Combinatorial Optimization of Matrix-Vector Multiplication in Finite Element Assembly

It has been shown that combinatorial optimization of matrix-vector multiplication can lead to faster evaluation of finite element stiffness matrices. Based on a graph model characterizing relationships between rows, an efficient set of operations can be generated to perform matrix-vector multiplication for this problem. We improve the graph model by extending the set of binary row relationships and solve this combinatorial optimization problem optimally for the binary row relationships implemented, yielding significantly improved results over previous published graph models. We also extend the representation by using hypergraphs to model more complicated row relationships, expressing a three-row relationship with a three-vertex hyperedge, for example. Our initial greedy algorithm for this hypergraph model has yielded significantly better results than the graph model for many matrices.