ACM Transactions on Mathematical Software最新文献

Version 5.99 of the empirical Gramian framework – emgr – completes a development cycle which focused on parametric model order reduction of gas network models while preserving compatibility to the previous development for the application of combined state and parameter reduction for neuroscience network models. Secondarily, new features concerning empirical Gramian types, perturbation design, and trajectory post-processing, as well as a Python version in addition to the default MATLAB / Octave implementation, have been added. This work summarizes these changes, particularly since emgr version 5.4, see Himpe, 2018 [Algorithms 11(7): 91], and gives recent as well as future applications, such as parameter identification in systems biology, based on the current feature set.

IFISS is an established MATLAB finite element software package for studying strategies for solving partial differential equations (PDEs). IFISS3D is a new add-on toolbox that extends IFISS capabilities for elliptic PDEs from two to three space dimensions. The open-source MATLAB framework provides a computational laboratory for experimentation and exploration of finite element approximation and error estimation, as well as iterative solvers. The package is designed to be useful as a teaching tool for instructors and students who want to learn about state-of-the-art finite element methodology. It will also be useful for researchers as a source of reproducible test matrices of arbitrarily large dimension.

For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced using approximations to a distribution’s inverse cumulative distribution function. These approximations are designed to be computationally inexpensive, and much cheaper than library functions which are exact to within machine precision, and thus highly suitable for use in Monte Carlo simulations. The approximation errors they introduce can then be eliminated through use of the multilevel Monte Carlo method. Two approximations are presented for the Gaussian distribution: a piecewise constant on equally spaced intervals, and a piecewise linear using geometrically decaying intervals. The errors of the approximations are bounded and the convergence demonstrated, and the computational savings measured for C and C++ implementations. Implementations tailored for Intel and Arm hardware are inspected, alongside hardware agnostic implementations built using OpenMP. The savings are incorporated into a nested multilevel Monte Carlo framework with the Euler-Maruyama scheme to exploit the speed ups without losing accuracy, offering speed ups by a factor of 5–7. These ideas are empirically extended to the Milstein scheme, and the non-central χ² distribution for the Cox-Ingersoll-Ross process, offering speed ups of a factor of 250 or more.

One can simulate low-precision floating-point arithmetic via software by executing each arithmetic operation in hardware and then rounding the result to the desired number of significant bits. For IEEE-compliant formats, rounding requires only standard mathematical library functions, but handling subnormals, underflow, and overflow demands special attention, and numerical errors can cause mathematically correct formulae to behave incorrectly in finite arithmetic. Moreover, the ensuing implementations are not necessarily efficient, as the library functions these techniques build upon are typically designed to handle a broad range of cases and may not be optimized for the specific needs of rounding algorithms. CPFloat is a C library for simulating low-precision arithmetics. It offers efficient routines for rounding, performing mathematical computations, and querying properties of the simulated low-precision format. The software exploits the bit-level floating-point representation of the format in which the numbers are stored and replaces costly library calls with low-level bit manipulations and integer arithmetic. In numerical experiments, the new techniques bring a considerable speedup (typically one order of magnitude or more) over existing alternatives in C, C++, and MATLAB. To our knowledge, CPFloat is currently the most efficient and complete library for experimenting with custom low-precision floating-point arithmetic.

Task-based programming models have succeeded in gaining the interest of the high-performance mathematical software community because they relieve part of the burden of developing and implementing distributed-memory parallel algorithms in an efficient and portable way.In increasingly larger, more heterogeneous clusters of computers, these models appear as a way to maintain and enhance more complex algorithms. However, task-based programming models lack the flexibility and the features that are necessary to express in an elegant and compact way scalable algorithms that rely on advanced communication patterns. We show that the Sequential Task Flow paradigm can be extended to write compact yet efficient and scalable routines for linear algebra computations. Although, this work focuses on dense General Matrix Multiplication, the proposed features enable the implementation of more complex algorithms. We describe the implementation of these features and of the resulting GEMM operation. Finally, we present an experimental analysis on two homogeneous supercomputers showing that our approach is competitive up to 32,768 CPU cores with state-of-the-art libraries and may outperform them for some problem dimensions. Although our code can use GPUs straightforwardly, we do not deal with this case because it implies other issues which are out of the scope of this work.