Topological Optimization of the Evaluation of Finite Element Matrices

TL;DR

This paper introduces a topological framework that optimizes finite element matrix evaluations by reducing computational complexity through graph-based algorithms, significantly decreasing operation counts.

Contribution

It proposes a novel topological approach using graph theory and minimum spanning trees to optimize the evaluation of finite element matrices, which is a new methodology in this area.

Findings

Significant reduction in operation count for common multilinear forms

Introduction of complexity-reducing relations as a new concept

Efficient algorithms for constructing the optimization graph

Abstract

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization.