%0 Journal Article %1 Shroff_1993 %A Shroff, Gautam M. %A Keller, Herbert B. %D 1993 %I Society for Industrial & Applied Mathematics (SIAM) %J SIAM Journal on Numerical Analysis %K 65h10-systems-of-nonlinear-algebraic-equations %N 4 %P 1099--1120 %R 10.1137/0730057 %T Stabilization of Unstable Procedures: The Recursive Projection Method %V 30 %X Fixed-point iterative procedures for solving nonlinear parameter dependent problems can con- verge 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 continua- tion 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.

@article{Shroff_1993, abstract = {Fixed-point iterative procedures for solving nonlinear parameter dependent problems can con- verge 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 continua- tion 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. }, added-at = {2019-10-04T06:59:19.000+0200}, author = {Shroff, Gautam M. and Keller, Herbert B.}, biburl = {https://www.bibsonomy.org/bibtex/2c2b97d14b519b680531ac05a686c6d95/gdmcbain}, doi = {10.1137/0730057}, interhash = {dc4015a635b17d8d2d615cf9af3e173d}, intrahash = {c2b97d14b519b680531ac05a686c6d95}, journal = {{SIAM} Journal on Numerical Analysis}, keywords = {65h10-systems-of-nonlinear-algebraic-equations}, month = aug, number = 4, pages = {1099--1120}, publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})}, timestamp = {2019-10-04T06:59:19.000+0200}, title = {Stabilization of Unstable Procedures: The Recursive Projection Method}, volume = 30, year = 1993 }