ACM Transactions on Mathematical Software最新文献

The hp-adaptive finite element method (FEM) – where one independently chooses the mesh size (h) and polynomial degree (p) to be used on each cell – has long been known to have better theoretical convergence properties than either h- or p-adaptive methods alone. However, it is not widely used, owing at least in parts to the difficulty of the underlying algorithms and the lack of widely usable implementations. This is particularly true when used with continuous finite elements.

Herein, we discuss algorithms that are necessary for a comprehensive and generic implementation of hp-adaptive finite element methods on distributed-memory, parallel machines. In particular, we will present a multi-stage algorithm for the unique enumeration of degrees of freedom (DoFs) suitable for continuous finite element spaces, describe considerations for weighted load balancing, and discuss the transfer of variable size data between processes. We illustrate the performance of our algorithms with numerical examples, and demonstrate that they scale reasonably up to at least 16 384 Message Passing Interface (MPI) processes.

We provide a reference implementation of our algorithms as part of the open-source library deal.II.

We consider the computation of the Euclidean (or L2) norm of an n-dimensional vector in floating-point arithmetic. We review the classical solutions used to avoid spurious overflow or underflow and/or to obtain very accurate results. We modify a recently published algorithm (that uses double-word arithmetic) to allow for a very accurate solution, free of spurious overflows and underflows. To that purpose, we use a double-word square-root algorithm of which we provide a tight error analysis. The returned L2 norm will be within very slightly more than 0.5 ulp from the exact result, which means that we will almost always provide correct rounding.

We present a robust technique to build a topologically optimal all-hexahedral layer on the boundary of a model with arbitrarily complex ridges and corners. The generated boundary layer mesh strictly respects the geometry of the input surface mesh, and it is optimal in the sense that the hexahedral valences of the boundary edges are as close as possible to their ideal values (local dihedral angle divided by 90°). Starting from a valid watertight surface mesh (all-quad in practice), we build a global optimization integer programming problem to minimize the mismatch between the hexahedral valences of the boundary edges and their ideal values. The formulation of the integer programming problem relies on the duality between boundary hexahedral configurations and triangulations of the disk, which we reframe in terms of integer constraints. The global problem is solved efficiently by performing combinatorial branch-and-bound searches on a series of sub-problems defined in the vicinity of complicated ridges/corners, where the local mesh topology is necessarily irregular because of the inherent constraints in hexahedral meshes. From the integer solution, we build the topology of the all-hexahedral layer, and the mesh geometry is computed by untangling/smoothing. Our approach is fully automated, topologically robust, and fast.

MQSI is a Fortran 2003 subroutine for constructing monotone quintic spline interpolants to univariate monotone data. Using sharp theoretical monotonicity constraints, first and second derivative estimates at data provided by a quadratic facet model are refined to produce a univariate C² monotone interpolant. Algorithm and implementation details, complexity and sensitivity analyses, usage information, a brief performance study, and comparisons with other spline approaches are included.

We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations. The implementation rests on Krawczyk’s method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as a powerful new tool in numerical algebraic geometry, which can make certification the default and not just an option.

The robust scale estimator Q_n developed by Croux and Rousseeuw [3], for the computation of which they provided a deterministic algorithm, has proven to be very useful in several domains including in quality management and time series analysis. It has interesting mathematical (50% breakdown, 82% Asymptotic Relative Efficiency) and computing (O(nlogn) time, O(n) space) properties. While working on a faster algorithm to compute Q_n, we have discovered an error in the computation of the d constant, and as a consequence in the d_n constants that are used to scale the statistic for consistency with the variance of a normal sample. These errors have been reproduced in several articles including in the International Standard Organisation 13,528 [12] document. In this article, we fix the errors and present a new approach, which includes a new algorithm, allowing computations to run 1.3 to 4.5 times faster when n grows from 10 to 100,000.

We present the new software OpDiLib, a universal add-on for classical operator overloading AD tools that enables the automatic differentiation (AD) of OpenMP parallelized code. With it, we establish support for OpenMP features in a reverse mode operator overloading AD tool to an extent that was previously only reported on in source transformation tools. We achieve this with an event-based implementation ansatz that is unprecedented in AD. Combined with modern OpenMP features around OMPT, we demonstrate how it can be used to achieve differentiation without any additional modifications of the source code; neither do we impose a priori restrictions on the data access patterns, which makes OpDiLib highly applicable. For further performance optimizations, restrictions like atomic updates on adjoint variables can be lifted in a fine-grained manner. OpDiLib can also be applied in a semi-automatic fashion via a macro interface, which supports compilers that do not implement OMPT. We demonstrate the applicability of OpDiLib for a pure operator overloading approach in a hybrid parallel environment. We quantify the cost of atomic updates on adjoint variables and showcase the speedup and scaling that can be achieved with the different configurations of OpDiLib in both the forward and the reverse pass.

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption.

We present the new features available in the recent release of SuperLU_DIST, Version 8.1.1. SuperLU_DIST is a distributed-memory parallel sparse direct solver. The new features include (1) a 3D communication-avoiding algorithm framework that trades off inter-process communication for selective memory duplication, (2) multi-GPU support for both NVIDIA GPUs and AMD GPUs, and (3) mixed-precision routines that perform single-precision LU factorization and double-precision iterative refinement. Apart from the algorithm improvements, we also modernized the software build system to use CMake and Spack package installation tools to simplify the installation procedure. Throughout the article, we describe in detail the pertinent performance-sensitive parameters associated with each new algorithmic feature, show how they are exposed to the users, and give general guidance of how to set these parameters. We illustrate that the solver’s performance both in time and memory can be greatly improved after systematic tuning of the parameters, depending on the input sparse matrix and underlying hardware.