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A Simple Mesh Generator in MATLAB
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| Pengarang | : | Per-Olof Persson |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 329-345 |
| Abstrak | : | Creating a mesh is the first step in a wide range of applications, including scientific computing and computer graphics. An unstructured simplex mesh requires a choice of meshpoints (vertex nodes) and a triangulation. We want to offer a short and simple MATLAB code, described in more detail than usual, so the reader can experiment (and add to the code) knowing the underlying principles. We find the node locations by solving for equilibrium in a truss structure (using piecewise linear force-displacement relations) and reset the topology by the Delaunay algorithm. The geometry is described implicitly by its distance function. In addition to being much shorter and simpler than other meshing techniques, our algorithm typically produces meshes of very high quality. We discuss ways to improve the robustness and the performance, but our aim here is simplicity. Readers can download (and edit) the codes from http://math.mit.edu/~persson/mesh. |
An Invitation to Hermite's Integration and Summation: A Comparison between Hermite's and Simpson's Rules
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 311-328 |
| Abstrak | : | Hermite's rule for numerical integration is presented and compared with the more familiar Simpson's rule. Through several examples, the power of Hermite's rule for numerical summation is then demonstrated. It is established that Hermite's rule surpasses Simpson's rule due to better error estimates and due to the fact that Simpson's rule is not easily applicable to numerical summation. For these reasons, it is argued that Hermite's rule should replace Simpson's rule in the undergraduate curriculum. The derivation of Hermite's rule is accessible to undergraduates, and the topic should be included in the undergraduate curriculum. |
Testing of Clustering
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 285-308 |
| Abstrak | : | In this work we study the problem of clustering with respect to the diameter and the radius costs: We say that a set X of points in is $(k,b)-clusterable with respect to the diameter cost if X can be partitioned into k subsets (clusters) so that the distance between every pair of points in each cluster is at most b. In the case of the radius cost we require that all points that belong to the same cluster be at a distance of at most b for some common central point. Here we approach the problem of clustering from within the framework of property testing. In property testing, the goal is to determine whether a given object has a particular property or whether it should be modified significantly so that it obtains the property. In the context of clustering, testing takes on the following form: The algorithm is given parameters k, b, , and , and it can sample from the set of points X. The goal of the algorithm is to distinguish between the case when X is (k,b)-clusterable and the case when X is -far from being -clusterable. By -far from being -clusterable we mean that more than points should be removed from X so that it becomes -clusterable. In this work we describe and analyze algorithms that use a sample of size polynomial in k and and independent of |X|. (The dependence on and on the dimension, d, of the points varies with the different algorithms.) Such algorithms may be especially useful when the set of points X is very large and it may not even be feasible to observe all of it. Our algorithms can also be used to find approximately good clusterings. Namely, these are clusterings of all but an -fraction of the points in X that have optimal (or close to optimal) cost. The benefit of our algorithms is that they construct an implicit representation of such clusterings in time independent of |X|. That is, without actually having to partition all points in X, the implicit representation can be used to answer queries concerning the cluster to which any given point belongs. |
Stochastic Models That Separate Fractal Dimension and the Hurst Effect
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 269-282 |
| Abstrak | : | Fractal behavior and long-range dependence have been observed in an astonishing number of physical, biological, geological, and socioeconomic systems. Time series, profiles, and surfaces have been characterized by their fractal dimension, a measure of roughness, and by the Hurst coefficient, a measure of long-memory dependence. Both phenomena have been modeled and explained by self-affine random functions, such as fractional Gaussian noise and fractional Brownian motion. The assumption of statistical self-affinity implies a linear relationship between fractal dimension and Hurst coefficient and thereby links the two phenomena. This article introduces stochastic models that allow for any combination of fractal dimension and Hurst coefficient. Associated software for the synthesis of images with arbitrary, prespecified fractal properties and power-law correlations is available. The new models suggest a test for self-affinity that assesses coupling and decoupling of local and global behavior. |
Link Analysis: Hubs and Authorities on the World Wide Web
| Pengarang | : | Chris H. Q. Ding |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 256-268 |
| Abstrak | : | Ranking the tens of thousands of retrieved webpages for a user query on a Web search engine such that the most informative webpages are on the top is a key information retrieval technology. A popular ranking algorithm is the HITS algorithm of Kleinberg. It explores the reinforcing interplay between authority and hub webpages on a particular topic by taking into account the structure of the Web graphs formed by the hyperlinks between the webpages. In this paper, we give a detailed analysis of the HITS algorithm through a unique combination of probabilistic analysis and matrix algebra. In particular, we show that to first-order approximation, theranking given by the HITS algorithm is the same as the ranking by counting inbound and outbound hyperlinks. Using Web graphs of different sizes, we also provide experimental results to illustrate the analysis. |
Geometric Fourier Analysis of the Conformal Camera for Active Vision
| Pengarang | : | Subekti, Sukono |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 230-255 |
| Abstrak | : | Suppose one intends to design an active vision system that should perform some artificial intelligence functions. For instance, it should recognize a planar object (or a three-dimensional object containing a piece of a planar surface) in a dynamic scene. Ideally, such a system should be built upon some data model representing visual inputs and algorithms storing, processing, and analyzing visual information that are well adapted to image transformations produced by different perspectives between planar objects and the imaging system. In spite of its importance, this problem remained unsolved until recently. In this article, building on the author's work, projective Fourier analysis for patterns is constructed in the framework of geometric Fourier analysis on groups and homogeneous spaces. It is done by identifying in the conformal camera the group , which gives image projective transformations by acting through linear-fractional mappings on the image plane---homogeneous under the group action. This analysis is being implemented in perspectively adapted digital image processing, and its basic components are tested for binary images in computer simulations. It is recognized that the data model of digital image representation developed in the article is explicitly designed for foveated sensors, the use of which in active vision systems is presently limited due to the lack of such a data model. |
In the Garden of Branching Processes
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 202-229 |
| Abstrak | : | The current paper surveys and develops numerical methods for Markovian multitype branching processes in continuous time. Particular attention is paid to the calculation of means, variances, extinction probabilities, and marginal distributions in the presence of a Poisson stream of immigrant particles. The Poisson process assumption allows for temporally complex patterns of immigration and facilitates application of marked Poisson processes and Campbell's formulas. The methods and formulas derived are applied to four models: two population genetics models, a model for vaccination against an infectious disease in a community of households, and a model for the growth of resistant HIV virus in patients undergoing drug therapy. |
A Linear Complementarity Problem with a P-Matrix
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 46 (No. 2) |
| Halaman | : | 189-201 |
| Abstrak | : | In this paper, we present an application of a linear complementarity problem where M is a P-matrix but, in general, is neither an H-matrix nor a positive definite matrix. This application occurs originally in [J. Rohn, Linear Algebra Appl., 126 (1989), pp. 39--78], which is less known to the LCP community. Its focus is in computing the exact interval enclosures of the components of the solution set of an interval linear system. We extend the idea such that the calculations can be done by a computer with rigorous error control. |
Combinatorial and Hodge Laplacians: Similarities and Differences
| Pengarang | : | Emily Ribando-Gros, Rui Wang, Jiahui Chen, Yiying Tong, Guo-Wei Wei |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 3) |
| Halaman | : | 575-601 |
| Abstrak | : | As key subjects in spectral geometry and combinatorial graph theory, respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in their realization of diffusion and minimization of harmonic measures. It is believed that they also both associate with vector calculus, through the gradient, curl, and divergence, as argued in the popular usage of “Hodge Laplacians on graphs” in the literature. Nevertheless, these Laplacians are intrinsically different in their domains of definitions and applicability to specific data formats, hindering any in-depth comparison of the two approaches. For example, the spectral decomposition of a vector field on a simple point cloud using combinatorial Laplacians defined on some commonly used simplicial complexes does not give rise to the same curl-free and divergence-free components that one would obtain from the spectral decomposition of a vector field using either the continuous Hodge Laplacians defined on differential forms in manifolds or the discretized Hodge Laplacians defined on a point cloud with boundary in the Eulerian representation or on a regular mesh in the Eulerian representation. To facilitate the comparison and bridge the gap between the combinatorial Laplacian and Hodge Laplacian for the discretization of continuous manifolds with boundary, we further introduce boundary-induced graph (BIG) Laplacians using tools from discrete exterior calculus (DEC). BIG Laplacians are defined on discrete domains with appropriate boundary conditions to characterize the topology and shape of data. The similarities and differences among the combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined. Through an Eulerian representation of 3D domains as level-set functions on regular grids, we show experimentally the conditions for the convergence of BIG Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes. |
Operator Learning Using Random Features: A Tool for Scientific Computing
| Pengarang | : | Nicholas H. Nelsen, Andrew M. Stuart |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 66 (No. 3) |
| Halaman | : | 535-571 |
| Abstrak | : | Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing, which may often be framed in terms of operators mapping between spaces of functions. Building on the classical random features methodology for scalar regression, this paper introduces the function-valued random features method. This leads to a supervised operator learning architecture that is practical for nonlinear problems yet is structured enough to facilitate efficient training through the optimization of a convex, quadratic cost. Due to the quadratic structure, the trained model is equipped with convergence guarantees and error and complexity bounds, properties that are not readily available for most other operator learning architectures. At its core, the proposed approach builds a linear combination of random operators. This turns out to be a low-rank approximation of an operator-valued kernel ridge regression algorithm, and hence the method also has strong connections to Gaussian process regression. The paper designs function-valued random features that are tailored to the structure of two nonlinear operator learning benchmark problems arising from parametric partial differential equations. Numerical results demonstrate the scalability, discretization invariance, and transferability of the function-valued random features method. |