Anderson Acceleration

Anderson acceleration, also known in quantum chemistry as Pulay mixing or direct inversion in the iterative subspace (DIIS), is a technique for accelerating the convergence of a fixed-point iteration. It has been widely used in electronic structure computations, but does not seem to be well known to numerical analysts.

Anderson’s original paper is from 1965 and is well cited, as Google Scholar shows: and65-gs.jpg I learned about Anderson acceleration in the minisymposium Anderson Acceleration and Applications organized by Tim Kelley at the SIAM Conference on Computational Science and Engineering in Salt Lake City in March 2015. Tim gave an excellent overview of the topic in the opening talk. The slides for that talk are available on Tim’s website.

PhD student Nataša Strabić and I have shown that Anderson acceleration works very well for speeding up the alternating projections method for computing the nearest correlation matrix. It typically gives a reduction in the number of iterations by a factor at least 2 for the standard nearest correlation matrix problem and by at least a factor 3 when additional constraints are imposed on the matrix (specified elements fixed and a lower bound on the smallest eigenvalue). In some cases the reduction is by a factor of as much as 25. Since the overhead of Anderson acceleration is small, significant speedups are obtained.

In my 2013 post The Nearest Correlation Matrix I included a MATLAB code nearcorr.m. In place of this I now recommend our new accelerated code nearcorr_aa.m, which is available from the repository anderson-accel-ncm on GitHub. Our paper describing this work is available on MIMS EPrints.

For me this project is an excellent illustration of the importance of going to conferences in order to learn of new ideas and new developments.

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