Reducing Inter-Process Communication Overhead in Parallel Sparse Matrix-Matrix Multiplication

Parallel sparse matrix-matrix multiplication algorithms (PSpGEMM) spend most of their running time on inter-process communication. In the case of distributed matrix-matrix multiplications, much of this time is spent on interchanging the partial results that are needed to calculate the final product matrix. This overhead can be reduced with a one-dimensional distributed algorithm for parallel sparse matrix-matrix multiplication that uses a novel accumulation pattern based on the logarithmic complexity of the number of processors (i.e., where is the number of processors). This algorithm's MPI communication overhead and execution time were evaluated on an HPC cluster, using randomly generated sparse matrices with dimensions up to one million by one million. The results showed a reduction of inter-process communication overhead for matrices with larger dimensions compared to another one dimensional parallel algorithm that takes run-time complexity for accumulating the results.

Efficient Computation of Highly Oscillatory Integrals by Using QTT Tensor Approximation

We propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter ${\omega \ge 0}$, typically varying in a large interval. Our approach is based, for a fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain ω-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals which makes them difficult to evaluate. Furthermore, they have to be approximated typically in large intervals. Here we use the quantized-tensor train (QTT) approximation method for functional M-vectors of logarithmic complexity in M in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions.