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71
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). In typical applications, ..."
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Cited by 94 (2 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.
Theory of Algorithms for Unconstrained Optimization
, 1992
"... this article I will attempt to review the most recent advances in the theory of unconstrained optimization, and will also describe some important open questions. Before doing so, I should point out that the value of the theory of optimization is not limited to its capacity for explaining the behavio ..."
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Cited by 84 (1 self)
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this article I will attempt to review the most recent advances in the theory of unconstrained optimization, and will also describe some important open questions. Before doing so, I should point out that the value of the theory of optimization is not limited to its capacity for explaining the behavior of the most widely used techniques. The question
Parallel LagrangeNewtonKrylovSchur methods for PDEconstrained optimization. Part I: The KrylovSchur solver
 SIAM J. Sci. Comput
, 2000
"... Abstract. Large scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. The stateoftheart for such problems is reduced quasiNewton sequential quadratic programming (SQP) methods. These methods take full advantage of existin ..."
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Cited by 72 (11 self)
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Abstract. Large scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. The stateoftheart for such problems is reduced quasiNewton sequential quadratic programming (SQP) methods. These methods take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this twopart article we propose a new method for steadystate PDEconstrained optimization, based on the idea of full space SQP with reduced space quasiNewton SQP preconditioning. The basic components of the method are: Newton solution of the firstorder optimality conditions that characterize stationarity of the Lagrangian function; Krylov solution of the KarushKuhnTucker (KKT) linear systems arising at each Newton iteration using a symmetric quasiminimum residual method; preconditioning of the KKT system using an approximate state/decision variable decomposition that replaces the forward PDE Jacobians by their own preconditioners, and the decision space Schur complement (the reduced Hessian) by a BFGS approximation or by a twostep stationary method. Accordingly, we term the new method LagrangeNewtonKrylov Schur (LNKS). It is fully parallelizable, exploits the structure of available parallel algorithms for the PDE forward problem, and is locally quadratically convergent. In the first part of the paper we investigate the effectiveness of the KKT linear system solver. We test the method on two optimal control problems in which the flow is described by the steadystate Stokes equations. The
The Adaptive Multilevel Finite Element Solution of the PoissonBoltzmann Equation on Massively Parallel Computers
 J. COMPUT. CHEM
, 2000
"... Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element soluti ..."
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Cited by 43 (14 self)
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Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element solution of the PoissonBoltzmann equation for a microtubule on the NPACI IBM Blue Horizon supercomputer. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600,000 atoms, and has a net charge of1800 e. PoissonBoltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.
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.
Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element ..."
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Cited by 41 (24 self)
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ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2 and 3manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2 and 3manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unitybased method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.
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
Parallel Multiscale GaussNewtonKrylov Methods For Inverse Wave Propagation
, 2002
"... One of the outstanding challenges of computational science and engineering is largescale nonlinear parameter estimation of systems governed by partial differential equations. These are known as inverse problems,in contradistinction to the forward problems that usually characterize largescale simu ..."
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Cited by 31 (12 self)
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One of the outstanding challenges of computational science and engineering is largescale nonlinear parameter estimation of systems governed by partial differential equations. These are known as inverse problems,in contradistinction to the forward problems that usually characterize largescale simulation. Inverse problems are significantly more difficult to solve than forward problems, due to illposedness, large dense illconditioned operators, multiple minima, spacetime coupling, and the need to solve the forward problem repeatedly. We present a parallel algorithm for inverse problems governed by timedependent PDEs, and scalability results for an inverse wave propagation problem of determining the material field of an acoustic medium. The difficulties mentioned above are addressed through a combination of total variation regularization, preconditioned matrixfree GaussNewtonKrylov iteration, algorithmic checkpointing, and multiscale continuation. We are able to solve a synthetic inverse wave propagation problem though a pelvic bone geometry involving 2.1 million inversion parameters in 3 hours on 256 processors of the Terascale Computing System at the Pittsburgh Supercomputing Center.
Gradient Method With Retards And Generalizations
 SIAM Journal on Numerical Analysis
, 1999
"... . A generalization of the steepest descent and other methods for solving a large scale symmetric positive definitive system Ax = b is presented. Given a positive integer m, the new iteration is given by x
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Cited by 28 (5 self)
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.<F3.81e+05> A generalization of the steepest descent and other methods for solving a large scale symmetric positive definitive system<F3.441e+05> Ax<F3.81e+05> =<F3.441e+05> b<F3.81e+05> is presented. Given a positive integer<F3.441e+05><F3.81e+05> m, the new iteration is given by<F3.441e+05> x<F2.521e+05><F2.733e+05> k+1<F3.81e+05> =<F3.441e+05> x<F2.521e+05> k<F3.59e+05><F3.441e+05><F3.81e+05><F3.441e+05> #(x<F2.521e+05><F2.733e+05><F2.521e+05><F2.733e+05> #(k)<F3.81e+05><F3.441e+05> )(Ax<F2.521e+05> k<F3.59e+05> <F3.441e+05><F3.81e+05> b), where<F3.441e+05><F3.81e+05><F3.441e+05> #(x<F2.521e+05><F2.733e+05><F2.521e+05><F2.733e+05> #(k)<F3.81e+05> ) is the steepest descent step at a previous iteration<F3.441e+05><F3.81e+05><F3.441e+05><F3.81e+05> #(k)<F3.59e+05> #<F3.441e+05> {k,<F3.59e+05><F3.81e+05><F3.441e+05> k1, . . . ,<F3.81e+05><F3.59e+05><F3.81e+05><F3.441e+05> max{0,<F3.59e+05><F3.441e+05><F3.59e+05><F3.81e+05> km}}. The global convergence to the solution of the p...
Inexact Newton Methods for Solving Nonsmooth Equations
 Journal of Computational and Applied Mathematics
, 1999
"... This paper investigates inexact Newton methods for solving systems of nonsmooth equations. We define two inexact Newton methods for locally Lipschitz functions and we prove local (linear and superlinear) convergence results under the assumptions of semismoothness and BDregularity at the solution. W ..."
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Cited by 24 (9 self)
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This paper investigates inexact Newton methods for solving systems of nonsmooth equations. We define two inexact Newton methods for locally Lipschitz functions and we prove local (linear and superlinear) convergence results under the assumptions of semismoothness and BDregularity at the solution. We introduce a globally convergent inexact iteration function based method. We discuss implementations and we give some numerical examples.