Mark Hoemmen

James Demmel, Laura Grigori, Mark Hoemmen et al.

We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. We prove optimality by extending known lower bounds on communication bandwidth for sequential and parallel matrix multiplication to provide latency lower bounds, and show these bounds apply to the LU and QR decompositions. We not only show that our QR algorithms attain these lower bounds (up to polylogarithmic factors), but that existing LAPACK and ScaLAPACK algorithms perform asymptotically more communication. We also point out recent LU algorithms in the literature that attain at least some of these lower bounds.

James Demmel, Laura Grigori, Mark Hoemmen et al.

We present parallel and sequential dense QR factorization algorithms for tall and skinny matrices and general rectangular matrices that both minimize communication, and are as stable as Householder QR. The sequential and parallel algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms, including LAPACK and ScaLAPACK, for example up to 6.7x over ScaLAPACK. The parallel algorithm for general rectangular matrices is estimated to show significant speedups over ScaLAPACK, up to 22x over ScaLAPACK.

Patrick G. Bridges, Kurt B. Ferreira, Michael A. Heroux et al.

Energy increasingly constrains modern computer hardware, yet protecting computations and data against errors costs energy. This holds at all scales, but especially for the largest parallel computers being built and planned today. As processor counts continue to grow, the cost of ensuring reliability consistently throughout an application will become unbearable. However, many algorithms only need reliability for certain data and phases of computation. This suggests an algorithm and system codesign approach. We show that if the system lets applications apply reliability selectively, we can develop algorithms that compute the right answer despite faults. These "fault-tolerant" iterative methods either converge eventually, at a rate that degrades gracefully with increased fault rate, or return a clear failure indication in the rare case that they cannot converge. Furthermore, they store most of their data unreliably, and spend most of their time in unreliable mode. We demonstrate this for the specific case of detected but uncorrectable memory faults, which we argue are representative of all kinds of faults. We developed a cross-layer application / operating system framework that intercepts and reports uncorrectable memory faults to the application, rather than killing the application, as current operating systems do. The application in turn can mark memory allocations as subject to such faults. Using this framework, we wrote a fault-tolerant iterative linear solver using components from the Trilinos solvers library. Our solver exploits hybrid parallelism (MPI and threads). It performs just as well as other solvers if no faults occur, and converges where other solvers do not in the presence of faults. We show convergence results for representative test problems. Near-term future work will include performance tests.

James Demmel, Laura Grigori, Mark Hoemmen et al.

We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. Our first algorithm, Tall Skinny QR (TSQR), factors m-by-n matrices in a one-dimensional (1-D) block cyclic row layout, and is optimized for m >> n. Our second algorithm, CAQR (Communication-Avoiding QR), factors general rectangular matrices distributed in a two-dimensional block cyclic layout. It invokes TSQR for each block column factorization.

Matthias Mayr, Alexander Heinlein, Christian Glusa et al.

Trilinos is a community-developed, open-source software framework that facilitates building large-scale, complex, multiscale, multiphysics simulation code bases for scientific and engineering problems. Since the Trilinos framework has undergone substantial changes to support new applications and new hardware architectures, this document is an update to ``An Overview of the Trilinos project'' by Heroux et al. (ACM Transactions on Mathematical Software, 31(3):397-423, 2005). It describes the design of Trilinos, introduces its new organization in product areas, and highlights established and new features available in Trilinos. Particular focus is put on the modernized software stack based on the Kokkos ecosystem to deliver performance portability across heterogeneous hardware architectures. This paper also outlines the organization of the Trilinos community and the contribution model to help onboard interested users and contributors.