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Analysis of an Adaptive Safeguarded Newton-Anderson Algorithm of Depth One with Applications to Fluid Problems

Matt Dallas, Sara Pollock, Leo G. Rebholz

Abstract

The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We present an adaptive safeguarding scheme with a tunable parameter, which we call adaptive gamma-safeguarding, that one can use in tandem with Anderson acceleration to improve the performance of Newton's method when solving problems at or near singular points. The key features of adaptive gamma-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems automatically, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. The result is a flexible algorithm that performs well for singular and nonsingular problems, and can recover convergence from both standard Newton and Newton-Anderson with the right parameter choice. This leads to faster local convergence compared to both Newton's method, and Newton- Anderson without safeguarding, with effectively no additional computational cost. We demonstrate three strategies one can use when implementing Newton-Anderson and gamma-safeguarded Newton-Anderson to solve parameter-dependent problems near singular points. For our benchmark problems, we take two parameter-dependent incompressible flow systems: flow in a channel and Rayleigh-Benard convection.

Analysis of an Adaptive Safeguarded Newton-Anderson Algorithm of Depth One with Applications to Fluid Problems

Abstract

The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We present an adaptive safeguarding scheme with a tunable parameter, which we call adaptive gamma-safeguarding, that one can use in tandem with Anderson acceleration to improve the performance of Newton's method when solving problems at or near singular points. The key features of adaptive gamma-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems automatically, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. The result is a flexible algorithm that performs well for singular and nonsingular problems, and can recover convergence from both standard Newton and Newton-Anderson with the right parameter choice. This leads to faster local convergence compared to both Newton's method, and Newton- Anderson without safeguarding, with effectively no additional computational cost. We demonstrate three strategies one can use when implementing Newton-Anderson and gamma-safeguarded Newton-Anderson to solve parameter-dependent problems near singular points. For our benchmark problems, we take two parameter-dependent incompressible flow systems: flow in a channel and Rayleigh-Benard convection.
Paper Structure (17 sections, 5 theorems, 14 equations, 14 figures, 2 algorithms)

This paper contains 17 sections, 5 theorems, 14 equations, 14 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The $\gamma$-safeguarding algorithm
  3. Adaptive $\gamma$-safeguarding
  4. Numerics
  5. Test Problems
  6. General Discussion of Results
  7. Asymptotic Safeguarding
  8. Results for Channel Flow Model
  9. Results for Rayleigh-Bénard Model
  10. Preasymptotic Safeguarding
  11. Results for Channel Flow Model
  12. Results for Rayleigh-Bénard Model
  13. Increasing Anderson Depth
  14. Results for Channel Flow Model
  15. Results for Rayleigh-Bénard Model
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $f\in C^3$, $\dim N=1$, $\hat{D}$ nonsingular, and $x_k,x_{k-1}\in \hat{W}$ so that $x_{k+1} = (x_k+w_{k+1})^{\alpha}$ is well-defined. If $P_Nw_{k+1}\neq P_Nw_k$ and $| 1-\gamma_{k+1}|\,\|P_Ne_k\|\neq |\gamma_{k+1}|\,\|P_Ne_{k-1}\|\neq 0$, then for sufficiently small $\hat{\sigma}>0$ and $\hat{ where $\hat{\gamma}:=(P_Nw_{k+1})^T(P_Nw_{k+1}-P_Nw_k)/\|P_Nw_{k+1}-P_Nw_k\|^2$.

Figures (14)

  • Figure 1: Meshes used for benchmark problems. Top: Mesh used for Rayleigh-Bénard model. Bottom: mesh used for flow in a channel.
  • Figure 2: Solutions to channel flow problem \ref{['coanda-model']} for different $\mu$. Top: Representative symmetric solution for $\mu = 1.0$. Middle: Representative asymmetric solution with positive vertical velocity upon exiting the narrow channel for $\mu = 0.9$. Bottom: Representative asymmetric solution with negative vertical velocity upon exiting the narrow channel for $\mu = 0.9$.
  • Figure 3: Velocity streamlines for velocity $u$ from Model \ref{['rayben-model']} showing transition from one eddy to two eddies. Top Left: Ri = 3.0. Top Right: Ri = 3.1. Bottom Left: Ri = 3.2. Bottom Right: Ri = 3.4.
  • Figure 4: Comparison of $\mathop{\mathrm{\gamma\text{NAA}(\hat{r})}}\nolimits$ applied asymptotically with Newton and NA applied to Model \ref{['coanda-model']} with $\mu = 0.96$.
  • Figure 5: Comparison of $\mathop{\mathrm{\gamma\text{NAA}(\hat{r})}}\nolimits$ applied asymptotically with Newton and NA applied to Model \ref{['coanda-model']} with $\mu = 0.94$.
  • ...and 9 more figures

Theorems & Definitions (8)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.1
  • proof