Parallel Computation of the Minimal Elements of a Poset

Computing the minimal elements of a partially ordered finite set (poset) is a fundamental problem in combinatorics with numerous applications such as polynomial expression optimization, transversal hypergraph generation and redundant component removal, to name a few. We propose a divide-and-conquer algorithm which is not only cache-oblivious but also can be parallelized free of determinacy races. We have implemented it in Cilk++ targeting multicores. For our test problems of sufficiently large input size our code demonstrates a linear speedup on 32 cores.National Science Foundation (U.S.). (Grant number CNS-0615215)National Science Foundation (U.S.). (Grant number CCF- 0621511

Automatic Generation of Fast and Certified Code for Polynomial Evaluation

International audienceDesigning an efficient floating-point implementation of a function based on polynomial evaluation requires being able to find an accurate enough evaluation program, exploiting at most the target architecture features. This article introduces CGPE, a tool dealing with the generation of fast and certified codes for the evaluation of bivariate polynomials. First we discuss the issue underlying the evaluation scheme combinatorics before giving an overview of the CGPE tool. The approach we propose consists in two steps: the generation of evaluation schemes by using some heuristics so as to quickly find some of low latency; and the selection that mainly consists in automatically checking their scheduling on the given target and validating their accuracy. Then, we present on-going development and ideas for possible improvements of the whole process. Finally, we illustrate the use of CGPE on some examples, and show how it allows us to generate fast and certified codes in a few seconds and thus to reduce the development time of libms like FLIP

Efficient Evaluation of Large Polynomials

In scientific computing, it is often required to evaluate a polynomial expression (or a matrix depending on some variables) at many points which are not known in advance or with coordinates containing “symbolic expressions”. In these circumstances, standard evaluation schemes, such as those based on Fast Fourier Transforms do not apply. Given a polynomial f expressed as the sum of its terms, we propose an algorithm which generates a representation of f optimizing the process of evaluating f at some points. In addition, this evaluation of f can be done efficiently in terms of data locality and parallelism. We have implemented our algorithm in the Cilk++ concurrency platform and our implementation achieves nearly linear speedup on 16 cores with large enough input. For some large polynomials, the generated schedule can be evaluated at least 10 times faster than the schedules produced by other available software solutions. Moreover, our code can handle much larger input polynomials

New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with better characteristics. There is therefore clear interest in searching for better cubature rules. Here we present a number of new cubature rules on the triangle, exhibiting full or rotational symmetry, that improve on those available in the literature either in terms of number of points or in terms of quality. These rules were obtained by determining and implementing minimal orthonormal polynomial bases that can express the symmetries of the cubature rules. As shown in specific benchmark examples, this results in significantly better performance of the employed algorithm.Comment: 12 pages, 1 figur

Reduced Order and Surrogate Models for Gravitational Waves

We present an introduction to some of the state of the art in reduced order and surrogate modeling in gravitational wave (GW) science. Approaches that we cover include Principal Component Analysis, Proper Orthogonal Decomposition, the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order Quadratures, and Compressed Likelihood evaluations. We divide the review into three parts: representation/compression of known data, predictive models, and data analysis. The targeted audience is that one of practitioners in GW science, a field in which building predictive models and data analysis tools that are both accurate and fast to evaluate, especially when dealing with large amounts of data and intensive computations, are necessary yet can be challenging. As such, practical presentations and, sometimes, heuristic approaches are here preferred over rigor when the latter is not available. This review aims to be self-contained, within reasonable page limits, with little previous knowledge (at the undergraduate level) requirements in mathematics, scientific computing, and other disciplines. Emphasis is placed on optimality, as well as the curse of dimensionality and approaches that might have the promise of beating it. We also review most of the state of the art of GW surrogates. Some numerical algorithms, conditioning details, scalability, parallelization and other practical points are discussed. The approaches presented are to large extent non-intrusive and data-driven and can therefore be applicable to other disciplines. We close with open challenges in high dimension surrogates, which are not unique to GW science.Comment: Invited article for Living Reviews in Relativity. 93 page

Magnetic field and ion-optical simulations for the optimization of the Super-FRS

The growing demand in the field of discovering and investigating exotic nuclei by means of fragment separators yields challenging restrictions on future facilities. The main task of a fragment separator is the in-flight separation of many different species of nuclides, produced with an ion beam on a target. To achieve the best resolution and capture of rare nuclei, maximal beam illumination of the apertures of the ion-optical elements is required. Many fragment separators have a wide operation range of the magnetic rigidity Brho. Moreover, frequent changes of Brho are required during experiments. Often magnets are operated in the saturation region of the iron yokes, leading to local changes of the magnetic field (B-field) distributions and the corresponding particle trajectories. In such cases it is important to have a fast ion-optical model with good predictability, which considers the real field distributions and the saturation. This thesis describes the development of a general approach to provide a fast and accurate ion-optical model (Taylor transfer map) of large aperture magnets starting from simulated or measured 3D B-field distributions. To produce highly accurate transfer map, a B-field has to be represented by 3D polynomials. It is crucial that the whole transversal aperture is described by a single polynomial, whereas many polynomials might be used in the longitudinal direction. High non-uniformity of the B-field makes this problem more complicated, especially for the regions near the pole shoe ends. The problem was solved by means of a combination of the Surface Integration Helmholtz Method (SIHM) and the Least Squares (LS) method. The approach was extended further for obtaining the B-field polynomial representation dependent on both: the coordinates and the excitation current. This representation allows to produce Brho dependent transfer maps, which can be useful for the optimization of the separator settings for different experiments. The method was tested using the analytical field model, based on a configuration of thin wires and a Biot-Savart law, resulting in a high stability against the errors in the input B-field. The rigidity dependent transfer maps were generated for the normal conducting dipole of the Super-FRS preseparator. The ion-optical study of the preseparator in the separator as well as in the spectrometer modes were conducted