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Orthogonal Projections of the Identity: Spectral Analysis and Applications to Approximate Inverse Preconditioning
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 1) |
| Halaman | : | 66-75 |
| Abstrak | : | Many strategies for constructing different structures of sparse approximate inverse preconditioners for large linear systems have been proposed in the literature. In a more general framework, this paper analyzes the theoretical effectiveness of the optimal preconditioner (in the Frobenius norm) of a linear system over an arbitrary subspace of . For this purpose, the spectral analysis of the Frobenius orthogonal projections of the identity matrix onto the linear subspaces of is performed. This analysis leads to a simple, general criterion: The effectiveness of the optimal approximate inverse preconditioners (parametrized by any vectorial structure)\ improves at the same time as the smallest singular value (or the smallest eigenvalue's modulus) of the corresponding preconditioned matrices increases to 1. |
Reliability Allocation for Networks and Systems
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 1) |
| Halaman | : | 43-65 |
| Abstrak | : | Abstrak tidak tersedia. |
Convergence Analysis of Krylov Subspace Iterations with Methods from Potential Theory
| Pengarang | : | Arno B. J. Kuijlaars |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 48 (No. 1) |
| Halaman | : | 3-40 |
| Abstrak | : | Krylov subspace iterations are among the best-known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix. This leads to an extremal problem in polynomial approximation theory: How small can a monic polynomial of a given degree be on the spectrum? This survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices. It is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure. Then, depending on the number of iterations, the Lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region, which is characterized by means of a constrained equilibrium problem from potential theory. The same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems. |
Modeling Basketball Free Throws
| Pengarang | : | Armin Baur |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 47 (No. 4) |
| Halaman | : | 775-798 |
| Abstrak | : | This paper presents a mathematical model for basketball free throws. It is intended to be a supplement to an existing calculus course and could easily be used as a basis for a calculus project. Students will learn how to apply calculus to model an interesting real-world problem, from problem identification all the way through to interpretation and verification. Along the way we will introduce topics such as optimization (univariate and multiobjective), numerical methods, and differential equations. |
The Effect of Dispersal Patterns on Stream Populations
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 47 (No. 4) |
| Halaman | : | 749-772 |
| Abstrak | : | Individuals in streams are constantly subject to predominantly unidirectional flow. The question of how these populations can persist in upper stream reaches is known as the "drift paradox." We employ a general mechanistic movement-model framework and derive dispersal kernels for this situation. We derive thin- as well as fat-tailed kernels. We then introduce population dynamics and analyze the resulting integrodifferential equation. In particular, we study how the critical domain size and the invasion speed depend on the velocity of the stream flow. We give exact conditions under which a population can persist in a finite domain in the presence of stream flow, as well as conditions under which a population can spread against the direction of the flow. We find a critical stream velocity above which a population cannot persist in an arbitrarily large domain. At exactly the same stream velocity, the invasion speed against the flow becomes zero; for larger velocities, the population retreats with the flow. |
A Software Package for Lie Algebraic Computations
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 47 (No. 4) |
| Halaman | : | 722-745 |
| Abstrak | : | The paper presents a computer algebra package that facilitates Lie algebraic symbolic computations required in the solution of a variety of problems, such as the solution of right-invariant differential equations evolving on Lie groups. Lie theory is a powerful tool, helpful in the analysis and design of modern nonlinear control laws, nonlinear filters, and the study of particle dynamics. The practical application of Lie theory often results in highly complex symbolic expressions that are difficult to handle efficiently without the aid of a computer software tool. The aim of the package is to facilitate and encourage further research relying on Lie algebraic computations. |
Dominance of Cyclic Solutions and Challenges in the Scheduling of Robotic Cells
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 47 (No. 4) |
| Halaman | : | 709-721 |
| Abstrak | : | We consider the problem of scheduling operations in bufferless robotic cells that produce identical parts. Maximizing the long-term average throughput of parts is an important problem in both theory and practice. We define an appropriate state space required to analyze this problem and show that cyclic schedules which repeat a fixed sequence of robot moves indefinitely are the only ones that need to be considered. For the different classes of robotic cells studied in the literature, we discuss the current state of knowledge with respect to cyclic schedules. Finally, we discuss the importance of two fundamental open problems concerning optimal cyclic schedules, special cases for which these problems have been solved, and attempts to solve the general case. |
What Color Is Your Jacobian? Graph Coloring for Computing Derivatives
| Pengarang | : | Assefaw Hadish Gebremedhin |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 47 (No. 4) |
| Halaman | : | 629-705 |
| Abstrak | : | Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertex coloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrix estimation problems. The framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance-2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrix as an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here. |
Hyperasymptotics and the Linear Boundary Layer Problem: Why Asymptotic Series Diverge
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 47 (No. 3) |
| Halaman | : | 553-575 |
| Abstrak | : | The simplest problem with boundary layers, , is used to illustrate (i) why the perturbation series in powers of is asymptotic but divergent, (ii) why the optimally truncated expansion is "superasymptotic" in the sense that that error is proportional to , and (iii) how to obtain an improved "hyperasymptotic" approximation. |
A Simply Stabilized Running Model
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 47 (No. 3) |
| Halaman | : | 519-549 |
| Abstrak | : | The spring-loaded inverted pendulum (SLIP), or monopedal hopper, is an archetypal model for running in numerous animal species. Although locomotion is generally considered a complex task requiring sophisticated control strategies to account for coordination and stability, we show that stable gaits can be found in the SLIP with both linear and "air" springs, controlled by a simple fixed-leg reset policy. We first derive touchdown-to-touchdown Poincaré maps under the common assumption of negligible gravitational effects during the stance phase. We subsequently include and assess these effects and briefly consider coupling to pitching motions. We investigate the domains of attraction of symmetric periodic gaits and bifurcations from the branches of stable gaits in terms of nondimensional parameters. |