Anderson Acceleration with Truncated Gram-Schmidt: Numerical Methods for Nonlinear Equations

Core Concepts

Anderson Acceleration with Truncated Gram-Schmidt (AATGS) offers advantages over classical AA in solving symmetric linear problems.

Abstract

Anderson Acceleration (AA) enhances convergence of fixed-point iterations.
AATGS reduces memory and computational costs for symmetric linear problems.
Convergence analysis shows equivalence of AATGS to classical AA in the linear case.
Limited-depth AATGS(m) simplifies orthogonalization process for non-symmetric nonlinear problems.
Restarting strategy monitors errors and improves accuracy in numerical computations.
Experiments demonstrate AATGS performance on Bratu problem with low nonlinearity.

Stats

Anderson Acceleration (AA) has found applications in scientific computing and machine learning.
The full-depth AA(∞) is essentially equivalent to the GMRES method applied to a specific linear system.
The convergence rate of AA(m) is not worse than that of the underlying fixed point iteration.
AA(m) can improve convergence rate by a factor τj ≤ 1 equal to the ratio of certain norms.
The r-linear convergence factor of AA(m) depends on initial conditions and coefficients θj oscillate during convergence.
Superlinearly and sublinearly converging fixed point iterations have been studied for AA(m).

Quotes

Key Insights Distilled From

Anderson Acceleration with Truncated Gram-Schmidt

by Ziyuan Tang,... at arxiv.org 03-25-2024

https://arxiv.org/pdf/2403.14961.pdf

Deeper Inquiries

How does AATGS compare to other numerical methods for solving nonlinear equations

AATGS, or Anderson Acceleration with Truncated Gram-Schmidt, offers several advantages when compared to other numerical methods for solving nonlinear equations. One key benefit is its ability to exploit the symmetry of the Jacobian matrix in certain cases, leading to a reduction in memory and computational costs. The short-term recurrence feature of AATGS allows for more efficient updates and can improve convergence rates, especially in situations where the problem exhibits linear characteristics or near-symmetry. Additionally, AATGS has shown promising results in reducing numerical instabilities and improving overall performance on both symmetric and non-symmetric problems.

What are potential drawbacks or limitations of using Anderson Acceleration with Truncated Gram-Schmidt

While Anderson Acceleration with Truncated Gram-Schmidt (AATGS) offers many benefits, there are potential drawbacks or limitations associated with its use. One limitation is that AATGS may encounter numerical instabilities when applied to certain types of problems, particularly those that deviate significantly from linearity or symmetry. The reliance on short-term recurrences could lead to issues such as error propagation over multiple iterations if not carefully monitored. Additionally, the need for parameter tuning and monitoring thresholds like η in the restarting strategy adds complexity to the implementation of AATGS.

How can the concepts discussed in this content be applied to real-world optimization problems

The concepts discussed in this context can be applied to real-world optimization problems by leveraging Anderson Acceleration with Truncated Gram-Schmidt (AATGS) as an efficient algorithm for enhancing convergence rates and reducing computational costs. In practical optimization scenarios where iterative methods are used to solve complex nonlinear equations or systems, implementing AATGS can lead to improved efficiency and accuracy. By exploiting symmetries within optimization models or utilizing short-term recurrences effectively, practitioners can achieve faster convergence and better performance when tackling real-world optimization challenges across various domains such as engineering design optimizations, financial modeling optimizations, machine learning algorithms refinement etc.