Fast 3D Block Parallelisation for the Matrix Multiplication Prefix Problem - Application in Quantum Control

For exploiting the power of supercomputers like the HLRB-II cluster, developing parallel algorithms becomes increasingly important. The matrix prefix problem belongs to a class of issues lending themselves for parallelisation. We compare the tree-based parallel prefix scheme, which is adapted from a recursive approach, with a sequential multiplication scheme where only the individual matrix multiplications are parallelised. We show that this fine-grain approach outperforms the parallel prefix scheme by a factor of 2−3 and also leads to less memory requirements. Unlike the tree-based scheme, the fine-grain approach enables many options in the choice of the number of parallel processors and shows a better speedup performance when increasing the matrix sizes. The usage of the fine-grain approach in a quantum control algorithm instead of the coarse-grain approach allows us both to deal with systems of higher dimensions and to choose a finer discretisation. Introduction: The Prefix Problem In general, the prefix problem is given as follows: Let ◦ be a binary operator and A a set, which is closed under the operator ◦. Furthermore, let ◦ be associative, i.e. the identity (a ◦ b) ◦ c = a ◦ (b ◦ c) holds for all a,b,c ∈ A . Then, for given elements x1, . . . ,xM ∈A , the prefix problem means the computation of all the products yi = x1 ◦ · · · ◦ xi . (1) Analogously, the suffix problem amounts to computing of all the products K. Waldherr, T. Huckle and T. Auckenthaler Dept. of Computer Science, TU Munich, D-85748 Garching, Germany, e-mail: [email protected] U. Sander and T. Schulte-Herbrüggen Dept. of Chemistry, TU Munich, D-85747 Garching, Germany, e-mail: [email protected]