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Stabilization of Unstable Procedures: The Recursive Projection Method

Shroff, Gautam M. and Keller, Herbert B. (1993) Stabilization of Unstable Procedures: The Recursive Projection Method. SIAM Journal on Numerical Analysis, 30 (4). pp. 1099-1120. ISSN 0036-1429.

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Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a “black-box” time integration scheme is stabilized, enabling it to compute unstable steady states. The RPM can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes.

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Additional Information:© 1993 Society for Industrial and Applied Mathematics. Received by the editors June 3, 1991; accepted for publication (in revised form) September 11, 1992. This work was supported in part by the National Science Foundation Cooperative Agreement CCR-9120008 and the Department of Energy Project Agreement No. DE-FG03-89ER25073. The government has certain rights on this material.
Funding AgencyGrant Number
NSF Cooperative AgreementCCR-9120008
Department of Energy (DOE) Project AgreementDE-FG03-89ER25073
Subject Keywords:stabilization procedures; projection methods; stabilized continuation
Classification Code:AMS Subject Headings: 65B99, 65M10, 65N20
Record Number:CaltechAUTHORS:20120307-153620928
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Official Citation:Stabilization of Unstable Procedures: The Recursive Projection Method Gautam M. Shroff and Herbert B. Keller, SIAM J. Numer. Anal. 30, 1099 (1993), DOI:10.1137/0730057
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29630
Deposited By: Jason Perez
Deposited On:12 Mar 2012 21:47
Last Modified:26 Dec 2012 14:55

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