# Performance of Low Synchronization Orthogonalization Methods in Anderson Accelerated Fixed Point Solvers

@article{Lockhart2021PerformanceOL, title={Performance of Low Synchronization Orthogonalization Methods in Anderson Accelerated Fixed Point Solvers}, author={Shelby Lockhart and David J. Gardner and Carol S. Woodward and Stephen J. Thomas and Luke N. Olson}, journal={ArXiv}, year={2021}, volume={abs/2110.09667} }

Anderson Acceleration (AA) is a method to accelerate the convergence of fixed point iterations for nonlinear, algebraic systems of equations. Due to the requirement of solving a least squares problem at each iteration and a reliance on modified Gram-Schmidt for updating the iteration space, AA requires extra costly synchronization steps for global reductions. Moreover, the number of reductions in each iteration depends on the size of the iteration space. In this work, we introduce three low…

## References

SHOWING 1-10 OF 26 REFERENCES

Considerations on the implementation and use of Anderson acceleration on distributed memory and GPU-based parallel computers

- Computer Science
- 2016

Performance results show that for sufficiently large problems a GPU implementation of Anderson acceleration can provide a significant performance increase over CPU versions due to the GPU’s higher memory bandwidth.

Low-Synch Gram-Schmidt with Delayed Reorthogonalization for Krylov Solvers

- Mathematics, Computer ScienceArXiv
- 2021

Strong-scaling results are presented for finding eigenvalue-pairs of nonsymmetric matrices and a new variant of CGS2 that requires only one reduction per iteration is applied to the Arnoldi-QR iteration.

Anderson Acceleration for Fixed-Point Iterations

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2011

It is shown that, on linear problems, Anderson acceleration without truncation is “essentially equivalent” in a certain sense to the generalized minimal residual (GMRES) method and the Type 1 variant in the Fang-Saad Anderson family is similarly essentially equivalent to the Arnoldi (full orthogonalization) method.

On the Influence of the Orthogonalization Scheme on the Parallel Performance of GMRES

- Computer ScienceEuro-Par
- 1998

It is shown that the iterative classical Gram-Schmidt method overcomes its three competitors in speed and in parallel scalability while keeping robust numerical properties.

Two classes of multisecant methods for nonlinear acceleration

- Computer Science, MathematicsNumer. Linear Algebra Appl.
- 2009

This work presents two classes of multisecant methods which allows to take into account a variable number of secant equations at each iteration of a fixed point iteration, whereby a fixed Point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’.

Solving linear least squares problems by Gram-Schmidt orthogonalization

- Mathematics
- 1967

AbstractA general analysis of the condition of the linear least squares problem is given. The influence of rounding errors is studied in detail for a modified version of the Gram-Schmidt…

Rounding error analysis of the classical Gram-Schmidt orthogonalization process

- Mathematics, Computer ScienceNumerische Mathematik
- 2005

It is shown that, provided the initial set of vectors has numerical full rank, two iterations of the classical Gram-Schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level.

The Effects of Loss of Orthogonality on Large Scale Numerical Computations

- Computer ScienceICCSA
- 2018

A nice theoretical indicator of loss of orthogonality and linear independence is discussed and it is shown how it leads to a related higher dimensional orthog onality that can be used to analyze and prove the effectiveness of such algorithms.

Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization

- Mathematics
- 1976

Numerically stable algorithms are given for updating the GramSchmidt QR factorization of an m X n matrix A (m > n) when A is modified by a matrix of rank one, or when a row or column is inserted or…

Iterative Procedures for Nonlinear Integral Equations

- Mathematics, Computer ScienceJACM
- 1965

A procedure is synthesized to offset some of the disadvantages of these t e c h n i q u e s in this context; however, the procedure is not restricted to this pt~rticular class of s y s t e m s of nonlinear equations.