Parallelisation of partial differential equations via representation theory
Sheehan Olver
TL;DR
The paper demonstrates that discretisations of PDEs which respect discrete symmetries can be decoupled into independent, parallelizable linear systems by constructing symmetry-adapted bases from irreducible representations. Using detailed crash-course material on representation theory for groups like $D_4$ and $O_h$, it shows how to obtain block-diagonal forms via explicit bases or numerical decomposition, with Schur's lemma guaranteeing sparsity and decoupling. In practice, this yields significant reductions in the effective system size: for the square, 6 blocks under $D_4$ (vs 4 under simple parity), and for the cube, 20 blocks under $O_h$ (vs 8 under $S_4\times C_2$), enabling parallel solves and substantial speedups, especially in high dimensions or with permutation symmetries. The framework extends to multi-particle Schrödinger problems, where permutation and negation symmetries yield even larger decouplings, and it points to future applications in vector-valued PDEs, preconditioning, and spectral methods, with notable potential in Boson/Fermion computations and high-dimensional settings.
Abstract
Incorporating symmetries into the numerical solution of differential equations has been a mainstay of research over the last 40 years, however, one aspect is less known and under-utilised: discretisations of partial differential equations that commute with symmetry actions (like rotations, reflections or permutations) can be decoupled into independent systems solvable in parallel by incorporating knowledge from representation theory. We introduce this beautiful subject via a crash course in representation theory focussed on hands-on examples for the symmetry groups of the square and cube, and its utilisation in the construction of so-called symmetry-adapted bases. Schur's lemma, which is not well-known in applied mathematics, plays a powerful role in proving sparsity of resulting discretisations and thereby showing that partial differential equations do indeed decouple. Using Schrödinger equations as a motivating example, we demonstrate that a symmetry-adapted basis leads to a significant increase in the number of independent linear systems. Counterintuitively, the effectiveness of this approach is in fact greater for partial differential equations with less symmetries, for example a Schrödinger equation where the potential is only invariant under permutations, but not under rotations or reflections. We also explore this phenomenon as the dimension of the partial differential equation becomes large, hinting at the potential for significant savings in high-dimensions.