Showing papers by "Tony F. Chan published in 1984"

Nonlinearly Preconditioned Krylov Subspace Methods for Discrete Newton Algorithms

Abstract: We propose an algorithm for implementing Newton's method for a general nonlinear system $f(x) = 0$ where the linear systems that arise at each step of Newton's method are solved by a preconditioned Krylov subspace iterative method. The algorithm requires only function evaluations and does not require the evaluation or storage of the Jacobian matrix. Matrix-vector products involving the Jacobian matrix are approximated by directional differences. We develop a framework for constructing preconditionings for this inner iterative method which do not reference the Jacobian matrix explicitly. We derive a nonlinear SSOR type preconditioning which numerical experiments show to be as effective as the linear SSOR preconditioning that uses the Jacobian explicitly.

On the existence and computation of $LU$-factorizations with small pivots

Abstract: Let A be an n by n matrix which may be singular with a one-dimensional null space, and consider the LU-factorization of A. When A is exactly singular, we show conditions under which a pivoting strategy will produce a zero n th pivot. When A is not singular, we show conditions under which a pivoting strategy will produce an nth pivot that is O(G,,) or O(K(A)), where ,, is the smallest singular value of A and K(A) is the condition number of A. These conditions are expressed in terms of the elements of A in general but reduce to conditions on the elements of the singular vectors corresponding to ,, when A is nearly or exactly singular. They can be used to build a 2-pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices. As an example, we exhibit an LU-factorization of the n by n upper triangular matrix

Deflation Techniques and Block-Elimination Algorithms for Solving Bordered Singular Systems

Abstract: In numerical continuation methods for solving parametrized nonlinear systems, one often has to solve linear systems with matrices of the following form: \[ M = \left[ {\begin{array}{*{20}c} A & b \\ {c^T } & d \\ \end{array} } \right] \] where A may become singular but M is well conditioned. If A has special structures (e.g. sparseness, special data structure, special solver), then direct Gaussian elimination on M with pivoting will destroy the structures in A. An often used method that does exploit structures in A is the block-elimination (BE) algorithm which involves solving two systems with A for each system with M. In this paper, we show that the BE algorithm may become unstable and inaccurate when A is nearly singular. We then propose a stable variant which employs deflation techniques for solving the two systems with A. The deflation techniques can be viewed as working in coordinate systems orthogonal to the approximate null vectors of A, enabling an accurate representation of the solution to be computed. The extra work amounts to a few (e.g. 2) more backsolves with A. Backward error bounds and numerical results are presented.

Deflated Decomposition of Solutions of Nearly Singular Systems

Abstract: When solving the linear system $Ax = b$, where A may be nearly singular and b is not consistent with A, one is often interested in computing a deflated solution, i.e., a unique solution to a nearby singular but consistent system $A_s x_d = \tilde b$. Kelley [14], [15] has considered deflated solutions with $A_s $ corresponding to setting a small pivot of the LU-factorization of A to zero. Stewart [22] proposed an iterative algorithm for computing a deflated solution with $A_s $ corresponding to setting the smallest singular value of A to zero. Kelley’s approach explicitly uses submatrices of the LU-factors whereas Stewart’s approach is implicit in that it only involves solving systems with A. In this paper, we generalize the concept of a deflated solution to that of a deflated decomposition, which expresses the solution x in terms of $x_d $ and the null vectors of $A_s $. We treat such decompositions in a uniform framework that includes the approaches of Kelley and Stewart and introduce some new deflated ...

Newton-Like Pseudo-Arclength Methods for Computing Simple Turning Points

Abstract: We present a new method for computing simple turning points of nonlinear equations of the form $G(u,\lambda ) = 0$ which is based on applying Newton's method to the characterization ${{d\lambda (\sigma )} / {d\sigma }} = 0$, where $\sigma $ is a pseudo-arclength parameter used in a continuation method for following the solution paths. The method is quadratically convergent and needs only one starting point on the solution path. Second derivatives of G (or difference approximations of them) have to be computed but the method is relatively insensitive to their values and they also give rise to a more accurate second order predictor in the continuation method. We present a chord-Newton variant for improving the efficiency of the algorithm which requires only one factorization of a Jacobian matrix. We also present a damped-Newton variant for improving the robustness and the global convergence of the algorithm. Results of numerical experiments on two standard nonlinear elliptic problems of Simpson's [SIAM J. Numer. Anal., 12 (1975), pp. 439–451] show that the new algorithm compares favorably with the best of the existing methods in terms of efficiency and robustness.

An Approximate Newton Method for Coupled Nonlinear Systems

Abstract: We propose an approximate Newton method for solving the coupled nonlinear system $G(u,t) = 0$ and $N(u,t) = 0$ where $u \in R^n $, $t \in R^m $, $G:R^n \times R^m \mapsto R^n $ and $N:R^n \times R^m \mapsto R^m $. The method involves applying the basic iteration S of a general solver for the equation $G(u,t) = 0$, withtfixed. It is therefore well suited for problems for which such a solver already exists or can be implemented more efficiently than a solver for the coupled system. We derive conditions for S under which the method is locally convergent. Basically, if S is sufficiently contractive for G, then convergence for the coupled system is guaranteed. Otherwise, we show how to construct an $\hat S$ from S for which convergence is assured. These results are applied to continuation methods where N represents a pseudo-arclength condition. We show that under certain conditions the algorithm converges if S is convergent for G. Numerical results are given for a two-level nonlinear multi-grid solver applied ...

Techniques for Large Sparae Systems Arising from Continuation Methods

Abstract: 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,λ) = 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 Gu 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.