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Topological Techniques for Efficient Rigorous Computations in Dynamics
, 2001
"... This paper is an expository article on using topological methods for the efficient, rigorous computation of dynamical systems. Of course, since its inception the computer has been used for the purpose of simulating nonlinear models. However, in recent years there has been a rapid development in nume ..."
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Cited by 27 (10 self)
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This paper is an expository article on using topological methods for the efficient, rigorous computation of dynamical systems. Of course, since its inception the computer has been used for the purpose of simulating nonlinear models. However, in recent years there has been a rapid development in numerical methods specifically designed to study of these models from a dynamical systems point of view, i.e. with a particular emphasis on the structures which capture the longterm or asymptotic states of the system. At the risk of greatly simplifying these results, this work has followed two themes: indirect methods and direct methods. The indirect methods are most closely associated with simulations and as such are extremely important because they tend to be the cheapest computationally. The emphasis is on developing numerical schemes whose solutions exhibit the same dynamics as the original system, e.g. if one is given a Hamiltonian system, then it is reasonable to want a numerical method that preserves the integrals of the original system. A comprehensive introduction to these questions can be found in [61]. The direct methods focus on the development of numerical techniques that find particular dynamical structures, e.g. fixed points, periodic orbits, heteroclinic orbits, invariant manifolds, etc., and are often associated with continuation methods (see [7, 15, 14] and references therein). To paraphrase Poincare, these techniques provide us with a window into the rich structures that nonlinear systems exhibit. There is no question that these methods are essential. However, they cannot capture the full dynamics. As pointed out in [61, p. xiii] a fundamental question for the indirect method, that requires a positive answer, is "Assume that the differential equation has a parti...
Globalization Techniques for Newton–Krylov Methods and Applications to the FullyCoupled
 Solution of the Navier–Stokes Equations, Tech. Report Sand20041777, Sandia National Laboratories
, 2003
"... Abstract. A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are usually glo ..."
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Cited by 14 (5 self)
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Abstract. A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are usually globalized, i.e., augmented with auxiliary procedures (globalizations) that improve the likelihood of convergence from a starting point that is not near a solution. In recent years, globalized Newton–Krylov methods have been used increasingly for the fully coupled solution of largescale problems. In this paper, we review several representative globalizations, discuss their properties, and report on a numerical study aimed at evaluating their relative merits on largescale two and threedimensional problems involving the steadystate Navier–Stokes equations. Key words. Newton’s method, inexact Newton methods, Newton iterative methods, Newton–Krylov methods, globalized Newton methods, backtracking, line search, trustregion methods, dogleg methods, fully coupled solution methods, Navier–Stokes equations
ARNOLDI AND JACOBIDAVIDSON METHODS FOR GENERALIZED EIGENVALUE PROBLEMS Ax = λBx WITH SINGULAR B
, 2007
"... In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem Ax = λBx are needed. If exact linear solves with A − σB are available, implicitly restarted Arnoldi with purification is a common approach for problems where B is positive semidefinite. In this p ..."
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Cited by 8 (0 self)
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In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem Ax = λBx are needed. If exact linear solves with A − σB are available, implicitly restarted Arnoldi with purification is a common approach for problems where B is positive semidefinite. In this paper, a new approach based on implicitly restarted Arnoldi will be presented that avoids most of the problems due to the singularity of B. Secondly, if exact solves are not available, JacobiDavidson QZ will be presented as a robust method to compute a few specific eigenvalues. Results are illustrated by numerical experiments.
Implementation of extended systems using symbolic algebra
 in Continuation Methods in Fluid Dynamics, Notes on Numerical Fluid Mechanics
, 2000
"... Following the pioneering work of Keller and others in the 1970s and ’80s, numerical techniques for solving nonlinear systems of equations that exhibit bifurcations have been developed to the point where they can potentially be applied to a wide range of problems arising in continuum mechanics. The ..."
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Cited by 3 (1 self)
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Following the pioneering work of Keller and others in the 1970s and ’80s, numerical techniques for solving nonlinear systems of equations that exhibit bifurcations have been developed to the point where they can potentially be applied to a wide range of problems arising in continuum mechanics. The central idea is to augment the discretised governing equations with one or more conditions so that the ‘extended system ’ characterises a particular bifurcation point. By computing paths of singular points the behaviour of the system under investigation can be mapped out in a comprehensive fashion. Of considerable practical difficulty when implementing these methods is that they require the evaluation of derivatives of the discretised equations with respect to both the independent variables and the parameters. The higher the codimension of the singularity being sought, the higher the order of the derivatives required. Evaluating these derivatives is both tedious and error prone. An efficient method for computing the necessary derivatives for discretisations based on the Galerkin finiteelement method will be presented that takes advantage of a symbolic algebra package. Our method makes it possible to deal with complicated nonlinearities in a very straightforward manner. We demonstrate the complexity of systems that may be addressed by considering Marangoni convection in a twodimensional domain with a deformable free surface. 1 1
Periodic motion in highsymmetric flow
 In Proceedings of the IUTAM meeting on Elementary Vortices and Coherent Structures. Kyoto
, 2004
"... Abstract We investigate unstable periodic motion embedded in isotropic turbulence with high symmetry. Several orbits of different period are continued from the regime of weak turbulence into developed turbulence. The orbits of short period diverge from the turbulent state as the Reynolds number incr ..."
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Cited by 2 (1 self)
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Abstract We investigate unstable periodic motion embedded in isotropic turbulence with high symmetry. Several orbits of different period are continued from the regime of weak turbulence into developed turbulence. The orbits of short period diverge from the turbulent state as the Reynolds number increases but the orbit of longest period we analysed, about five eddyturnover times, represents several average values of the turbulence well. In particular we measure the energy dissipation rate and the largest Lyapunov exponent as a function of the viscosity. At the largest microscale Reynolds number attained in the continuation we compare the energy spectra of periodic and turbulent motion. The results suggest that periodic motion of a sufficiently long period can represent turbulence in a statistical sense.
JACOBIAN FREE COMPUTATION OF LYAPUNOV EXPONENTS
"... Abstract. The purpose of this paper is to present new algorithms to approximate Lyapunov exponents of nonlinear differential equations, without using Jacobian matrices. We first derive first order methods for both continuous and discrete QR approaches, and then second order methods. Numerical testin ..."
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Abstract. The purpose of this paper is to present new algorithms to approximate Lyapunov exponents of nonlinear differential equations, without using Jacobian matrices. We first derive first order methods for both continuous and discrete QR approaches, and then second order methods. Numerical testing is given, showing considerable savings with respect to existing implementations. 1.
A New Algorithm for Continuation and Bifurcation Analysis of Large Scale Free Surface Flows
, 2004
"... This thesis presents a new algorithm to find and follow particular solutions of parameterized nonlinear systems. Important applications often arise after spatial discretization of time dependent PDEs. We embed a block eigenvalue solver in a continuation framework for the computation of some specific ..."
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This thesis presents a new algorithm to find and follow particular solutions of parameterized nonlinear systems. Important applications often arise after spatial discretization of time dependent PDEs. We embed a block eigenvalue solver in a continuation framework for the computation of some specific eigenvalues of large Jacobian matrices that depend on one or more parameters. The new approach is then employed to study the behavior of an industrial process referred to as coating. Stability analysis of the discretized system that models this process is important because it provides alternatives for changing parameters in order to improve the quality of the final product or to increase productivity. Experiments on several problems show the reliability of the new approach in the accurate detection of critical points. Further analysis of twodimensional coating flow problems reveals that computational results are competitive with those of previous continuation approaches. As a byproduct, one obtains information about the stability of the process with no additional cost. Due to the size and structure of the matrices generated in threedimensional free surface flow applications, it is necessary to use a general iterative linear solver, such as GMRES. However, GMRES displays a very slow rate of convergence as a consequence of the poor conditioning in the coefficient matrices. To speed up GMRES convergence, we developed and implemented a scalable approximate sparse inverse preconditioner. Numerical experiments demonstrate that this preconditioner greatly improves the convergence of the method. Results illustrate the effectiveness of the preconditioner on very large fr...
techniques to a model of a kidney nephron
"... Abstract — Numerical continuation and bifurcation techniques are applied to a delay equation model of a nephron, the main functioning unit of a kidney. The effect of different forms for the delay on the dynamics are considered. While qualitative behavioural similarities occur, significant quantitati ..."
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Abstract — Numerical continuation and bifurcation techniques are applied to a delay equation model of a nephron, the main functioning unit of a kidney. The effect of different forms for the delay on the dynamics are considered. While qualitative behavioural similarities occur, significant quantitative differences emerge. For some forms of the delay, the ‘linear chain trick ’ enables the model to be written as a system of ordinary differential equations and the continuation and bifurcation package AUTO may be used. However, for a discrete delay, recent developments in the numerical solution of functional differential equations are necessary. We discuss the use of DDEBIFTOOL in this case. I.
1 Background Parallel Methods for PDE Eigenvalue Problems GR/M59075/01
"... The numerical solution of large sparse eigenvalue problems arising from discretised PDEs is an important problem in computational mathematics with many and varied applications. It arises, for example, in stability assessment in structural mechanics and in the computation of resonances in electromagn ..."
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The numerical solution of large sparse eigenvalue problems arising from discretised PDEs is an important problem in computational mathematics with many and varied applications. It arises, for example, in stability assessment in structural mechanics and in the computation of resonances in electromagnetics. In this project our main target application was in fluid flow problems governed by the incompressible NavierStokes equations, where Hopf bifurcations are typically detected by linearised stability analysis. With respect to our target application, the PI had worked for many years with K.A. Cliffe of SERCO Assurance on iterative methods for eigenvalues, based on shiftinvert strategies, but only when the systems were small enough to admit a direct inner solver (e.g. [5]). These fail to work in the case of practically important model problems such as the stability of 2D flow in an expanding pipe to 3D perturbations. The main aims of the present project were (i) to produce a numerical analysis of eigenvalue iterations based on inexact inner solves and (ii) to build parallel software based on this analysis (using domain decomposition methods for the inexact inner solves) to compute eigenvalues of linear PDEs, in particular the linearised NavierStokes equations. When the project started there was relatively little analysis on eigenvalue iterative methods with inexact inner solves. However during the course of the project there has been a considerable growth of interest in this
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"... Multiple solutions and stability of con®ned convective and swirling ¯ows a continuing challenge ..."
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Multiple solutions and stability of con®ned convective and swirling ¯ows a continuing challenge