Large sparse continuation problems
This article begins by reviewing continuation methods in general, using clear and careful notation; this includes a brief review of literature on exploitation of matrix structure in large continuation problems. The next section deals with incorporation of a nonlinear conjugate gradient method for the corrector phase (as opposed to using a linear conjugate gradient method for a Newton's method-based corrector, as other authors have done). In this method, the line searches are taylored to the fact that we can adjust the predictor steplength so that higher-order terms in the objective function are negligible. Despite the context of a solution manifold instead of isolated solutions, several convergence results are presented, and it is conjectured that the scheme is locally superlinearly convergent. Practicalities, such as preconditioners and use of higher-order predictors to offset the somewhat slower convergence of the conjugate gradient method, are discussed. Handling bifurcation in this context is discussed; local perturbations are used, since determinant sign changes are not readily available from the nonlinear conjugate gradient method. Numerical experiments dealing with nonlinear eigenvalue problems corresponding to buckled plates of various shapes are given; solutions are presented graphically. The reference list is sufficiently complete to guide a beginner through a thorough self-study of the underlying concepts.
scientific article; zbMATH DE number 4128307
scientific article