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Preasymptotic Stability and Homogeneous Approximations of Hybrid Dynamical Systems
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| Pengarang | : | Rafal Goebel |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 52 (No. 1) |
| Halaman | : | 87-109 |
| Abstrak | : | Hybrid dynamical systems are systems that combine features of continuous-time dynamical systems and discrete-time dynamical systems, and can be modeled by a combination of differential equations or inclusions, difference equations or inclusions, and constraints. Preasymptotic stability is a concept that results from separating the conditions that asymptotic stability places on the behavior of solutions from issues related to existence of solutions. In this paper, techniques for approximating hybrid dynamical systems that generalize classical linearization techniques are proposed. The approximation techniques involve linearization, tangent cones, homogeneous approximations of functions and set-valued mappings, and tangent homogeneous cones, where homogeneity is considered with respect to general dilations. The main results deduce preasymptotic stability of an equilibrium point for a hybrid dynamical system from preasymptotic stability of the equilibrium point for an approximate system. Further results relate the degree of homogeneity of a hybrid system to the Zeno phenomenon that can appear in the solutions of the system. |
Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems
| Pengarang | : | S. L. Mitchell |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 52 (No. 1) |
| Halaman | : | 57-86 |
| Abstrak | : | The work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems. These include standard test problems, both with one and two phase changes, which have exact solutions that enable us to test the accuracy of the approximate solutions. We also consider situations where no analytical solution is available and compare these to numerical solutions. It is popular to use a quadratic profile as an approximation of the temperature, but we show that a cubic profile, seldom considered in the literature, is far more accurate in most circumstances. In addition, the refined integral method can give greater improvement still, and we develop a variation on this method which turns out to be optimal in some cases. We assess which integral method is better for various problems, showing that it is largely dependent on the specified boundary conditions. |
Numerical Methods for Electronic Structure Calculations of Materials
| Pengarang | : | Yousef Saad |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 52 (No. 1) |
| Halaman | : | 3-54 |
| Abstrak | : | The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scientific computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful but approximate versions of this equation, which allow one to study nontrivial systems, took about five or six decades to develop. In particular, the last two decades saw a flurry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as density functional theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well. |
Energy Considerations for Multiphase Fluids with Variable Density and Surface Tension
| Pengarang | : | Yue Lei |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51 (No. 2) |
| Halaman | : | 423-431 |
| Abstrak | : | We examine the energy budget associated with a multiphase fluid system in the presence of a fluid–fluid interface with variable surface tension. Such an interface is found in thermo-capillary systems and also describes solutions of miscible liquids separated from a third fluid by an interface. We incorporate local density variations and derive an energy budget from the Navier–Stokes and advection-diffusion equations. In addition to the usual potential, kinetic, surface, and dissipated energy, three additional terms are found which correspond to transfers from thermodynamic internal energy to surface, potential, or kinetic energy, respectively. Energy transfers from internal energy to surface energy are found to be comparable to viscously dissipated energy in thermo-capillary systems. Taking into account such transfers is essential to assessing the validity of numerical simulations of such systems. |
A Problem with the Assessment of an Iris Identification System
| Pengarang | : | Michel Dekking |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51 (No. 2) |
| Halaman | : | 417-422 |
| Abstrak | : | Most probability and statistics textbooks are loaded with dice, coins, and balls in urns. These are perfect metaphors for actual phenomena where uncertainty plays a role. However, students will greatly appreciate a real-life example. In this paper we examine the mathematics of an implementation of an iris recognition system. We show that the determination of a crucial spread parameter is made on implicit assumptions that are not fulfilled. Some elementary probability theory calculations show that this leads in general to an optimistic assessment of the reliability of the identification system. Then we use a famous inequality to quantify this optimism. |
Hat Guessing Games
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51 (No. 2) |
| Halaman | : | 399-413 |
| Abstrak | : | Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of n players tries to guess the color of the hat he or she is wearing by looking at the colors of the hats worn by some of the other players. In this paper we consider several variants of the problem, united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case. We also summarize what is currently known about the worst-case analysis of deterministic hat guessing problems with a finite number of players. |
Multigrid Methods for PDE Optimization
| Pengarang | : | Alfio Borzì |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51 (No. 2) |
| Halaman | : | 361-395 |
| Abstrak | : | Research on multigrid methods for optimization problems is reviewed. Optimization problems considered include shape design, parameter optimization, and optimal control problems governed by partial differential equations of elliptic, parabolic, and hyperbolic type. |
l1 Trend Filtering
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51 (No. 2) |
| Halaman | : | 339-360 |
| Abstrak | : | The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on Hodrick–Prescott (H-P) filtering, a widely used method for trend estimation. The proposed trend filtering method substitutes a sum of absolute values (i.e., norm) for the sum of squares used in H-P filtering to penalize variations in the estimated trend. The trend filtering method produces trend estimates that are piecewise linear, and therefore it is well suited to analyzing time series with an underlying piecewise linear trend. The kinks, knots, or changes in slope of the estimated trend can be interpreted as abrupt changes or events in the underlying dynamics of the time series. Using specialized interior-point methods, trend filtering can be carried out with not much more effort than H-P filtering; in particular, the number of arithmetic operations required grows linearly with the number of data points. We describe the method and some of its basic properties and give some illustrative examples. We show how the method is related to regularization-based methods in sparse signal recovery and feature selection, and we list some extensions of the basic method. |
On Nonobtuse Simplicial Partitions
| Pengarang | : | Trambak Banerjee |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51 (No. 2) |
| Halaman | : | 317-335 |
| Abstrak | : | This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics. |
A Discrete Invitation to Quantum Filtering and Feedback Control
| Pengarang | : | Luc Bouten |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51 (No. 2) |
| Halaman | : | 239-316 |
| Abstrak | : | The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods. |