On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions

Griewank, Andreas

doi:10.1137/1027141

Cited by 109 publications

(92 citation statements)

References 39 publications

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“…Then, based on Reddien's advances [41,42], Decker and Kelley [6,7] precise the convergence rate for singular problems of first and second orders for maps between Banach spaces. Griewank and Osborne [14,16,15] propose generalizations and precise convergence domains.…”

Section: Introductionmentioning

confidence: 99%

“…As for the deflation techniques, the main idea is the construction of a system that admits a simple solution in place of the singular solution of the original system, hence Newton's operator can be used. For double zeroes, this technique is explored in [56,57,15,22,23,33]. Tsuchiya [52] is based on [57] and treats multiple zeroes having corank 1.…”

Section: Introductionmentioning

confidence: 99%

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On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

et al. 2006

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Abstract. Isolated multiple zeroes or clusters of zeroes of analytic maps with several variables are known to be difficult to locate and approximate. This article is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the eighties. This theory restricts to simple zeroes, i.e., where the map has corank zero. In this article we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeroes and a fast algorithm for approximating them, with quadratic convergence. In case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

et al. 2006

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“…These methods work well for problems with wellconditioned Jacobians at the solution (they have a quadratic convergence rate), but they face difficulties when the Jacobian is singular or nearly singular at the solution. Many authors have analyzed the behavior of Newton's method on singular problems and have proposed acceleration techniques as remedies (see, e.g., Decker, Keller, and Kelley [14]; Decker and Kelley [15,16,17]; Griewank [29]; Griewank and Osborne [31]; Kelley and Suresh [39]; and Reddien [47]). Their collective analysis shows that Newton's method without acceleration is locally q-linearly convergent with the ratio of the norm of consecutive residuals converging to 1 2 .…”

Section: Introductionmentioning

confidence: 99%

Robust large-scale parallel nonlinear solvers for simulations.

Bader¹,

Pawlowski²,

Kolda

2005

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This report documents research to develop robust and efficient solution techniques for solving large-scale systems of nonlinear equations. The most widely used method for solving systems of nonlinear equations is Newton's method. While much research has been devoted to augmenting Newton-based solvers (usually with globalization techniques), little has been devoted to exploring the application of different models. Our research has been directed at evaluating techniques using different models than Newton's method: a lower order model, Broyden's method, and a higher order model, the tensor method. We have developed large-scale versions of each of these models and have demonstrated their use in important applications at Sandia.Broyden's method replaces the Jacobian with an approximation, allowing codes that cannot evaluate a Jacobian or have an inaccurate Jacobian to converge to a solution. Limitedmemory methods, which have been successful in optimization, allow us to extend this approach to large-scale problems. We compare the robustness and efficiency of Newton's method, modified Newton's method, Jacobian-free Newton-Krylov method, and our limited-memory Broyden method. Comparisons are carried out for large-scale applications of fluid flow simulations and electronic circuit simulations. Results show that, in cases where the Jacobian was inaccurate or could not be computed, Broyden's method converged in some cases where Newton's method failed to converge. We identify conditions where Broyden's method can be more efficient than Newton's method.We also present modifications to a large-scale tensor method, originally proposed by Bouaricha, for greater efficiency, better robustness, and wider applicability. Tensor methods are an alternative to Newton-based methods and are based on computing a step based on a local quadratic model rather than a linear model. The advantage of Bouaricha's method is that it can use any existing linear solver, which makes it simple to write and easily portable. However, the method usually takes twice as long to solve as Newton-GMRES on general problems because it solves two linear systems at each iteration. In this paper, we discuss modifications to Bouaricha's method for a practical implementation, including a special globalization technique and other modifications for greater efficiency. We present numerical results showing computational advantages over Newton-GMRES on some realistic problems.We further discuss a new approach for dealing with singular (or ill-conditioned) matrices. In particular, we modify an algorithm for identifying a turning point so that an increasingly ill-conditioned Jacobian does not prevent convergence.iv

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“…We will discuss the approximate size of the linear rates, rk m after the statement of Theorem 1.3. The behavior of iterative methods applied to singular problems in more than one variable is more complex [2], [4]- [9], [12]- [16], [19]- [21], [24], [25], [29], [30]. To begin with, it is no longer the case that F'(x) remains nonsingular in a deleted neighborhood of the root.…”

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confidence: 99%