TY - JOUR
T1 - Two classes of multisecant methods for nonlinear acceleration
AU - Fang, Haw Ren
AU - Saad, Yousef
PY - 2009/3
Y1 - 2009/3
N2 - Many applications in science and engineering lead to models that require solving large-scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi-Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allow to take into account a variable number of secant equations at each iteration. The first is the Broyden-like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola-Nevanlinna-type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed 'mixing'.
AB - Many applications in science and engineering lead to models that require solving large-scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi-Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allow to take into account a variable number of secant equations at each iteration. The first is the Broyden-like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola-Nevanlinna-type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed 'mixing'.
KW - Anderson mixing
KW - Broyden's methods
KW - Fixed point problems
KW - Quasi-Newton methods
KW - Self-consistent field (SCF) iteration
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U2 - 10.1002/nla.617
DO - 10.1002/nla.617
M3 - Article
AN - SCOPUS:65749083859
SN - 1070-5325
VL - 16
SP - 197
EP - 221
JO - Numerical Linear Algebra with Applications
JF - Numerical Linear Algebra with Applications
IS - 3
ER -