LOCA 1.0 Library of Continuation Algorithms: Theory and Implementation Manual

Salinger, Andrew G.; Bou-Rabee, Nawaf; Burroughs, Elizabeth A.; Pawlowski, Roger P.; Lehoucq, Richard B.; Romero, Louis A.; Wilkes, Edward D.

doi:10.2172/800778

Cited by 73 publications

(103 citation statements)

References 12 publications

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“…This capability has been verified and validated for numerous fluid flow applications and has demonstrated parallel scaling to millions of unknowns [12,13], and is briefly described in Section 2.3. The LOCA (Library of Continuation Algorithms) library [14] has also been interfaced with the MPSalsa code for directly calculating bifurcations [15]. A Newtonbased algorithm in LOCA is used to converge directly to the instability, converging the parameter value and solution simultaneously.…”

Section: 0604x10 5 =mentioning

confidence: 99%

“…6 is solved using Arnoldi's method with a version of the P_ARPACK software [10,11] driven by software in the LOCA library for performing the generalized Cayley transformation [14]. The approximate matrix inversions are solved using the Aztec package, exactly the same as in the Newton iterations.…”

Section: Linear Stability Analysis Algorithmsmentioning

confidence: 99%

“…A set of Newton algorithms for directly locating and tracking a bifurcation points has been implemented as part of the LOCA library at Sandia National Labs [14]. These have been developed to be relatively non invasive to simplify implementation within an application code and to work with codes that are based on iterative linear solvers.…”

Section: Hopf Bifurcation Tracking Algorithmmentioning

confidence: 99%

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Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Salinger

Lehoucq

Pawlowski

et al. 2002

Numerical Methods in Fluids

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Summary:Stability analysis algorithms coupled with a robust Newton-Krylov steady state iterative solver are used to understand the behavior of the 2D model problem of thermal convection in a 8:1 differentially heated cavity. Parameter continuation methods along with bifurcation and linear stability analysis are used to study transition from steady to transient flow as a function of Rayleigh number. To carry out this study the steady state form of the governing PDEs is discretized using a Galerkin/Least Squares Finite Element formulation, and solved on parallel computers using a fully coupled Newton method and preconditioned Krylov iterative linear solvers. Linear stability analysis employing a large scale eigenvalue capability is used to determine the stability of the steady solutions. The boundary between steady and time dependent flows is determined by a Hopf bifurcation tracking capability that is used to directly track the instability with respect to the aspect ratio of the system and with respect to mesh resolution. The effect of upwinding stabilization terms in the finite element formulation on the computed value of critical Rayleigh number is investigated. The Hopf bifurcation signaling the onset of flow is determined to occur at a critical Rayleigh number of .

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Section: 0604x10 5 =mentioning

confidence: 99%

Section: Linear Stability Analysis Algorithmsmentioning

confidence: 99%

Section: Hopf Bifurcation Tracking Algorithmmentioning

confidence: 99%

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Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Salinger

Lehoucq

Pawlowski

et al. 2002

Numerical Methods in Fluids

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“…More information about continuation methods (for both real and homotopy parameters) can be found from Salinger et al [192].…”

Section: Homotopymentioning

confidence: 99%

Sensitivity technologies for large scale simulation.

Collis¹,

Bartlett²,

Smith³

et al. 2005

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“…We discuss a practical algorithm in LOCA [54,55] for solving this system. It is called a bordered algorithm, and it may be derived from a block elimination procedure.…”

Section: Bordered Algorithmmentioning

confidence: 99%

Robust large-scale parallel nonlinear solvers for simulations.

Bader¹,

Pawlowski²,

Kolda

2005

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This report documents research to develop robust and efficient solution techniques for solving large-scale systems of nonlinear equations. The most widely used method for solving systems of nonlinear equations is Newton's method. While much research has been devoted to augmenting Newton-based solvers (usually with globalization techniques), little has been devoted to exploring the application of different models. Our research has been directed at evaluating techniques using different models than Newton's method: a lower order model, Broyden's method, and a higher order model, the tensor method. We have developed large-scale versions of each of these models and have demonstrated their use in important applications at Sandia.Broyden's method replaces the Jacobian with an approximation, allowing codes that cannot evaluate a Jacobian or have an inaccurate Jacobian to converge to a solution. Limitedmemory methods, which have been successful in optimization, allow us to extend this approach to large-scale problems. We compare the robustness and efficiency of Newton's method, modified Newton's method, Jacobian-free Newton-Krylov method, and our limited-memory Broyden method. Comparisons are carried out for large-scale applications of fluid flow simulations and electronic circuit simulations. Results show that, in cases where the Jacobian was inaccurate or could not be computed, Broyden's method converged in some cases where Newton's method failed to converge. We identify conditions where Broyden's method can be more efficient than Newton's method.We also present modifications to a large-scale tensor method, originally proposed by Bouaricha, for greater efficiency, better robustness, and wider applicability. Tensor methods are an alternative to Newton-based methods and are based on computing a step based on a local quadratic model rather than a linear model. The advantage of Bouaricha's method is that it can use any existing linear solver, which makes it simple to write and easily portable. However, the method usually takes twice as long to solve as Newton-GMRES on general problems because it solves two linear systems at each iteration. In this paper, we discuss modifications to Bouaricha's method for a practical implementation, including a special globalization technique and other modifications for greater efficiency. We present numerical results showing computational advantages over Newton-GMRES on some realistic problems.We further discuss a new approach for dealing with singular (or ill-conditioned) matrices. In particular, we modify an algorithm for identifying a turning point so that an increasingly ill-conditioned Jacobian does not prevent convergence.iv

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LOCA 1.0 Library of Continuation Algorithms: Theory and Implementation Manual

Cited by 73 publications

References 12 publications

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Sensitivity technologies for large scale simulation.

Robust large-scale parallel nonlinear solvers for simulations.

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