Numerical methods for nonlinear equations

Abstract

This article is about numerical methods for the solution of nonlinear equations. We consider both the fixed-point form $\mathbf{x}=\mathbf{G}(\mathbf{x})$ and the equations form $\mathbf{F}(\mathbf{x})=0$ and explain why both versions are necessary to understand the solvers. We include the classical methods to make the presentation complete and discuss less familiar topics such as Anderson acceleration, semi-smooth Newton’s method, and pseudo-arclength and pseudo-transient continuation methods.

On Quasi‐Newton methods in fast Fourier transform‐based micromechanics

“…One such algorithm is Anderson acceleration which was recently applied by Shantraj et al and Chen et al in the context of FFT‐based micromechanics. A general discussion of the scheme and its implementation is found, for example, in Walker and Ni or Kelley . Eyert and Fang and Saad pointed out the relation of Anderson acceleration to Quasi‐Newton schemes and identified it as a generalized form of Broyden's second method.…”

“…The initial value and upper bound for the forcing term are set to γ 0 =γ max =0.75. Choice 3 is given by γ n =0.1, that is, the forcing term is set to a constant value, corresponding to a modest accuracy for solving the linear system. Kelley suggests this choice as a simple forcing‐term strategy which works well in practice. Choice 4 sets the forcing term to a low constant value of γ n =5×10 −5 , corresponding to a high accuracy. The accuracy is chosen so that the Newton‐CG scheme converges in one step for the linear elastic case.…”

“…A general discussion of the scheme and its implementation is found, for example, in Walker and Ni 59 or Kelley. 51 Eyert 60 and Fang and Saad 29 pointed out the relation of Anderson acceleration to Quasi-Newton schemes and identified it as a generalized form of Broyden's second method. Recently, Evans et al 31 provided a proof that Anderson acceleration improves the convergence rate of linearly converging fixed-point methods.…”

“…Choice 3 is given by n = 0.1, that is, the forcing term is set to a constant value, corresponding to a modest accuracy for solving the linear system. Kelley 51 suggests this choice as a simple forcing-term strategy which works well in practice. 4.…”

Benchmarking results for the Newton–Anderson method

“…D2. Chandrasekhar H-equation from [16]. The Chandrasekhar H-equation from radiative transfer (see [16,26,27] and the references therein), is given by…”