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26
Parallel NewtonKrylovSchwarz Algorithms For The Transonic Full Potential Equation
, 1998
"... We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finite ..."
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Cited by 42 (27 self)
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We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finitedifference Newton method and a Krylov space iterative method, with a twolevel overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and economical for this class of mixed elliptichyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fillin in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributedmemory parallel computer.
Nonlinearly preconditioned inexact Newton algorithms
 SIAM J. Sci. Comput
, 2000
"... Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at loc ..."
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Cited by 35 (14 self)
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Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of �F �, especiallyfor problems with unbalanced nonlinearities, because the methods do not have builtin machineryto deal with the unbalanced nonlinearities. To find the same solution u ∗ , one maywant to solve instead an equivalent nonlinearlypreconditioned system F(u ∗ ) = 0 whose nonlinearities are more balanced. In this paper, we propose and studya nonlinear additive Schwarzbased parallel nonlinear preconditioner and show numericallythat the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails. Key words. nonlinear preconditioning, inexact Newton methods, Krylov subspace methods, nonlinear additive Schwarz, domain decomposition, nonlinear equations, parallel computing, incompressible
Analysis of Inexact TrustRegion SQP Algorithms
 RICE UNIVERSITY, DEPARTMENT OF
, 2000
"... In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximatio ..."
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Cited by 17 (2 self)
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In this paper we extend the design of a class of compositestep trustregion SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trustregion SQP method or from approximations of firstorder derivatives. Accuracy requirements in our trustregion SQP methods are adjusted based on feasibility and optimality of the iterates. Our accuracy requirements are stated in general terms, but we show how they can be enforced using information that is already available in matrixfree implementations of SQP methods. In the absence of inexactness our global convergence theory is equal to that of Dennis, ElAlem, Maciel (SIAM J. Optim., 7 (1997), pp. 177207). If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trustregion methods with inexact gradient information fo...
Spectral residual method without gradient information for solving largescale nonlinear systems: Theory and experiments
, 2004
"... Abstract. A fully derivativefree spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for ..."
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Cited by 14 (3 self)
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Abstract. A fully derivativefree spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than wellknown NewtonKrylov methods for largescale problems is also presented. 1.
Analysis of Inexact TrustRegion InteriorPoint SQP Algorithms
, 1995
"... In this paper we analyze inexact trustregion interiorpoint (TRIP) sequential quadratic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applicati ..."
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Cited by 11 (7 self)
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In this paper we analyze inexact trustregion interiorpoint (TRIP) sequential quadratic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonlinear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is expensive. Often, the solution of linear systems and derivatives are computed inexactly yielding nonzero residuals. This paper analyzes the effect of the inexactness onto the convergence of TRIP SQP and gives practical rules to control the size of the residuals of these inexact calculations. It is shown that if the size of the residuals is of the order of both the size of the constraints and the trustregion radius, t...
Consistent Initial Condition Calculation For DifferentialAlgebraic Systems
, 1995
"... In this paper we describe a new algorithm for the calculation of consistent initial conditions for a class of systems of differentialalgebraic equations which includes semiexplicit indexone systems. We consider initial condition problems of two typesone where the differential variables are speci ..."
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Cited by 9 (1 self)
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In this paper we describe a new algorithm for the calculation of consistent initial conditions for a class of systems of differentialalgebraic equations which includes semiexplicit indexone systems. We consider initial condition problems of two typesone where the differential variables are specified, and one where the derivative vector is specified. The algorithm requires a minimum of additional information from the user. We outline the implementation in a generalpurpose solver DASPK for differentialalgebraic equations, and present some numerical experiments which illustrate its effectiveness.
On the Convergence Theory of TrustRegionBased Algorithms for EqualityConstrained Optimization
, 1995
"... In this paper we analyze incxact trust region interior point (TRIP) sequential quadr tic programming (SOP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applicati ..."
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Cited by 8 (0 self)
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In this paper we analyze incxact trust region interior point (TRIP) sequential quadr tic programming (SOP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonhnear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of hncarizcd equations is expensive. Often, the solution of hncar systems and derivatives arc computed incxactly yielding nonzero residuals. This paper
Practical quasiNewton methods for solving nonlinear systems
, 2000
"... Practical quasiNewton methods for solving nonlinear systems are surveyed. The definition of quasiNewton methods that includes Newton 's method as a particular case is adopted. However, especial emphasis is given to the methods that satisfy the secant equation at every iteration, which are call ..."
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Cited by 8 (2 self)
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Practical quasiNewton methods for solving nonlinear systems are surveyed. The definition of quasiNewton methods that includes Newton 's method as a particular case is adopted. However, especial emphasis is given to the methods that satisfy the secant equation at every iteration, which are called here, as usually, secant methods. The leastchange secant update (LCSU) theory is revisited and convergence results of methods that do not belong to the LCSU family are discussed. The family of methods reviewed in this survey includes Broyden 's methods, structured quasiNewton methods, methods with direct updates of factorizations, rowscaling methods and columnupdating methods. Some implementation features are commented. The survey includes a discussion on global convergence tools and linearsystem implementations of Broyden's methods. In the final section, practical and theoretical perspectives of this area are discussed. 1 Introduction In this survey we consider nonlinear ...
Solving Nonlinear Systems of Equations With Simple Constraints
, 1996
"... A general algorithm for solving F (x) = 0, x 2 \Omega\Gamma where F : IR n ! IR n is a differentiable function and\Omega is closed and convex, is introduced. The new method uses the inexactNewton approach with a globalization procedure. Conditions are given that ensure global convergence ..."
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Cited by 6 (2 self)
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A general algorithm for solving F (x) = 0, x 2 \Omega\Gamma where F : IR n ! IR n is a differentiable function and\Omega is closed and convex, is introduced. The new method uses the inexactNewton approach with a globalization procedure. Conditions are given that ensure global convergence to a solution of the problem. Numerical experiments where \Omega is an n\Gammadimensional box are presented. Keywords. Nonlinear systems, inexact  Newton method, global convergence, convex constraints, box constraints. AMS: 65H10 Department of Mathematics, Federal University of Santa Catarina, Florian'opolis, Santa Catarina, Brazil. This author was supported by FAPESP (Grant 9037246). y Department of Applied Mathematics, IMECCUNICAMP, University of Campinas, CP 6065, 13081970 Campinas SP, Brazil (martinez@ime.unicamp.br). This author was supported by FAPESP (Grant 9037246), CNPq, FINEP and FAEPUNICAMP. z Department of Mathematics, IMECCUNICAMP, University of Campinas, CP 6...
Solving Nonlinear Systems Of Equations By Means Of QuasiNewton Methods With A Nonmonotone Strategy
, 1997
"... A nonmonotone strategy for solving nonlinear systems of equations is introduced. The idea consists of combining efficient local methods with an algorithm that reduces monotonically the squared norm of the system in a proper way. The local methods used are Newton's method and two quasiNewton algorith ..."
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Cited by 5 (2 self)
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A nonmonotone strategy for solving nonlinear systems of equations is introduced. The idea consists of combining efficient local methods with an algorithm that reduces monotonically the squared norm of the system in a proper way. The local methods used are Newton's method and two quasiNewton algorithms. Global iterations are based on recently introduced boxconstrained minimization algorithms. We present numerical experiments. 1 INTRODUCTION Given F : IR n ! IR n ; F = (f 1 ; : : : ; f n ) T , our aim is to find solutions of F (x) = 0: (1) We assume that F is well defined and has continuous partial derivatives on an open set of IR n . J(x) denotes the Jacobian matrix of partial derivatives of F (x). We are mostly interested in problems where n is large and J(x) is structurally sparse. This means that most entries of J(x) are zero for all x in the domain of F . The package Nightingale has been developed at the Department of Applied Mathematics of the University of Campinas for...