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The Mathematics of Atmospheric Dispersion Modeling
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| Pengarang | : | John M.Stockie |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 53 (No. 2) |
| Halaman | : | 349-372 |
| Abstrak | : | The Gaussian plume model is a standard approach for studying the transport of airborne contaminants due to turbulent diffusion and advection by the wind. This paper reviews the assumptions underlying the model, its derivation from the advection-diffusion equation, and the key properties of the plume solution. The results are then applied to solving an inverse problem in which emission source rates are determined from a given set of ground-level contaminant measurements. This source identification problem can be formulated as an overdetermined linear system of equations that is most easily solved using the method of least squares. Various generalizations of this problem are discussed, and we illustrate our results with an application to the study of zinc emissions from a smelting operation. |
A Mathematical Model for the Control and Eradication of a Wood Boring Beetle Infestation
| Pengarang | : | Stephen A. Gourley |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 53 (No. 2) |
| Halaman | : | 321-345 |
| Abstrak | : | We propose a mathematical model for an infestation of a wooded area by a beetle species in which the larvae develop deep in the wood of living trees. Due to the difficulties of detection, we presume that only a certain proportion of infested trees will be detected and that detection, if it happens, will occur only after some delay, which could be long. An infested tree once detected is immediately cut down and burned. The model is stage structured and contains a second time delay, which is the development time of the beetle from egg to adult. There is a delicate interplay between the two time delays due to the possibility in one case for a larva to mature even in a tree destined for destruction. We present conditions sufficient for infestation eradication and discuss the significance of the conditions, particularly in terms of the proportion of infested trees that need to be detected and removed. If the infestation is successfully eradicated, there are always a number of trees that completely escape infestation, and we compute lower bounds and an approximation for this number. Finally, we present the results of some numerical simulations. |
Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples
| Pengarang | : | Rodrigo B.Platte |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 53 (No. 2) |
| Halaman | : | 308-318 |
| Abstrak | : | It is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions. To avoid instability, one must settle for root-exponential convergence. The proof combines a Bernstein inequality of 1912 with an estimate due to Coppersmith and Rivlin in 1992. |
Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Algorithm
| Pengarang | : | Steven J. Brams |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 53 (No. 2) |
| Halaman | : | 291-307 |
| Abstrak | : | We analyze a class of proportional cake-cutting algorithms that use a minimal number of cuts ( if there are n players) to divide a cake that the players value along one dimension. While these algorithms may not produce an envy-free or efficient allocation—as these terms are used in the fair-division literature—one, divide-and-conquer (D&C), minimizes the maximum number of players that any single player can envy. It works by asking players successively to place marks on a cake—valued along a line—that divide it into equal halves (when n is even) or nearly equal halves (when n is odd), then halves of these halves, and so on. Among other properties, D&C ensures players of at least shares, as they each value the cake, if and only if they are truthful. However, D&C may not allow players to obtain proportional, connected pieces if they have unequal entitlements. Possible applications of D&C to land division are briefly discussed. |
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
| Pengarang | : | N. Halko |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 53 (No. 2) |
| Halaman | : | 217-288 |
| Abstrak | : | Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an matrix. (i) For a dense input matrix, randomized algorithms require floating-point operations (flops) in contrast to for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data. |
Searching for Rare Growth Factors Using Multicanonical Monte Carlo Methods
| Pengarang | : | Tobin A. Driscoll |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 4) |
| Halaman | : | 673-692 |
| Abstrak | : | The growth factor of a matrix quantifies the amount of potential error growth possible when a linear system is solved using Gaussian elimination with row pivoting. While it is an easy matter [N. J. Higham and D. J. Higham, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 155–164] to construct examples of matrices having any growth factor up to the maximum of , the weight of experience and analysis [N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996], [L. N. Trefethen and R. S. Schreiber, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 335–360], [L. N. Trefethen and I. D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997] suggest that matrices with exponentially large growth factors are exceedingly rare. Here we show how to conduct numerical experiments on random matrices using a multicanonical Monte Carlo method to explore the tails of growth factor probability distributions. Our results suggest, for example, that the occurrence of an matrix with a growth factor of 40 is on the order of a once-in-the-age-of-the-universe event. |
A Sum of Squares Approximation of Nonnegative Polynomials
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 4) |
| Halaman | : | 651-669 |
| Abstrak | : | We show that every real nonnegative polynomial f can be approximated as closely as desired (in the -norm of its coefficient vector) by a sequence of polynomials that are sums of squares. The novelty is that each has a simple and explicit form in terms of f and. |
Uniform Asymptotics Applied to Ultrawideband Pulse Propagation
| Pengarang | : | Natalie A. Cartwright |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 4) |
| Halaman | : | 628-648 |
| Abstrak | : | A canonical problem of central importance in the theory of ultrawideband pulse propagation through temporally dispersive, absorptive materials is the propagation of a Heaviside step-function signal through a medium that exhibits anomalous dispersion. This problem is rich in the use of asymptotic theory. Sommerfeld and Brillouin provided the first (qualitatively accurate but quantitatively inaccurate) closed-form approximations of the dynamic evolution of this waveform through a single-resonance Lorentz model dielectric based upon Debye's method of steepest descent. An improved approximation has since been provided by Oughstun and Sherman using modern, uniform asymptotic methods that rely upon the saddle-point method. An accurate, uniform asymptotic approximation describing the dynamical evolution of the unit step-function modulated sine wave signal through a single-resonance Lorentz model dielectric is presented here based upon their work. This refined asymptotic description results in a continuous evolution of the propagated field for all space-time points. |
Adaptive Polynomial Interpolation on Evenly Spaced Meshes
| Pengarang | : | Feldman, Allan,Nation, Molly Trendell |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 4) |
| Halaman | : | 604-627 |
| Abstrak | : | The problem of oscillatory polynomial interpolants arising from equally spaced mesh points is considered. It is shown that by making use of adaptive approaches the oscillations may be contained and the resulting polynomials are data-bounded and monotone on each interval. This is achieved at the cost of using a different polynomial on each subinterval. Computational results for a number of challenging functions including a number of problems similar to Runge's function with as many as 511 points per interval are shown. |
Revisiting Hypergraph Models for Sparse Matrix Partitioning
| Pengarang | : | Cevdet Aykanat |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 4) |
| Halaman | : | 595-603 |
| Abstrak | : | We provide an exposition of hypergraph models for parallelizing sparse matrix-vector multiplies. Our aim is to emphasize the expressive power of hypergraph models. First, we set forth an elementary hypergraph model for the parallel matrix-vector multiply based on one-dimensional (1D) matrix partitioning. In the elementary model, the vertices represent the data of a matrix-vector multiply, and the nets encode dependencies among the data. We then apply a recently proposed hypergraph transformation operation to devise models for 1D sparse matrix partitioning. The resulting 1D partitioning models are equivalent to the previously proposed computational hypergraph models and are not meant to be replacements for them. Nevertheless, the new models give us insights into the previous ones and help us explain a subtle requirement, known as the consistency condition, of hypergraph partitioning models. Later, we demonstrate the flexibility of the elementary model on a few 1D partitioning problems that are hard to solve using the previously proposed models. We also discuss extensions of the proposed elementary model to two-dimensional matrix partitioning. |