Techniques for Large Sparse Systems Arising from Continuation Methods

摘要

We survey numerical techniques for solving the nonlinear and linear systems arising from applying continuation methods to tracing solution manifolds of parameterized nonlinear systems of the form G(u,lambda) = 0. We concentrate on large and sparse problems, e.g. discretizations of partial differential equations, for which this part of the computation dominates the overall cost. The basic issue is a tradeoff of the exploitation of the sparsity structure of the Jacobian G/sub u/ and the numerical treatment of its singularity. Among the techniques to be discussed are: Newton and quasi-Newton methods, low rank correction methods, implicit deflation techniques, Krylov subspace iterative methods and multi-grid methods. (ERA citation 09:027648)