Numerical methods for parameter continuation problems

Abstract

This thesis is concerned with numerical procedures suitable for the study of highdimensional dynamical systems that arise from the discretisation of partial differential equations. In this context, we discuss several areas that might arise in a more complete programme of computer-based analysis of physical systems. Model building Galerkin discretisation is a typical step in the model-building phase of engineering analysis. We discuss a simple generalisation of Ainsworth's formulation (M. Ainsworth, Essential boundary conditions and multi-point constraints in finite element analysis, Comput. Meth. Appl. Mech. Eng. 190 (48) (2001) 6323-6339) for the imposition of general linear constraints on Galerkin weak forms, leading to an explicit expression for the constraint-reduced residual and corresponding -tangent matrices arising from nonlinear evolutionary problems. Continuation with variable-order predictors Numerical continuation is a means of computing discrete approximations to equilibrium branches of parameterised dynamical systems. We implement the multi-step prediction scheme of Lundberg & Poore (Variable order Adams-Bashforth predictors with an error-stepsize control for continuation methods, SIAM J. Sci. Stat. Comput. 12 (3) (1991) 695-723), and describe an interface accommodating iterative correctors suitable for large-scale problems. We also experiment with high-order Hermite interpolants, and compare the two schemes with a popular first-order model. Bifurcation points Points at which bifurcations occur along a continuation curve are distinguished in the sense that they are associated with a transition in stability or loss of local solution uniqueness. We review various direct formulations - fully-extended, minimallyextended, and reduced - for the computation of codimension-one bifurcation points; we then detail an implementation and high-level interface, based on algorithmic differentiation software, for evaluation of the augmented residuals and -tangent operators required for iterative computation of codimension-one bifurcation points. The cost of numerical continuation - of equilibria and bifurcation points - is dominated by that of bordered linear systems of equations. We survey various blockelimination procedures for preconditioning these systems and, as a device for testing software for the computation of bifurcation points of arbitrary codimension, we discuss a simple procedure for the manufacture of artificial systems whose tangent matrices at a prescribed equilibrium point have a specified Jordan block. Eigendecompositions Bifurcations are characterised by the eigenvalues of the system's tangent matrices. We review issues that arise in the computation of smoothly-varying eigendecompositions (cf. Bindel et al. , Continuation of invariant subspaces in large bifurcation problems, SIAM J. Sci. Comput. 30 (2 (2008) 637--65) and eigenpair tracking for the purpose of detecting bifurcation along a continuation curve. We experiment with a new acceleration scheme for the generalised Sylvester equation AXB + BXC = D, which arises in the update of partial eigendecompositions. The resulting scheme is found to be of limited utility. Further to earlier discussion on multi-step (state) predictors, we experiment briefly with multi-step subspace predictors - that is, interpolation on the Grassmann manifold of subspaces. We noted no significant difference in performance between interpolants constructed in the tangent space, which respect the curved structure of the manifold, and Euclidean interpolants that do not.