Applications of group theory in numerical analysis? - MathOverflow

Applications of group theory in numerical analysis?

2012-07-25T09:58:22Z

Are applications of group theory known to exist in numerical analysis? One particular aspect I am curious about is whether matrix groups have been successfully used to derive algorithms. Also, are there aspects of numerical analysis that would have been difficult to conceive or derive WITHOUT the help of group theory?

Answer by Mark Sapir for Applications of group theory in numerical analysis?

2012-07-25T10:09:22Z

To solve PDE exactly, you can use the "differential Galois theory" which first was studied by Sofus Lie and more recently by Ovsiannikov and his followers. See, for example Ovsyannikov, L. V. The group analysis purposes. Modern group analysis, V (Johannesburg, 1994). Lie Groups Appl. 1 (1994), no. 1, 193–202. I am sure that similar ideas can work in numerical analysis.

Answer by PaPiro for Applications of group theory in numerical analysis?

2012-07-25T11:01:01Z

I recommend P. Winternitz, Group Theory and Numerical Analysis, AMS, 2005.

From Google Books:

The Workshop on Group Theory and Numerical Analysis brought together scientists working in several different but related areas. The unifying theme was the application of group theory and geometrical methods to the solution of differential and difference equations. The emphasis was on the combination of analytical and numerical methods and also the use of symbolic computation.

Also, P.J. Oliver, Applications of Lie Groups to Differential Equations, Springer, 2000.

From Amazon:

A solid introduction to applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented such that graduates and researchers can readily learn to use them.

Answer by Alexander Chervov for Applications of group theory in numerical analysis?

2012-07-25T11:54:19Z

Alain Connes& K mentions some "Butcher group" ( http://arxiv.org/abs/hep-th/9904044 ):

... We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.

Answer by Denis Serre for Applications of group theory in numerical analysis?

2012-07-25T12:13:12Z

As far as DFT and FFT is considered as a part of Numerical Analysis, finite abelian groups apply, especially ${\mathbb Z}/2^n{\mathbb Z}$.

Answer by quid for Applications of group theory in numerical analysis?

2012-07-25T12:41:45Z

There is a (recent) approach to the problem of fast matrix multiplication that involves representation theory of finite (nonabelian) groups. (Roughly, it reduces/transfers the problem of the existence of fast algorithms to certain question on representations of finite groups.) This was pioneered by Cohn and Umans, see for example Group theoretic algorithms for matrix multiplication by Cohn, Kleinberg, Szegedy, Umans for more details.

However, as incidentally recently mentioned on MO by one of the authors these algorithms are not / cannot (yet) be used in practise.

Answer by jmbr for Applications of group theory in numerical analysis?

2013-01-12T20:05:06Z

An example of ideas from Lie groups being used for the analysis of ODE integrators can be found in Reich, S. (1999). Backward error analysis for numerical integrators. SIAM Journal on Numerical Analysis, 36(5), 1549–1570..