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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
A restricted additive Schwarz preconditioner for general sparse linear systems
 SIAM J. Sci. Comput
, 1999
"... Abstract. We introduce some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems and show, by numerical examples, that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost w ..."
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Cited by 129 (24 self)
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Abstract. We introduce some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems and show, by numerical examples, that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost when implemented on distributed memory computers. This is especially true for harder problems such as indefinite complex linear systems and systems of convectiondiffusion equations from threedimensional compressible flows. Both sequential and parallel results are reported. Key words. Overlapping domain decomposition, preconditioner, iterative method, sparse matrix AMS(MOS) subject classifications. 65N30, 65F10
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
An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms
 SIAM J. NUMER. ANAL
, 2001
"... Convergence results for the restrictive additive Schwarz (RAS) method of Cai and Sarkis [SIAM J. Sci. Comput., 21 (1999), pp. 792–797] for the solution of linear systems of the form Ax = b are provided using an algebraic view of additive Schwarz methods and the theory of multisplittings. The linear ..."
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Cited by 20 (9 self)
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Convergence results for the restrictive additive Schwarz (RAS) method of Cai and Sarkis [SIAM J. Sci. Comput., 21 (1999), pp. 792–797] for the solution of linear systems of the form Ax = b are provided using an algebraic view of additive Schwarz methods and the theory of multisplittings. The linear systems studied are usually discretizations of partial differential equations in two or three dimensions. It is shown that in the case of A symmetric positive definite, the projections defined by the methods are not orthogonal with respect to the inner product defined by A, and therefore the standard analysis cannot be used here. The convergence results presented are for the class of Mmatrices (and more generally for Hmatrices) using weighted max norms. Comparison between different versions of the RAS method are given in terms of these norms. A comparison theorem with respect to the classical additive Schwarz method makes it possible to indirectly get quantitative results on rates of convergence which otherwise cannot be obtained by the theory. Several RAS variants are considered, including new ones and twolevel schemes.
Convergence theory of restricted multiplicative Schwarz methods
 IN PREPARATION
, 2003
"... Convergence results for the restricted multiplicative Schwarz (RMS) method, the multiplicative version of the restricted additive Schwarz (RAS) method for the solution of linear systems of the form Ax = b, are provided. An algebraic approach is used to prove convergence results for nonsymmetric Mm ..."
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Cited by 10 (6 self)
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Convergence results for the restricted multiplicative Schwarz (RMS) method, the multiplicative version of the restricted additive Schwarz (RAS) method for the solution of linear systems of the form Ax = b, are provided. An algebraic approach is used to prove convergence results for nonsymmetric Mmatrices. Several comparison theorems are also established. These theorems compare the asymptotic rate of convergence with respect to the amount of overlap, the exactness of the subdomain solver, and the number of domains. Moreover, comparison theorems are given between the RMS and RAS methods as well as between the RMS and the classical multiplicative Schwarz method.
Coarse grid acceleration of a parallel block preconditioner
"... A block preconditioner is considered in a parallel computing environment. This preconditioner has good parallel properties, however, the convergence deteriorates when the number of blocks increases. Two different techniques are studied to accelerate the convergence: overlapping at the interfaces and ..."
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Cited by 9 (4 self)
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A block preconditioner is considered in a parallel computing environment. This preconditioner has good parallel properties, however, the convergence deteriorates when the number of blocks increases. Two different techniques are studied to accelerate the convergence: overlapping at the interfaces and using a coarse grid correction. It appears that the latter technique is indeed scalable, so the wall clock time is constant when the number of blocks increases. Furthermore, the method is easily added to an existing solution code. © 2001 Elsevier Science B.V. All rights reserved.
Flow simulation around a micro air vehicle in a plume characterization scenario
 AIAA paper 20046598, American Institute of Aeronautics and Astronautics, 3rd Unmanned Unlimited Technical Conference, Workshop and Exhibit
, 2004
"... Numerical simulation of flow around a recently developed micro air vehicle (MAV) at the University of Colorado is presented. The vehicle design is so a network of such vehicles could be used for plume characterization in urban areas. The MAV network will map out the concentration of a plume and calc ..."
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Cited by 4 (3 self)
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Numerical simulation of flow around a recently developed micro air vehicle (MAV) at the University of Colorado is presented. The vehicle design is so a network of such vehicles could be used for plume characterization in urban areas. The MAV network will map out the concentration of a plume and calculation of concentration gradient will guide the network toward the source of the contamination. Computational results of the aerodynamic characteristics of the Colorado fixed wing MAV are presented. A morphing wing concept based on biomimetic of bat’s flight is proposed. The adaptive wing suggested in this study changes shape according to the flight modes and conditions extending fully while loitering or in light winds and becoming more sweptback and flexed to cruise between locations or in stronger winds. This means that the wing is more adaptable and overall energy consumption of the vehicle will be reduced. 1
An explicit multimodel compressible flow formulation based on the full potential equation and the Euler equations
 In Proc. of the Eleventh Intl. Conference on Domain Decomposition Methods in Scientific and Engineering Computing
, 1999
"... The development of a multimodel formulation to simulate three dimensional compressible flows on parallel computers is presented. The goal is to reduce the overall time and memory required to simulate the flow by using locally selected cheaper and more computational efficient physical models without ..."
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Cited by 3 (2 self)
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The development of a multimodel formulation to simulate three dimensional compressible flows on parallel computers is presented. The goal is to reduce the overall time and memory required to simulate the flow by using locally selected cheaper and more computational efficient physical models without sacrificing the
Numerical simulation of flow around the Colorado micro aerial vehicle
 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS, 35TH AIAA FLUID DYNAMICS CONFERENCE AND EXHIBIT
, 2005
"... Micro aerial vehicles (MAVs) are distinguished by their small size, low aspect ratio, and low velocity. As a result, MAVs fly at low Reynolds number flow regimes with significant drag characteristics and strong tip vortices. This investigation is focused on the aerodynamic characteristics of a recen ..."
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Cited by 3 (1 self)
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Micro aerial vehicles (MAVs) are distinguished by their small size, low aspect ratio, and low velocity. As a result, MAVs fly at low Reynolds number flow regimes with significant drag characteristics and strong tip vortices. This investigation is focused on the aerodynamic characteristics of a recently developed MAV at the University of Colorado. The Colorado MAV has a flexible membrane wing with an aspect ratio of 1.2 and a chord of 0.27 m. Numerical simulations of the flow around the Colorado fixed wing MAV are presented using a steady state parallel compressible NavierStokes solver. The computational grid has 510,000 nodes and about 3 million tetrahedral elements. The maximum calculated lift coefficient is approximately 1.2. The airplane stall angle is at 30 ◦. The high stall angle is attributed to the enhanced lift from a low pressure region above the wing caused by strong tip vortices. Minimum drag coefficient was calculated to be 0.06 at 2 ◦ angle of attack. A laminar separation bubble is formed on the upper surface of the wing at moderate angle of attack. The drag increases rapidly as the angle of attack increases. A maximum aerodynamic efficiency of L/D = 4 is observed when flying at 10 m/s.