118,089 research outputs found

    Multivariate Design of Experiments for Engineering Dimensional Analysis

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    We consider the design of dimensional analysis experiments when there is more than a single response. We first give a brief overview of dimensional analysis experiments and the dimensional analysis (DA) procedure. The validity of the DA method for univariate responses was established by the Buckingham Π\Pi-Theorem in the early 20th century. We extend the theorem to the multivariate case, develop basic criteria for multivariate design of DA and give guidelines for design construction. Finally, we illustrate the construction of designs for DA experiments for an example involving the design of a heat exchanger

    Measuring the three-dimensional shear from simulation data, with applications to weak gravitational lensing

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    We have developed a new three-dimensional algorithm, based on the standard P3^3M method, for computing deflections due to weak gravitational lensing. We compare the results of this method with those of the two-dimensional planar approach, and rigorously outline the conditions under which the two approaches are equivalent. Our new algorithm uses a Fast Fourier Transform convolution method for speed, and has a variable softening feature to provide a realistic interpretation of the large-scale structure in a simulation. The output values of the code are compared with those from the Ewald summation method, which we describe and develop in detail. With an optimal choice of the high frequency filtering in the Fourier convolution, the maximum errors, when using only a single particle, are about 7 per cent, with an rms error less than 2 per cent. For ensembles of particles, used in typical NN-body simulations, the rms errors are typically 0.3 per cent. We describe how the output from the algorithm can be used to generate distributions of magnification, source ellipticity, shear and convergence for large-scale structure.Comment: 22 pages, latex, 11 figure

    Estimation of instrinsic dimension via clustering

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    The problem of estimating the intrinsic dimension of a set of points in high dimensional space is a critical issue for a wide range of disciplines, including genomics, finance, and networking. Current estimation techniques are dependent on either the ambient or intrinsic dimension in terms of computational complexity, which may cause these methods to become intractable for large data sets. In this paper, we present a clustering-based methodology that exploits the inherent self-similarity of data to efficiently estimate the intrinsic dimension of a set of points. When the data satisfies a specified general clustering condition, we prove that the estimated dimension approaches the true Hausdorff dimension. Experiments show that the clustering-based approach allows for more efficient and accurate intrinsic dimension estimation compared with all prior techniques, even when the data does not conform to obvious self-similarity structure. Finally, we present empirical results which show the clustering-based estimation allows for a natural partitioning of the data points that lie on separate manifolds of varying intrinsic dimension

    Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs

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    Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent beta, and high clustering that can be controlled via the temperature T. We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to T = 0. We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input. Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases

    Three Puzzles on Mathematics, Computation, and Games

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    In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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