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?
... We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.
On the Hopf Algebraic Structure of Lie Group Integrators H. Z. Munthe-Kaas, W. M. Wright
Hopf algebras of formal diffeomorphisms and numerical integration on manifolds Alexander Lundervold, Hans Munthe-Kaas
Hopf algebras are in some sense "almost groups":) so hopefully this should qualify.
In general it seems to me that these series of works by Connes, Kreimer, Broadhurst, Moscovich & K on the Hopf algebras in various fields of math is quite fascinating...
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.
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.
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.
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.
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..