A tight lower bound for required I/O when computing a matrix-matrix multiplication on a processor with two layers of memory is established. Prior work obtained weaker lower bounds by reasoning about the number of phases needed to perform C:=AB, where each phase is a series of operations involving S reads and writes to and from fast memory, and S is the size of fast memory. A lower bound on the number of phases was then determined by obtaining an upper bound on the number of scalar multiplications performed per phase. This paper follows the same high level approach, but improves the lower bound by considering C:=AB+C instead of C:=AB, and obtains the maximum number of scalar fused multiply-adds (FMAs) per phase instead of scalar additions. Key to obtaining the new result is the decoupling of the per-phase I/O from the size of fast memory. The new lower bound is 2mnk/S - 2S where S is the size of fast memory. The constant for the leading term is an improvement of a factor 4/2. A theoretical algorithm that attains the lower bound is given, and how the state-of-the-art Goto's algorithm also in some sense meets the lower bound is discussed.
Partitioning graphs is a common and useful operation in many areas, from parallel computing to VLSI design to sparse matrix algorithms. In this paper, we introduce Mongoose, a multilevel hybrid graph par- titioning algorithm and library. Building on previous work in multilevel partitioning frameworks and com- binatoric approaches, we introduce novel stall-reducing and stall-free coarsening strategies, as well as an efficient hybrid algorithm leveraging 1) traditional combinatoric methods and 2) continuous quadratic pro- gramming formulations. We demonstrate how this new hybrid algorithm outperforms either strategy in isolation, and we also compare Mongoose to METIS and demonstrate its effectiveness on large and social networking (power law) graphs.
SuiteSparse:GraphBLAS is a full implementation of the GraphBLAS standard, which defines a set of sparse matrix operations on an extended algebra of semirings using an almost unlimited variety of operators and types. When applied to sparse adjacency matrices, these algebraic operations are equivalent to computations on graphs. GraphBLAS provides a powerful and expressive framework for creating graph algorithms based on the elegant mathematics of sparse matrix operations on a semiring. An overview of the GraphBLAS specification is given, followed by a description of the key features of its implementation in the SuiteSparse:GraphBLAS package.
?Code Generation for Generally Mapped Finite Elements? includes performance results for the finite element methods discussed in that manuscript. The authors provided a Zenodo archive with the Firedrake components and dependencies used, as well as the scripts that generated the results. The software was installed on two similar platforms; then, new results were gathered and compared to the original results. After completing this process, the results have been deemed replicable by the reviewer.
There are several AD tools available, which all implement different strategies for the reverse mode of AD. The major strategies are primal value taping (implemented e.g. by ADOL-c) and Jacobi taping (implemented e.g. by adept and dco/c++). Especially for Jacobi taping, recent advances by using expression templates make this approach very attractive for large scale software. The current implementations are either closed source or miss essential features and flexibility. Therefore, we present the new AD tool CoDiPack (Code Differentiation Package) in this paper. It is specifically designed for a minimal memory consumption and optimal runtime, such that it can be used for the differentiation of large scale software. An essential part of the design of CoDiPack is the modular layout and the recursive data structures, which do not only allow the efficient implementation of the Jacobi taping approach, but will also enable other approaches like the primal value taping or new research ideas. We will also present the performance value of CoDiPack on a generic PDE example and on the SU2 code.
A paper by Karati and Sarkar at Asiacrypt'17 has pointed out the potential for Kummer lines in genus one, by observing that its SIMD-friendly arithmetic is competitive with the status quo. A more recent preprint explores the connection with (twisted) Edwards curves. In this paper we extend this work and significantly simplify their treatment. We show that their Kummer line is the \(x\)-line of a Montgomery curve translated by a point of order two, and exhibit a natural isomorphism to a twisted Edwards curve. Moreover, we show that the Kummer line presented by Gaudry and Lubicz can be obtained via the action of a point of order two on the \(y\)-line of an Edwards curve. The maps connecting these curves and lines are all very simple. As a result, a cryptographic implementation can use the arithmetic that is optimal for its instruction set at negligible cost.
IPscatt is a free toolbox in MATLAB that solves the inverse medium problem in scattering in two and three dimensions. It uses the Helmholtz equation as a physical model of scattering and provides point sources or plane waves as incident fields as well as simulated near or far field data with noise. Further, it can deal with real-world data in two dimensions from Institute Fresnel. A rapid variational regularization scheme minimizes the defect, i.e. the difference between the predicted data from the model and the given data, as well as penalty terms promoting sparsity, total variation and physical bounds. These regularization scheme relies on a primal-dual algorithm and was introduced in [F. Bürgel, K. S. Kazimierski, and A. Lechleiter 2017 Journal of Computational Physics, 339:130]. This paper provides a survey on the mathematical concepts, connects them with the implementation and shows the practical usage of the toolbox via guides.
The maximum flow problem is one of the most common network flow problems. This problem involves finding the maximum possible amount of flow between two designated nodes on a network with arcs having flow capacities. The push-relabel algorithm is one of the fastest algorithms to solve this problem. We present a shared memory parallel push-relabel algorithm. Graph coloring is used to avoid collisions between threads for concurrent push and relabel operations. In addition, excess values of target nodes are updated using atomic instructions to prevent race conditions. The experiments show that our algorithm is competitive for wide graphs with low diameters. Results from two different data sets are included, computer vision problems and DIMACS challenge problems. These are compared with existing push-relabel and pseudoflow implementations. We show that high speedup rates are possible using our coloring based parallelization technique on sparse networks. However, we also observe that the pseudoflow algorithm runs faster than the push-relabel algorithm on dense and long networks.
In this paper, we present an algorithm for the singular value decomposition (SVD) of a bidiagonal matrix by means of the eigenpairs of an associated symmetric tridiagonal matrix. The algorithm is particularly suited for the computation of a subset of singular values and corresponding vectors. We focus on a sequential version of the algorithm, and discuss special cases and implementation details. We use a large set of bidiagonal matrices to assess the accuracy of the implementation in single and double precision, as well as to identify potential shortcomings. We show that the algorithm can be up to three orders of magnitude faster than existing algorithms, which are limited to the computation of a full SVD. We also show time comparisons of an implementation that uses the strategy discussed in the paper as a building block for the computation of the SVD of general matrices.
In a recent paper, Lucambio Pérez and Prudente extended the Wolfe conditions for the vector-valued optimization. Here, we propose a line search algorithm for finding a step size satisfying the strong Wolfe conditions in the vector optimization setting. Well definedness and finite termination results are provided. We discuss practical aspects related to the algorithm and present some numerical experiments illustrating its applicability. Codes supporting this paper are written in Fortran 90 and are freely available for download.
Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite element transformations in FInAT and hence into the Firedrake finite element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly-enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using $C^1$ discretizations.
Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A Matlab implementation is provided to illustrate the computation and use of MDB-splines.
A scientific framework for simulations of large-scale networks, such as is required for the analysis of critical infrastructure interaction and interdependencies, is needed for applications on exascale computers. Such a framework must be able to manage heterogeneous physics and unstructured topology, and must be reusable. To this end we have developed DMNetwork, a class in PETSc that provides data and topology management and migration for network problems, along with multiphysics solvers to exploit the problem structure. It eases the application development cycle by providing the necessary infrastructure through simple abstrac- tions to define and query the network. This paper presents the design of the DMNetwork, illustrates its user interface, and demonstrates its ability to solve large network problems through the numerical simulation of a water pipe network with more than 2 billion variables on extreme-scale computers using up to 30,000 processor cores.