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We present efficient and scalable parallel algorithms for performing mathematical operations for low-rank tensors represented in the tensor train (TT) format. We consider algorithms for addition, elementwise multiplication, computing norms and inner products, orthonormalization, and rounding (rank truncation). These are the kernel operations for applications such as iterative Krylov solvers that exploit the TT structure. The parallel algorithms are designed for distributedmemory computation, and we propose a data distribution and strategy that parallelizes computations for individual cores within the TT format. We analyze the computation and communication costs of the proposed algorithms to show their scalability, and we present numerical experiments that demonstrate their efficiency on both shared-memory and distributed-memory parallel systems. For example, we observe better single-core performance than the existing MATLAB TT-Toolbox in rounding a 2GB TT tensor, and our implementation achieves a 34× speedup using all 40 cores of a single node. We also show nearly linear parallel scaling on larger TT tensors up to over 10,000 cores for all mathematical operations. [ABSTRACT FROM AUTHOR]

Sparse matrix-matrix multiplication (or SpGEMM) is a key primitive for many high-performance graph algorithms as well as for some linear solvers, such as algebraic multigrid. The scaling of existing parallel implementations of SpGEMM is heavily bound by communication. Even though 3D (or 2.5D) algorithms have been proposed and theoretically analyzed in the flat MPI model on Erdös--Rényi matrices, those algorithms had not been implemented in practice and their complexities had not been analyzed for the general case. In this work, we present the first implementation of the 3D SpGEMM formulation that exploits multiple (intranode and internode) levels of parallelism, achieving significant speedups over the state-of-the-art publicly available codes at all levels of concurrencies. We extensively evaluate our implementation and identify bottlenecks that should be subject to further research. [ABSTRACT FROM AUTHOR]

We consider the sequence of sparse matrix-matrix multiplications performed during the setup phase of algebraic multigrid. In particular, we show that the most commonly used parallel algorithm is often not the most communication-efficient one for all of the matrix-matrix multiplica- tions involved. By using an alternative algorithm, we show that the communication costs are reduced (in theory and practice), and we demonstrate the performance benefit for both model (structured) and more realistic unstructured problems on large-scale distributed-memory parallel systems. Our theoretical analysis shows that we can reduce communication by a factor of up to 5.4 for a model problem, and we observe in our empirical evaluation communication reductions of factors up to 4.7 for structured problems and 3.7 for unstructured problems. These reductions in communication translate to run-time speedups of factors up to 2.8 and 2.5, respectively. [ABSTRACT FROM AUTHOR]

In 1981 Hong and Kung proved a lower bound on the amount of communication (amount of data moved between a small, fast memory and large, slow memory) needed to perform dense, n-by-n matrix multiplication using the conventional O(n³) algorithm, where the input matrices were too huge to Bt in the small, East memory. In 2004 Irony, Toledo, and Tiskin gave a new proof of this result and extended it to the parallel case (where communication means the amount of data moved between processors). In both cases the lower bound may be expressed Ω(#arithinetic_operations/√M), where M is the size of the fast memory (or local memory in the parallel case). Here we generalize these results to a much wider variety of algorithms, including LU factorization. Cholesky factorization, LDLT factorization. QR factorization, the Gram--Schmidt algorithm, and algorithms for eigenvalues and singular values, i.e., essentially all direct methods of linear algebra. The proof works for dense or sparse matrices and for sequential or parallel algorithms. In addition to lower bounds on the amount of data moved (bandwidth cost), we get lower bounds on the number of messages required to move it (latency cost). We extend our lower bound technique to compositions of linear algebra operations (like Computing powers of a matrix) to decide whether it is enough to call a sequence of simpler optimal algorithms (like matrix multiplication) to minimize communication, or whether we can do better. We give examples of both. We also show how to extend our lower bounds to certain graph-theoretic problems. We point out recently designed algorithms that attain many of these lower bounds. [ABSTRACT FROM AUTHOR]