Showing papers in "Siam Review in 1999"

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

Abstract: A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.

Centroidal Voronoi Tessellations: Applications and Algorithms

Abstract: A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods for computing these tessellations, provide some analyses concerning both the tessellations and the methods for their determination, and, finally, present the results of some numerical experiments.

Electrical Impedance Tomography

Abstract: t. This paper surveys some of the work our group has done in electrical impedance tomography.

Mathematical Analysis of HIV-1 Dynamics in Vivo

Abstract: Mathematical models have proven valuable in understanding the dynamics of HIV-1 infection in vivo. By comparing these models to data obtained from patients undergoing antiretroviral drug therapy, it has been possible to determine many quantitative features of the interaction between HIV-1, the virus that causes AIDS, and the cells that are infected by the virus. The most dramatic finding has been that even though AIDS is a disease that occurs on a time scale of about 10 years, there are very rapid dynamical processes that occur on time scales of hours to days, as well as slower processes that occur on time scales of weeks to months. We show how dynamical modeling and parameter estimation techniques have uncovered these important features of HIV pathogenesis and impacted the way in which AIDS patients are treated with potent antiretroviral drugs.

Fast Marching Methods

Abstract: Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static Hamilton--Jacobi equations Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms They are optimal in the sense that the computational complexity of the algorithms is O(N log N), where N is the total number of points in the domain The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing first arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reflection calculations In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings; provide details of the computational schemes, including higher order versions; and demonstrate the techniques in a collection of different areas

Matrices, Vector Spaces, and Information Retrieval

Abstract: The evolution of digital libraries and the Internet has dramatically transformed the processing, storage, and retrieval of information. Efforts to digitize text, images, video, and audio now consume a substantial portion of both academic and industrial activity. Even when there is no shortage of textual materials on a particular topic, procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking. Recently developed information retrieval technologies are based on the concept of a vector space. Data are modeled as a matrix, and a user's query of the database is represented as a vector. Relevant documents in the database are then identified via simple vector operations. Orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself. The purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections.

The Discrete Cosine Transform

Abstract: Each discrete cosine transform (DCT) uses $N$ real basis vectors whose components are cosines In the DCT-4, for example, the $j$th component of $\boldv_k$ is $\cos (j + \frac{1}{2}) (k + \frac{1}{2}) \frac{\pi}{N}$ These basis vectors are orthogonal and the transform is extremely useful in image processing If the vector $\boldx$ gives the intensities along a row of pixels, its cosine series $\sum c_k \boldv_k$ has the coefficients $c_k=(\boldx,\boldv_k)/N$ They are quickly computed from a Fast Fourier Transform But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are We prove orthogonality in a different way Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix By varying the boundary conditions we get the established transforms DCT-1 through DCT-4 Other combinations lead to four additional cosine transforms The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform The centering also determines the period: $N-1$ or $N$ in the established transforms, $N-\frac{1}{2}$ or $N+ \frac{1}{2}$ in the other four The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices

Robust Parameter Estimation in Computer Vision

Abstract: Estimation techniques in computer vision applications must estimate accurate model parameters despite small-scale noise in the data, occasional large-scale measurement errors (outliers), and measurements from multiple populations in the same data set. Increasingly, robust estimation techniques, some borrowed from the statistics literature and others described in the computer vision literature, have been used in solving these parameter estimation problems. Ideally, these techniques should effectively ignore the outliers and measurements from other populations, treating them as outliers, when estimating the parameters of a single population. Two frequently used techniques are least-median of squares (LMS) [P. J. Rousseeuw, {J. Amer. Statist. Assoc., 79 (1984), pp. 871--880] and M-estimators [Robust Statistics: The Approach Based on Influence Functions, F. R. Hampel et al., John Wiley, 1986; Robust Statistics, P. J. Huber, John Wiley, 1981]. LMS handles large fractions of outliers, up to the theoretical limit of 50% for estimators invariant to affine changes to the data, but has low statistical efficiency. M-estimators have higher statistical efficiency but tolerate much lower percentages of outliers unless properly initialized. While robust estimators have been used in a variety of computer vision applications, three are considered here. In analysis of range images---images containing depth or X, Y, Z measurements at each pixel instead of intensity measurements---robust estimators have been used successfully to estimate surface model parameters in small image regions. In stereo and motion analysis, they have been used to estimate parameters of what is called the ''fundamental matrix,'' which characterizes the relative imaging geometry of two cameras imaging the same scene. Recently, robust estimators have been applied to estimating a quadratic image-to-image transformation model necessary to create a composite, ''mosaic image'' from a series of images of the human retina. In each case, a straightforward application of standard robust estimators is insufficient, and carefully developed extensions are used to solve the problem.

Solving Index-1 DAEs in MATLAB and Simulink

Abstract: This paper describes mathematical and software developments needed for the effective solution of differential algebraic equations of index 1 in the integrated computing environment MATLAB and the dynamic simulation package Simulink. The developments are applicable to other problem-solving environments and some are applicable to general scientific computation.

The Riemann Zeros and Eigenvalue Asymptotics

Abstract: Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP.

Parallel Multilevel series k -Way Partitioning Scheme for Irregular Graphs

Abstract: In this paper we present a parallel formulation of a multilevel k-way graph partitioning algorithm. A key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of the partitions produced by the serial multilevel k-way partitioning algorithm. In particular, the time taken by our parallel graph partitioning algorithm is only slightly longer than the time taken for re-arrangement of the graph among processors according to the new partition. Experiments with a variety of finite element graphs show that our parallel formulation produces high-quality partitionings in a short amount of time. For example, a 128-way partitioning of graphs with one million vertices can be computed in a little over two seconds on a 128-processor Cray T3D. Furthermore, the quality of the partitions produced is comparable (edge-cuts within 5%) to those produced by the serial multilevel k-way algorithm. Thus our parallel algorithm makes it feasible to perform frequent repartitioning of graphs in dynamic computations without compromising the partitioning quality.

Optimizing the Delivery of Radiation Therapy to Cancer Patients

Abstract: In the field of radiation therapy, much of the research is aimed at developing new and innovative techniques for treating cancer patients with radiation. In recent years, new treatment machines have been developed that provide a much greater degree of computer control than was available with the machines of previous generations. One innovation has been the development of an approach called "tomotherapy.'' Tomotherapy can be defined as computer-controlled rotational radiotherapy delivered using an intensity-modulated fan beam of radiation. The successful implementation of the new delivery techniques requires the development of a suitable approach for optimizing each patient's treatment plan. One of the challenges is to quantify optimality in radiation therapy. We have tested a variety of objective functions and constraints in pursuit of a formulation that performs well for a wide variety of disease sites. An additional challenge stems from the sizable amount of data and the large number of variables that are involved in each optimization. This paper presents several approaches to optimizing treatment plans in radiation therapy, and the advantages and disadvantages of a number of formulations are explored.

Stochastic Spatial Models

Abstract: In the models we will consider, space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to explain an idea of Durrett and Levin (1994): the behavior of these models can be predicted from the properties of the mean field ODE, i.e., the equations for the densities of the various types that result from pretending that all sites are always independent. We will illustrate this picture through a discussion of eight families of examples from statistical mechanics, genetics, population biology, epidemiology, and ecology. Some of our findings are only conjectures based on simulation, but in a number of cases we are able to prove results for systems with "fast stirring" by exploiting connections between the spatial model and an associated reaction diffusion equation.

On the Theory of Complex Rays

Abstract: The article surveys the application of complex-ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Sp{} in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterizations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex Wentzel--Kramers--Brilbuin expansion of these wavefields. Examples are given for several cases of physical importance.

Configuration Controllability of Simple Mechanical Control Systems

Abstract: In this paper we present a definition of ''configuration controllability'' for mechanical systems whose Lagrangian is kinetic energy with respect to a Riemannian metric minus potential energy. A computable test for this new version of controllability is derived. This condition involves an object that we call the symmetric product. Of particular interest is a definition of ''equilibrium controllability'' for which we are able to derive computable sufficient conditions. Examples illustrate the theory.

Green's Functions for Multiply Connected Domains via Conformal Mapping

Abstract: A method is described for the computation of the Green's function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz-Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K-1 slits. From the Green's function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iterations. By making the end of the strip jagged, the method can be generalized to weighted Green's functions and weighted approximations.

Ill-Conditioned Matrices Are Componentwise Near to Singularity

Abstract: For a square matrix normed to 1, the normwise distance to singularity is well known to be equal to the reciprocal of the condition number. In this paper we give an elementary and self-contained proof for the fact that an ill-conditioned matrix is also not far from a singular matrix in a componentwise sense. This is shown to be true for any weighting of the componentwise distance. In other words, for matrix inversion, "ill conditioned" means "nearly ill posed" in the normwise and also in the componentwise sense.

On the Reversion of an Asymptotic Expansion and the Zeros of the Airy Functions

Abstract: The general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared. The asymptotic theory is applied to obtain asymptotic expansions of the zeros of the Airy functions and their derivatives, and also of the associated values of the functions or derivatives. A Maple code is constructed to generate exactly the coefficients in these expansions. The only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available. The sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the Airy functions.

The Ring Loading Problem

Abstract: The following problem arose in the planning of optical communications networks which use bidirectional SONET rings. Traffic demands di,j are given for each pair of nodes in an $n$-node ring; each demand must be routed one of the two possible ways around the ring. The object is to minimize the maximum load on the cycle, where the load of an edge is the sum of the demands routed through that edge. We provide a fast, simple algorithm which achieves a load that is guaranteed to exceed the optimum by at most 3/2 times the maximum demand, and that performs even better in practice. En route we prove the following curious lemma: for any $x_1, \dots, x_n \in [0,1]$ there exist $y_1, \dots, y_n$ such that for each $k$, $|y_k|=x_k$ and $$ \left| \sum_{i=1}^k y_i - \sum_{i=k+1}^n y_i \right| \le 2. $$ [This article is reprinted here (with updates) from SIAM J. Discrete Math., 11 (1998), pp. 1--14. New developments include a $1+\varepsilon$ approximation algorithm and a variation of ring loading in the setting of wavelength division multiplexing; remarks added for this printing, about these and other issues, are enclosed in brackets.]

Derivation of Numerical Methods Using Computer Algebra

Abstract: The use of computer algebra systems in a course on scientific computation is demonstrated Various examples, such as the derivation of Newton's iteration formula, the secant method, Newton--Cotes and Gaussian integration formulas, as well as Runge--Kutta formulas, are presented For the derivations, the computer algebra system Maple is used

Integer Programming and Conway's Game of Life

Abstract: This article presents integer programming formulations for finding interesting patterns in Conway's game of Life, with accompanying exercises and solutions.

Periodic Folding of Thin Sheets

Abstract: When a thin sheet of a flexible material such as paper is fed from a horizontal spool towards a rough horizontal plane below it, the sheet folds on itself in a regular manner. We model this phenomenon as a free boundary problem for a nonlinearly elastic sheet, taking into account the stiffness and weight of the sheet and the height of the spool above the plane. By using a continuation scheme we solve the problem numerically and follow the evolution of one period of the fold for various values of the parameters. The results are found to agree well with observations of the folding of paper sheets.

Optimization Case Studies in the NEOS Guide

Abstract: We describe several of the case studies in the NEOS Guide, a site on the World Wide Web that contains informational and educational material about optimization. These studies show how optimization relates to practical applications. They guide the user through relevant details of the application, formulation, solution, and interpretation of the results. The studies use interactivity to build intuition, allowing users to define their own problems and examine the corresponding solutions. The studies can be used for assignments in optimization and operations research courses and as small self-guided units equivalent to one or two lecture classes.

On the Approximate and Null Controllability of the Navier--Stokes Equations

Abstract: This paper presents some known results on the approximate and null controllability of the Navier--Stokes equations All of them can be viewed as partial answers to a conjecture of J-L Lions

Bubbles in Wet, Gummed Wine Labels

Abstract: It is shown that bubbling on wine bottle labels is due to absorption of water from the glue, with subsequent hygroscopic expansion. Contrary to popular belief, most of the glue's water must be lost to the atmosphere rather than to the paper. A simple lubrication model is developed for spreading glue piles in the pressure chamber of the labelling machine. This model predicts a maximum rate for application of labels. Buckling theory shows that the current arrangement of periodic glue strips can indeed accommodate paper expansion. This project provides interesting applications of various areas of undergraduate mathematics, such as trigonometry, Maclaurin series, dimensional analysis, and fluid mechanics. It illustrates that simple mathematical modelling may provide insight into a complicated real-world problem.