerlNavierStokes iterative solvers applied to threedimensional calculations of a rotor
Domain Decomposition Based Algorithms for Inverse Problems
, 1999
"... Contents 1 Introduction 1 1.1 Inverse problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Methods used to solve such problems : : : : : : : : : : : : : : : : : : 3 1.3 Objectives : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.4 Outline of contents : : : : ..."
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Cited by 1 (0 self)
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Contents 1 Introduction 1 1.1 Inverse problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Methods used to solve such problems : : : : : : : : : : : : : : : : : : 3 1.3 Objectives : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.4 Outline of contents : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2 Domain Decomposition methods 6 2.1 Why Domain Decomposition ? : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Some Domain Decomposition methods : : : : : : : : : : : : : : : : : 6 2.3 Problem partitioning and decomposition hierarchy : : : : : : : : : : : 6 2.4 PETSc : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 3 A metal cutting problem 7 3.1 The dimensionless 2d nonlinear metal cutting problem : : : : : : : : 8 3.2 Problem partitionin
Parallel Mesh Partitioning on Distributed Memory Systems
 Parallel & Distributed Processing for Computational Mechanics. SaxeCoburg Publications
, 1999
"... We discuss the problem of deriving parallel mesh partitioning algorithms for mapping unstructured meshes to parallel computers. In itself this raises a paradox  we seek to find a high quality partition of the mesh, but to compute it in parallel we require a partition of the mesh. In fact, we ov ..."
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Cited by 1 (1 self)
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We discuss the problem of deriving parallel mesh partitioning algorithms for mapping unstructured meshes to parallel computers. In itself this raises a paradox  we seek to find a high quality partition of the mesh, but to compute it in parallel we require a partition of the mesh. In fact, we overcome this difficulty by deriving an optimisation strategy which can find a high quality partition even if the quality of the initial partition is very poor and then use a crude distribution scheme for the initial partition. The basis of this strategy is to use a multilevel approach combined with local refinement algorithms. Three such refinement algorithms are outlined and some example results presented which show that they can produce very high global quality partitions, very rapidly. The results are also compared with a similar multilevel serial partitioner and shown to be almost identical in quality. Finally we consider the impact of the initial partition on the results and demonstrate that the final partition quality is, modulo a certain amount of noise, independent of the initial partition.
Some observations on the l² convergence of the additive Schwarz preconditioned GMRES method
 NUMER. LINEAR ALGEBRA APPL
, 2002
"... Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l² norm, however, the optimal convergence theory is available only in the energy norm (or ..."
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Cited by 1 (0 self)
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Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l² norm, however, the optimal convergence theory is available only in the energy norm (or the equivalent Sobolev H¹ norm). Very little progress has been made in the theoretical understanding of the l² behavior of this very successful algorithm. To add to the difficulty in developing afull l² theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l² can not be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the EisenstatElmanSchultz theory [11], has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the SaadSchultz theory [18], is bounded from both above and below by constants multiplied by h −1/2.Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l² convergence theory and in other areas of domain decomposition methods.