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109
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 78 (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
SUNDIALS: Suite of Nonlinear and Differential/ Algebraic Equation Solvers
 ACM Trans. Math. Software
, 2005
"... SUNDIALS is a suite of advanced computational codes for solving largescale problems that can be modeled as a system of nonlinear algebraic equations, or as initialvalue problems in ordinary differential or differentialalgebraic equations. The basic versions of these codes are called KINSOL, CVOD ..."
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Cited by 69 (1 self)
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SUNDIALS is a suite of advanced computational codes for solving largescale problems that can be modeled as a system of nonlinear algebraic equations, or as initialvalue problems in ordinary differential or differentialalgebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact NewtonKrylov methods for solving largescale nonlinear systems; linear multistep methods for timedependent problems; a highly modular structure to allow incorporation of different 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 firstorder sensitivity analysis with respect to model parameters or initial conditions.
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 45 (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.
Globalized NewtonKrylovSchwarz algorithms and software for parallel implicit CFD
 Int. J. High Performance Computing Applications
, 1998
"... Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. 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 e ..."
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Cited by 39 (15 self)
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Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. 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 (ΨNKS) 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, ΨNKS 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 perprocessor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner. Two discouraging features of ΨNKS 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. We therefore distill several recommendations from our experience and from our reading of the literature on various algorithmic components of ΨNKS, and we describe a freely available, MPIbased portable parallel software implementation of the solver employed here. 1. Introduction. Disparate
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 (15 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
MPSalsaA finite element computer program for reacting flow problems. Part 2: User’s guide and examples
, 1996
"... ..."
Ultrascalable Implicit Finite Element Analyses in Solid Mechanics with over a Half a Billion Degrees of Freedom
 In ACM/IEEE Proceedings of SC2004: High Performance Networking and Computing
, 2004
"... We present a highly parallel finite element program, Olympus, equipped with an ultrascalable linear solver, Prometheus, applied to microFE bone modeling calculations on an IBM SP Power3. Scalability is demonstrated with scaled speedup studies of a nonlinear analyses of a vertebral body with over a ..."
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Cited by 22 (0 self)
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We present a highly parallel finite element program, Olympus, equipped with an ultrascalable linear solver, Prometheus, applied to microFE bone modeling calculations on an IBM SP Power3. Scalability is demonstrated with scaled speedup studies of a nonlinear analyses of a vertebral body with over a half of a billion degrees of freedom. We show parallel scalability with up to 4088 processors on the ACSI White machine. This work is significant in that, in the domain of unstructured implicit finite element analysis in solid mechanics with complex geometry, this is the first demonstration of a highly parallel, and e#cient, application of a mathematically optimal linear solution methodsmoothed aggregation algebraic multigrid.
A GrassmannRayleigh Quotient Iteration for Computing Invariant Subspaces
 SIAM REVIEW
, 2002
"... The classical Rayleigh quotient iteration (RQI) allows one to compute a onedimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a pdimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low com ..."
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Cited by 21 (9 self)
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The classical Rayleigh quotient iteration (RQI) allows one to compute a onedimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a pdimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.
Cubically convergent iterations for invariant subspace computations
 SIAM J. Matrix Anal. Appl
"... Abstract. We propose a Newtonlike iteration that evolves on the set of fixed dimensional subspaces of Rn and converges locally cubically to the invariant subspaces of a symmetric matrix. This iteration is compared in terms of numerical cost and global behavior with three other methods that display ..."
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Cited by 18 (7 self)
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Abstract. We propose a Newtonlike iteration that evolves on the set of fixed dimensional subspaces of Rn and converges locally cubically to the invariant subspaces of a symmetric matrix. This iteration is compared in terms of numerical cost and global behavior with three other methods that display the same property of cubic convergence. Moreover, we consider heuristics that greatly improve the global behavior of the iterations.
Efficient Minimization Method for a Generalized Total Variation Functional
, 2009
"... Replacing the ℓ² data fidelity term of the standard Total Variation (TV) functional with an ℓ¹ data fidelity term has been found to offer a number of theoretical and practical benefits. Efficient algorithms for minimizing this ℓ¹TV functional have only recently begun to be developed, the fastest of ..."
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Cited by 14 (4 self)
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Replacing the ℓ² data fidelity term of the standard Total Variation (TV) functional with an ℓ¹ data fidelity term has been found to offer a number of theoretical and practical benefits. Efficient algorithms for minimizing this ℓ¹TV functional have only recently begun to be developed, the fastest of which exploit graph representations, and are restricted to the denoising problem. We describe an alternative approach that minimizes a generalized TV functional, including both ℓ²TV and ℓ¹TV as special cases, and is capable of solving more general inverse problems than denoising (e.g. deconvolution). This algorithm is competitive with the graphbased methods in the denoising case, and is the fastest algorithm of which we are aware for general inverse problems involving a nontrivial forward linear operator.